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Havind problems calculating accruals and prepayments
i am really struggling calculating accruals and prepayments wtever i do i always messed up the months so can anyone help me understand
on 1st sep03 the partnership paid rates of £3000 covering 3 months to 30th nov 03
they calculate it as 3000*8/12
how they do it ????
On 31 August 2003 the partnership paid a telephone bill of £303.90, covering the three months to
31 July 2003. The bill for the following three months has not yet been received. Telephone bills are
treated as office expenses.
its same for that 303.90*8/12
can sumone explain it
i am really struggling calculating accruals and prepayments wtever i do i always messed up the months so can anyone help me understand
on 1st sep03 the partnership paid rates of £3000 covering 3 months to 30th nov 03
they calculate it as 3000*8/12
how they do it ????
On 31 August 2003 the partnership paid a telephone bill of £303.90, covering the three months to
31 July 2003. The bill for the following three months has not yet been received. Telephone bills are
treated as office expenses.
its same for that 303.90*8/12
can sumone explain it
What was the year end?
• was about to ask the same myself
if year end sept as i suspect you need to (prepayment) divded the payment by 3 then one month will be payment 2 months will be prepayment for next new year
ETB dont forget to include prepayment also as a debit
• If you're having trouble with accruals and pre-payments then on the AAT website there is an e-learning link for Unit 5. There's loads of options like depreciation and bad debts etc... Just scroll
down to accruals and pre-payments. I've found that link very helpful indeed.
Good Luck!!
If you're having trouble with accruals and pre-payments then on the AAT website there is an e-learning link for Unit 5. There's loads of options like depreciation and bad debts etc... Just
scroll down to accruals and pre-payments. I've found that link very helpful indeed.
Good Luck!!
Its brill used it myself am sure its what got me through
i am really struggling calculating accruals and prepayments wtever i do i always messed up the months so can anyone help me understand
on 1st sep03 the partnership paid rates of £3000 covering 3 months to 30th nov 03
they calculate it as 3000*8/12
how they do it ????
On 31 August 2003 the partnership paid a telephone bill of £303.90, covering the three months to
31 July 2003. The bill for the following three months has not yet been received. Telephone bills are
treated as office expenses.
its same for that 303.90*8/12
can sumone explain it
the 3000 is an prepayment of 1000 as your year end is sept...divide the 3000 into 3 to give you the month
the accrual 303.90 is 202.60 as again you divide it into 3 and x by 2 as you need 2 months worth
did this one yesterday
• thanks for all your help
i understand the prepayment but still didnt understand the accrual example how they calculate that :confused1:
• Your year end is September, however, you only have telephone expenditure in the P&L for upto July - as you need to have an amount for each month, you need to estimate a figure for August and
If you base this on your previous bill of £303.90, you will need to accrue two months expense.
Therefore, if your previous bill covers 3 months 303.90/3 = 101.30 average per month. £101.30 x 2months will give you an estimated charge of £202.60.
• thanks for all the help
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Instructions for Applying Statistical Testing - 2023 ACS 1-year and 2019-2023 ACS 5-year Data Releases: Technical Documentation - Survey ACS 2023 (1-Year Estimates) - Social Explorer
Once users have obtained standard errors for the basic estimates, there may be situations where users create derived estimates, such as percentages or differences that also require standard errors.
All methods in this section are
and users should be cautious in using them. This is because these methods do not consider the correlation or covariance between the basic estimates. They may be overestimates or underestimates of the
derived estimate's standard error depending on whether the two basic estimates are highly correlated in either the positive or negative direction. As a result, the approximated standard error may not
match direct calculations of standard errors or calculations obtained through other methods.
• Sum or Difference of Estimates
As the number of basic estimates involved in the sum or difference increases, the results of this formula become increasingly different from the standard error derived directly from the ACS
microdata. Care should be taken to work with the fewest number of basic estimates as possible. If there are estimates involved in the sum that are controlled in the weighting then the approximate
standard error can be tremendously different.
Here we define a proportion as a ratio where the numerator is a subset of the denominator, for example the proportion of persons 25 and over with a high school diploma or higher.
If the value under the square root sign is negative, then instead use
If P = 1 then use
If Q = 100% x P (a percent instead of a proportion), then SE(Q) = 100% x SE(P).
If the estimate is a ratio but the numerator is not a subset of the denominator, such as persons per household, per capita income, or percent change, then
For a product of two estimates - for example if users want to estimate a proportion's numerator by multiplying the proportion by its denominator - the standard error can be approximated as
Users may combine these procedures for complicated estimates. For example, if the desired estimate is
then SE(A+B+C) and SE(D+E) can be estimated first, and then those results used to calculate SE(P).
For examples of these formulas, please see any Accuracy of the Data document available on the ACS website at:
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Random Access Machine (RAM) Model
Date: January 7 2021
Summary: Modeling computational actions as single step actions
Keywords: ##zettel #random #access #machine #model #archive
Not Available
Table of Contents
Random Access Machine (RAM) Model: instructions are executed contiguously without concurrency. [1]
RAM concerns common real-world instructions executed at constant times:
1. Arithmetic:
□ Add
□ Subtract
□ Multiply
□ Divide
□ Remainder
□ Floor
□ Ceiling
2. Data Movement
3. Control flow
• The RAM model concerns integers and floating points for real number representation.
• A limit on the size of each word of data is assumed.
• The memory hierarchy (i.e. model caches or virtual memory on modern computers) is not modeled.
Algorithm analysis: predicting the resources an algorithm requires.Computational time is what is commonly measured. Memory, hardware, or bandwidth are sometimes also analyzed. [1]
Running time: the number of basic operations executed in an algorithm. In the RAM model, each executed line takes the constant time ci.
Generally, time taken by an algorithm grows in proportion to the size of the input. [1] Definitions for input size depend on what is studied:
• The number of items in the input
• The total number of bits needed to represent the input in binary notation
• Numbers of vertices and edges in a graph
Zelko, Jacob. Random Access Machine (RAM) Model. https://jacobzelko.com/01072021072613-random-access-model. January 7 2021.
[1] T. H. Cormen, Ed., Introduction to algorithms, 3rd ed. Cambridge, Mass: MIT Press, 2009.
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Math in Second Life
A screenshot of a location in Second Life.
This sim built by Henry Segerman http://www.segerman.org/ (aka Seifert Surface).
The metal ball above the bridge is made of 4 nested spheres, rotating on different axes. Behind the sphere is a tower of octahedrons. (A octahedron is a regular solid, having 8 faces, each face is a
equilateral triangle. (it's like two pyramids with their bases glued together)) On the right is a geodesic dome made of glass. Inside is a garden with mathematical sculptures.
The middle sculpture is a double spiral. This sculpture is based on the Equiangular Spiral. Most spirals seen in nature's growths, for example, plants and seashells, are equiangular spirals. (See:
Seashells photo gallery.)
This colorful spiral sculpture is a model of the Hopf bundle. You will need to have a phd in math to understand this one. For mere mortals, suffice it to say it is twisty.
hopf bundle. Standing in front is a Japan geisha in traditional Japanese attire the kimono.
The girl Eureka of the Japanese animation Eureka Seven. (The creator of this avatar is Yamiki Ayakashi)
The background blueish strip is the Moebius strip. Moebius strip is a example of a surface that has only one side. (If you are a ant crawling on the strip, you'll end up on the other side without
crossing any edge.) The rim of this strip, is a Trefoil knot. Tie a overhand knot, then connect the lose ends, and you have it!
This is called Alexander's Horned Sphere. As you can see, the linking of the circles continues on forever.
This shape is a fantastic discovery to mathematicians in 1924, in particular, of the branch topology. It is interesting because it is: “An Example of a Simply Connected Surface Bounding a Region
which is not Simply Connected.”. Here's some basic explanation:
On a piece of paper, draw a simply connected closed curve (for example, a circle). Once you draw a circle, it divides the region into two sections: inside, and outside. A theorem says that this will
always be so. (silly theorem, eh?) Furthermore, the two regions are similar in the sense that each is a rather coherent blob.
Now, if you have a sphere (aka ball) in space. This sphere also divides the 3D-space into two regions, the inner one, and the other one. Again, the inside region and outside region are similar in the
sense that both are coherent blobs.
However, now consider this shape called “Alexander's Horned Sphere”. It divides the space into the inner region and outter region. However, the outer region is no longer a coherent blob, yet the
inner region is still a un-pierced blob.
The above is a rough description on why it is interesting. To understand this exactly, a human animal needs to study math for about a decade. For a technical description, see: Alexander's horned
A shape similar to a stellated dodecahedron. This is actually a living animal. When it moves, the pointed tips flow freely like some sea creatures, making it extremely cute.
(This shape is not a stellated dodecahedron. Ask Seifert Surface or Bathsheba in-world if you want to know the detail)
A iterated function system styled fractal. Note how the pattern nests.
Another fractal. The outer shape is a tetrahedron. Bisect each of its edges, forms a octahedron in its center and 4 smaller tetrahedrons at each of the corners. Do this again on the tetrahedrons,
will form this shape.
This technique of forming fractals is based on the Sierpinski triangle.
This sequence shows how to turn a torus inside-out. (A torus is the name for the shape of a ring.)
Unless otherwise indicated, the objects shown in this page are created by Henry Segerman http://www.segerman.org/ (aka Seifert Surface in Second Life).
Other Websites
• “Evaluating Second Life for the Collaborative Exploration of 3D Fractals”, Paul Bourke, Computers and Graphics, 2008-09. [S:http://local.wasp.uwa.edu.au/~pbourke/papers/cg2008/index.html:S]
• “Evaluating Second Life as a tool for collaborative scientific visualisation”, by Paul Bourke. Computer Games and Allied Technology. Singapore, April 28-30, 2008. [S:http://local.wasp.uwa.edu.au/
• “Menger Sponge in Second Life” (2008-02) by Paul Bourke. [S:http://local.wasp.uwa.edu.au/~pbourke/fractals/menger_sl/index.html:S]
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A Video Analysis as an End of Unit Assessment
Dale Simnett, Peel District School Board
Part 1: The Case for the Humble Video Analysis
Friends, Peers, Physics Teachers,
The use of video in physics as a means of teaching and learning is as old as optic obscura. I’m sure most physics teachers have a video analysis project that they can dust off and give to their
students. Identifying the terrible physics in cartoons, superhero movies, or more recently, the Fast and Furious series, is a right of passage in physics teaching!
Observing the world, making measurements, and trying to make sense of what we see is at the heart of physics. Why not make that a goal in our teaching as well as our assessment? I propose using a
video analysis as an end of unit assessment.
Why should you try video analysis as an assessment? Here is my case.
They are easily produced.
Two classes writing at different times or different days? A student requiring writing at an alternative time? Students asking for test practice questions? An assessment can be produced with a
15-minute trip to YouTube. Take the following Grade 12 two-dimensional forces assessment — this is the entirety of the instructions for the assessment.
To create a new assessment requires finding another video on YouTube involving two-dimensional forces — snow boarding, roller coasters, anything with a ramp, YouTube can provide rich examples of
two-dimensional forces in a heartbeat. Just include the link and add an additional question that points students towards an interesting aspect of theory, and you’re finished. The ease of creating an
entirely new assessment means you can distribute them freely and students can practice to their heart’s content.
No more long-winded descriptions.
The best part of a video analysis is students are doing authentic physics where they start from observation and analyze real life. To help students understand a written question, I have incorporated
strategies such as collaborative group discussion of a problem, pretty diagrams, and flailing my hands at the front of the classroom on numerable occasions. English language learners, amongst others,
can still struggle to wade through the numbers and words used to describe a real-life situation. A video allows the student to develop their own description. There are students who perform
significantly better on their video analysis than on their standard problem-based assessments, I ask them why and consistently I hear “I know what is going on.”
Stuck on a Hill video
During assessments, I’ll play the video on repeat at the start of class for several minutes and let them have 5 minutes with a whiteboard to discuss what they saw. Then the whiteboards are whisked
away, and the assessment is handed out along with a rubric and good old fashion lined paper. Students can ask for the video to be replayed for further clarification.
Freedom from given values!
But where are the numbers? Students will look for values to plug into equations and they won’t find them. This may cause some consternation.
We know that measurement informs physics. For the situation, students are encouraged to think about what you could easily measure to use as a “given”. We don’t need a value, but we accept that a
value could be attained if we had a measuring tape, timer, radar gun or scale. Due to the potential for this value to vary, we will call these terms changeables. The challenge is to create an
equation with only changeables that represents the situation mathematically.
There are two added super bonuses.
First, being free from numbers allows students to start thinking about what the equation represents and how the changeables are related to each other. We can talk about proportionalities, what if’s
(what if this was heavier or the displacement required was greater) and further develop the ability to describe how the mathematical relationship represents real life.
Second, would you like to teach students programming? The ability to prepare an awesome spreadsheet by learning to “program” a formula into a cell may be the greatest gift you can give for their
future success (until A.I. ups its spreadsheet game). The day after the assessment, “program” the formula into Excel, iterate the equation by changing one changeable and plot the effect on another to
visualize relationships. This can take an assessment and turn it into a learning opportunity to develop a deeper understanding of the relationship between changeables.
(Note: Changeables was a poor attempt at mathematics humour. Variables would, of course, be the common parlance.)
Have I convinced you to give it a try? Next, let’s look at the success criteria for a video analysis.
Part 2: The Success Criteria for a Video Analysis
Draw diagrams to represent the situation.
Students are expected to create visual representations that accurately represent the situation. They start with a kinematics diagram to describe what they see and then layer on different diagrams to
analyze forces, energy, and more.
They are expected to use explanatory labels on these diagrams to explain the learning goals. In this case the learning goals involve Newton’s 2
and 3
laws. An additional “bonus” diagram is required to help explain a learning goal or a relationship between variables.
Rubric for using diagrams to analyze the video
Kinematics diagram for the truck getting stuck — Student Work
Using explanatory labels on force diagrams to explain Third Law — Student Work
Create a Mathematical Representation
During the term, students spend time looking at solved problems and breaking down the mathematical steps to analyze a situation. Collaboratively, they develop a problem-solving strategy that outlines
a step-by-step approach to analyzing a situation. Their mark is based on how well they can apply that problem-solving strategy to the question.
An additional requirement is that once they develop their equation, they pick one line of their analysis and describe how it relates to the learning goals listed or the additional question.
Rubric for using equations to analyze the video
Strategy and Mathematical Analysis — Note: The final line was used to explain Second Law
Explain the Learning Goals Using the Video
Meeting expectations involves using both diagrams and equations to relate the learning goals to the video and answer an additional question. If done well, an assessment won’t feature paragraphs of
text or word-for-word descriptions of theory, but instead would use explanatory labels on their mathematical analysis and diagrams to describe how the learning goals relate to the situation. The
learning goals below have been (partially) explained by labels in sections 1 and 2 above.
Rubric for relating video to theory
Learning Goals and Additional Question to Explain using Diagrams and Equations
Part 3: The Wrap-Up
In my classroom, I will do both traditional problem-based assessment and video analysis assessment. They complement each other nicely and have students develop a separate set of skills. Even in my
problem-based assessments, I try to incorporate a video that represents a similar situation to provide a visual representation. My final exam consists of a problem-based assessment for the first hour
and a video analysis for the second hour. Some students prefer one type of assessment while others prefer the other type. Either way, they are having fun doing physics!
For a collection of video analysis assessments and other problem-based learning resources,
follow this link to my google drive
. If you have any questions, feel free to contact me at
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What does transvalued mean?
Asked by: Prof. Helena Tromp Score: 4.4/5
13 votes
transitive verb. : to reevaluate especially on a basis that repudiates accepted standards.
What does Retice mean?
: a scale on transparent material (as in an optical instrument) used especially for measuring or aiming.
What does Purious mean?
adjective. full of fury, violent passion, or rage; extremely angry; enraged: He was furious about the accident. intensely violent, as wind or storms.
Can a person be specious?
Pleasing to the eye; externally fair or showy; appearing beautiful or charming; sightly; beautiful. Superficially fair, just, or correct; appearing well; apparently right; plausible; beguiling: as,
specious reasoning; a specious argument; a specious person or book.
What does spuriously mean?
1 : born to parents not married to each other. 2 : outwardly similar or corresponding to something without having its genuine qualities : false the spurious eminence of the pop celebrity. 3a : of
falsified or erroneously attributed origin : forged.
What does transvalue mean?
17 related questions found
What is the meaning of rectilinear motion?
: a linear motion in which the direction of the velocity remains constant and the path is a straight line.
What is social ridicule?
synonym study for ridicule
Ridicule, deride, mock, taunt imply making game of a person, usually in an unkind, jeering way. To ridicule is to make fun of, either sportively and good-humoredly, or unkindly with the intention of
humiliating: to ridicule a pretentious person.
Is mocking rude?
Mocking, imitating, and laughing at parents can be harmless fun, but it can also become an annoying behavior that undermines your authority. ... That's disrespect and an attempt to chip away at your
position of authority.
Why do people use mockery?
Mockery serves a number of social functions: Primitive forms of mockery represent the attempt to use aggression to protect oneself from engulfment, impingement or humiliation by diminishing the
perceived power and threat of the other.
How do you ridicule someone?
When you ridicule someone, you mock or make fun of them. They become the object of your ridicule or mockery. Your bad behavior might bring ridicule on your parents, who raised you to know better. The
word ridicule is related to ridiculous.
What are the 4 types of motions?
The four types of motion are:
• linear.
• rotary.
• reciprocating.
• oscillating.
Is an example of rectilinear motion?
Examples for Rectilinear Motion
The use of elevators in public places is an example of rectilinear motion. Gravitational forces acting on objects resulting in free fall is an example of rectilinear motion. Kids sliding down from a
slide is a rectilinear motion. The motion of planes in the sky is a rectilinear motion.
What is rectilinear motion and give an example?
Any motion in which objects take a straight path is known as a rectilinear motion. ... Planes in the sky that move in a straight line is in rectilinear motion. A ball rolling down an inclined plane
is considered to be in rectilinear motion. Skateboarders going down an inclined path are in rectilinear motion.
What is motion class 9?
An object is said to be in motion when its position changes with time. ... The shortest path/distance measured from the initial to the final position of an object is known as the displacement. •
Uniform motion: When an object covers equal distances in equal intervals of time, it is said to be in uniform motion.
What is difference between linear and rectilinear motion?
Answer: A body/ object is said to be in linear motion when it travels along a straight line or along a curve in a plane. Example- Athlete running along a straight path. ... In other words, when a
body travels only along a straight path, it is said to be in rectilinear motion.
What is the other name for rectilinear motion?
Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension.
What are 10 examples of rectilinear motion?
10 examples of rectilinear motion
• A boy walking on a straight road.
• A car moving in a straight road.
• Lifting something up and down.
• A boy pulling a toy towards him.
• moon walk step in dance.
• parade of army.
• fruit falling from a tree.
• Free fall of heavy object.
What are the types of rectilinear motion?
Rectilinear motion has three types: uniform motion (zero acceleration), uniformly accelerated motion (non-zero constant acceleration) and motion with non-uniform acceleration.
Which is an example of rotatory motion?
Rotatory motion is the motion that occurs when a body rotates on its own axis. ... While driving a car, the motion of wheels and the steering wheel about its own axis is an example of rotatory
What are the 10 types of motion?
Rotatory motion, rotatory motion, oscillatory motion, uniform circular and periodic motion, rectilinear motion, oscillatory motion and periodic motion.
What are the major types of motion?
In the world of mechanics, there are four basic types of motion. These four are rotary, oscillating, linear and reciprocating. Each one moves in a slightly different way and each type of achieved
using different mechanical means that help us understand linear motion and motion control.
Is it true that all motion is related Why?
All motions are relative to some frame of reference. Saying that a body is at rest, which means that it is not in motion, merely means that it is being described with respect to a frame of reference
that is moving together with the body.
What to say when someone is mocking you?
Say something like “wow, did you come up with that all by yourself” or “pardon me, but you seem to think that I care.” Try the “Yes, and…” technique. If someone is giving you a hard time about
something just respond by acknowledging their teasing and then inserting a joke.
Are you mocking me meaning?
Mocking = To make fun of someone. Ex. " I made fun of her laugh" = "I mocked her laugh." Sometimes you can mock a friend by making fun of the way they say words, but a lot of the time, when you are
mocking someone you are not being nice. " Are you kidding me?" = " Are you joking?" "
What does mocking someone mean?
: to laugh at or make fun of (someone or something) especially by copying an action or a way of behaving or speaking. : to criticize and laugh at (someone or something) for being bad, worthless, or
unimportant. mock.
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A Simple Method to Detect Score Similarity and Practical Implications for Its Use
There are numerous methods for detecting aberrant test taking behavior that can be applied by select users within the testing industry. One such behavior is collusion, and a method psychometricians
and other analysts within the credentialing industry use to detect it is a statistic known as a score similarity index (SSI).
SSI purports to identify pairs of examinees who have an unusually high number of identical correct/incorrect scores on a set of items. There are multiple methods that estimate the probability two
examinees colluded, such as Wollack’s^1 ω, generalized binomial model (GBT)^2 and a new application of residual correlations of persons known as B3.^3,4 While these methods are useful, they are
either computationally complex, require specialized software or take a very long time (hours if not days) to run on large sets of data (e.g., pairwise comparisons of 10,000 examinees on a 60-item
For users not in the credentialing field, such as classroom teachers, some of these methods may be difficult to implement due to the lack of software or familiarity with how to implement a method and
interpret the results. The purpose of this article is two-fold: 1) provide an approximation SSI (aSSI) method that fills this gap by exemplifying a method that is easily implemented, straightforward
to interpret and produces comparable results to other known SSI methods (i.e., Wollack’s ω, GBT and B3) and 2) provide guidance regarding the policies and procedures that should be in place when
implementing such a method.
Wollack’s ω, GBT and B3 are all estimation methods aimed at getting as close to “true” SSI as possible. In other words, while no SSI method will be perfect, all of these statistics do a decent job of
estimating the probability that two examinees share a given number of identical correct/incorrect scored responses on a set of items and tend to have Type I (false positive) and Type II (false
negative) error rates acceptable to testing programs. For that reason, these true SSI methods aim at flagging pairs of examinees with a high probability of collusion. The aSSI method has the same end
goal, but aims at approximating the results from the GBT method.
Why approximate the results of an estimation method? According to Welsh mathematician and philosopher Bertrand Russell,^5 “the behavior of large bodies can be calculated with a quite sufficient
approximation to the truth.” He continues, “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” Thus, it is reasonable to try to approximate true SSI via a
simpler method that can reach a wider audience than more complicated methods.
Brief Explanation of True SSI Methods
Wollack’s ω and GBT require the use of item response theory (IRT) to compute the expected agreement between two examinees. Then, a z-statistic is computed comparing the difference between the
observed and expected agreement. In brief, the difference between these methods is that ω estimates the expected agreement by summing the probabilities that the copier’s response (0,1) matches the
observed source’s response given the ability of the copier and the item’s IRT parameters. The GBT method estimates the expected agreement by summing the joint probabilities of matching scores (0,1)
between two examinees given the ability of each of the examinees and the item’s IRT parameters. For both methods, Bock’s nominal model is typically used to estimate the person and item parameters.
However, given that SSI uses dichotomously scored items, research presented in this study estimates the results of ω and GBT using both Bock’s nominal model (collapsed to a 0/1 model) as well as the
Rasch model. Both ω and GBT may be estimated using Zopluoglu’s “CopyDetect”^6 R package.
B3 also applies the Rasch model, but instead of a z-statistic, this method computes the correlation of the residuals for two examinees. B3 is much like Yen’s Q3 statistic,^7 which uses item residual
correlation values to help identify high item interdependence. B3 is the same statistic but focuses on the examinee residual correlations instead of item correlations. A high B3 value indicates two
examinees’ scores are not independent, in other words, the scores patterns are more similar than one would expect by chance. Unlike ω and GBT in which a small value (e.g., < 0.01) would lead to
flagging a pair of examinees, high values of B3 suggest aberrant behavior. Winsteps^8 readily produces person-residual correlations.
Explanation of aSSI
The aSSI method does not require IRT nor a sophisticated program and is less computationally intensive than ω, GBT, and B3. Like ω and GBT, aSSI is a z-score between Examinee 1 and Examinee 2:
(Equation 1) where M is the count of observed score matches, n is the number of items, p is E^*[12]/n and q = (1-p). E[12] is the adjusted expected value of the number of observed matches and is
computed as follows:
(Equation 2) where s[i] is the proportion correct score for person i and b is an adjustment to the magnitude of the correction. Based on recommendations by Smith,^9,10 this value is set at 12.5% for
this study.
While the denominator of Equation 1 has the same look as that of GBT, the expected value is an approximation of the expected value calculated in the true GBT method. In Equation 2, the first half of
the equation (left addend) estimates the independent probability of both examinees scoring their observed number of correct scores and the probability of both examinees scoring their observed number
of incorrect scores. The second half of the equation is an adjustment value for differences in the variability of the difficulty of the items on the exams. If the two examinees had identical scores,
then the adjustment would be n∙b or 0.125n. On the other extreme, if one examinee had a perfect score and the other examinee scored 0 points, then the adjustment would equal 0. Thus, the adjustment
value varies from 0 to 0.125n.
Like the true methods, there are assumptions to the aSSI method. The aSSI method assumes the data are approximately normally distributed, the items are independent and the items are dichotomously
scored. The method is fairly robust to violations of these assumptions and violations tend to make the results more conservative (i.e., lower type I error).
Example Computation of aSSI
Consider Examinee 1 and Examinee 2 who have the same correct/incorrect responses for 51 items on a 66-item exam. Examinee 1’s raw score is 38/66 and Examinee 2’s raw score is 35/66. Based on these
For a normal distribution, a z-score of 2.6911 is equivalent to a probability of < 0.50%. Based on their percent correct scores, the number of expected matched scores is approximately 40 items. Thus,
the expected probability of Examinees 1 and 2 matching on 51 items is unlikely, i.e., < 0.50% chance.
Previous work by Smith^9,10 demonstrated through a simulation study that aSSI results are comparable to those found by GBT. Therefore, the purpose of this study was to use real datasets to determine
the comparability of aSSI to ω (using both Bock’s nominal model and Rasch model), GBT (using both Bock’s nominal model and Rasch model) and B3.
Three real datasets with known security issues were used for the comparability study:
• Exam A: This exam contained 66 items, 416 examinees; content was found on a brain dump site
• Exam B: This exam contained 60 items, 1992 examinees; content was found on a brain dump site, where some of the items were mis-keyed
• Exam C: This exam contained 66 items, 1109 examinees; other security analyses, such as score by time and scored versus unscored analyses, indicated security concerns
A correlation matrix was used to compare the strength of the relationship between the values used to detect collusion for all possible pairs of examinees given each method. For example, the
probability of collusion for each pair of examinees based on ω (using Bock’s model) was correlated with the corresponding probability of collusion for each pair based on the aSSI method.^*
The results indicated aSSI performed just as well as ω, GBT and B3. Tables 1-3 show the correlations. The main findings of this comparison include:
• The strongest correlations among all three exams were between GBT-Rasch and aSSI
• The weakest correlations tended to involve B3
• aSSI had strong positive correlations (> 0.900) with both the ω-Rasch and GBT-Rasch methods
Table 1. Correlation of Flagged Pairs Using Different SSI Methods – Exam A
ω – Wollack’s Omega GBT- Generalized Binomial Test B3 – Person Residual Correlations aSSI- approximation Score Similarity Index
Table 2. Correlation of Flagged Pairs Using Different SSI Methods – Exam B
ω – Wollack’s Omega GBT- Generalized Binomial Test B3 – Person Residual Correlations aSSI- approximation Score Similarity Index
Table 3. Correlation of Flagged Pairs Using Different SSI Methods – Exam C
ω – Wollack’s Omega GBT- Generalized Binomial Test B3 – Person Residual Correlations aSSI- approximation Score Similarity Index
Identifying pairs of examinees who have colluded is a problem in the credentialing field as well as fields outside of credentialing (e.g., classroom assessments). As professionals in the field, part
of our responsibility is to reach out and educate those not in the testing field to ensure we provide the community with the resources and tools they need to help develop and validate results from
their own assessments.
Many of the current methods that identify if collusion has likely occurred involve statistical software packages not easily accessible nor simple to implement by those outside of the assessment
field. Thus, these methods effectively restrict the methods to a certain pool of users. The aSSI method presented in this article is one any individual could compute with a calculator and normal
distribution table or spreadsheet software. The results are directly interpretable as a probability, e.g., if the probability is < 0.01, then there is a strong possibility some form of collusion has
While thought needs to be given to the exact flagging threshold one applies (e.g., 0.001, 0.01), the method and results are accessible and understandable to a much wider audience than ω, BST and B3.
Practical Policy and Implementation Considerations
Score similarity indices, such as aSSI, are just one method to detect potential collusion among one or many pairs of examinees. Data forensic techniques, such as aSSI, may detect unusually similar
responses that should be investigated further, but one statistical method alone does not provide unequivocal and actionable evidence.^11 Multiple methods should be employed to detect potential
collusion and evaluate its impact on examination outcomes.^12
Jacobs, Judish and Murphy^13 discuss some general ways in which organizations may approach ethics violations (e.g., cheating, gaining pre-knowledge) in a legally defensible way. Based on their work,
guidance from the APA, AERA, & NCME Standards,^14 and recent work of others (e.g., Thompson, Weinstein and Schoenig;^15 Twing, Keen, Canto and Friess;^16 O’Leary and Owens,^17 several themes emerge
related to taking action based on data forensics:
1. Programs/schools should have a policy in place that examinees/students must agree to that indicates their exam results may be monitored for aberrant behavior as well as any actions that may be
taken. The policy could be published on such documents as a candidate agreement form or class syllabus.
2. Programs/schools should have a procedure in place for implementing a security policy fairly and consistently.
3. Programs/schools should make the policies and procedures related to violations of ethics code (e.g., cheating) transparent.
4. Programs/schools should gather multiple sources of evidence and/or suspicious results from data forensics analyses before taking action.
To this latter point, the results from aSSI alone may likely be insufficient to take action (e.g., canceling an exam score). However, grounds to take action become stronger if aSSI results showed
highly unusual behavior and a proctor observed collusion during the administration.
The results in this paper and those by Smith^9,10 suggest aSSI sufficiently approximates SSI and the simplicity of the method does not compromise the method’s effectiveness. In this paper, aSSI
strongly correlates with the ω and GBT methods (applying both Bock and Rasch models) as well as B3. As such, the results of this study indicate aSSI is a reasonable method to apply in order to
provide a layer of evidence that collusion has or has not likely occurred on an assessment. This method can be easily applied by both members and non-members of the credentialing industry. With the
ease of this method, any individual within the extended testing industry can readily apply this data forensic method and couple it with other evidence (e.g., statistical or non-statistical) to make a
stronger case for taking action against one or more examinees in accordance with an established and consistently applied set of policies and procedures.
*The results did not compare the number of flagged individuals because the flagging criteria may differ slightly for each method and the goal was not to identify the “best” method, but only the
comparability of the methods.
**The negative correlations with B3 are expected as B3 leverages residual correlations with no assumed underlying distribution and, therefore, no probability values. Higher B3 values indicate
collusion. The other statistics in these tables include assumed underlying distributions resulting in estimated probabilities.
1. Wollack, J.A. (1996). Detection of answer copying using item response theory. Dissertation Abstracts International, 57/05, 2015.
2. van der Linden, W. J., & Sotaridona, L. (2006). Detecting answer copying when the regular response process follows a known response model. Journal of Educational and Behavioral Statistics, 31(3),
3. Foley, B. P. (2019). Collusion Detection Using an Extension of Yen’s Q3 Statistic. Presented at the 8th Annual Conference on Test Security. Miami, FL.
4. Smith, R. W. (2019). Comparing B3 to Answer Similarity Index for Detecting Collusion. Presented at the 8th Annual Conference on Test Security. Miami, FL.
5. Russell, B. (1954). The Scientific Outlook. Third impression Great Britain: Unwin Brothers, Ltd. Available online at https://ia801606.us.archive.org/7/items/in.ernet.dli.2015.499767/
6. Zopluoglu C. (2018). CopyDetect: Computing response similarity indices for multiple-choice tests (R Package Version 1.3). https://cran.r-project.org/web/packages/CopyDetect/index.html
7. Yen W. M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30, 187-213.
8. Linacre, J. M. (2022). Winsteps® Rasch measurement computer program (Version 5.2.3). Portland, Oregon: Winsteps.com
9. Smith, R. W. (2021, October 6-7). A Practical Approximation of Response Similarity. Conference on Test Security, online.
10. Smith, R. W. (2022, April). Approximation answer and response similarity analyses: A practical approach [Paper presentation]. Annual meeting of the National Council on Measurement in Education
(NCME), San Diego, CA.
11. Foster, D. & Mulkey, J. (2019). Practical test security for professional credentialing programs. In J. Henderson (Ed.): The ICE Handbook (3^rd ed., pp. 415-446).
12. Hurtz, G. M. & J. A. Weiner. (2019). Analysis of test-taker profiles across a suite of statistical indices for detecting the presence and impact of cheating. Journal of Applied Testing
Technology, 20 (1). Available online at https://www.jattjournal.com/index.php/atp/article/view/140828
13. Jacobs, J., Judish, J., & Murphy, D. C. (2019). Certification law. In J. Henderson (Ed.). The ICE Handbook. 3rd ed., pp. 45-70.
14. AERA, APA, & NCME. (2014). Standards for Educational and Psychological Testing: National Council on Measurement in Education. Washington, DC: American Educational Research Association.
15. Thompson, C., Weinstein, M., & Schoenig, R. (2022). Opening keynote on Ogletree vs. Cleveland State University. Presented at the 2022 Conference on Test Security. Princeton, NJ.
16. Twing, J. S., Keen, J. M., Canto, P., Friess, B. (2022). Lessons learned from federal litigation of cheating involving a test preparation company: Security is a pre-requisite for validity.
Presented at the 2022 Conference on Test Security. Princeton, NJ.
17. O’Leary, L. & C. Owens. (2022). Simplifying Security: Deciphering Data Forensics into Accessible Actions. Presented at the 2022 Institute for Credentialing Excellence Conference. Savannah, GA.
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A lot of programming languages provide a contains/any method that is used to check whether there is at least one element in a sequence that satisfies a given predicate. For example:
Now, suppose you wanted to check whether every element of a sequence satisfies a given predicate. That’s pretty easy — you can invert both the condition and the result to achieve that. For example:
Functional programmers might prefer to use reduce instead:
However, both approaches are actualy less readable and the reduce approach introduces a hidden performance pitfall i.e. no short-circuiting. To address these problems, Swift 4.2 introduced a new
method in the standard library, called allSatisfy:
Now, imagine that our sequence of odd numbers was empty:
Wait… what?? There are clearly no numbers in oddNumbers, so there’s nothing to satisfy the predicate. So, the result of allSatisfy should be false instead, right? Well… no. To understand why, we need
to take a look at some math and logic.
A quantifier is basically something that transforms a statement asserting that a given thing has a certain property into a statement asserting the number (quantity) of things having that property.
For example, “all cars have wheels” or “the square of any natural number is non-negative”. Here, “all” and “any” are the quantifiers.
In math, quantifiers are used in a similar way, but instead of operating on statements, it operates on mathematical functions. Its purpose is to allow us to specify the number of elements in the
universe of discourse¹ that satisfy a mathematical function with at least one free variable.
Here’s a simple example:
0² = 0, 1² = 1 ∧ 2² = 4 ∧ 3² = 9 …, etc.
Here, we have a logical conjunction of propositions to argue that the square of any natural number is greater than or equal to zero². However, “…etc” cannot be interpreted as a logical conjunction,
at least in formal logic, as it is not a valid logical constant.
Now, it’s also not practical to write all the propositions because the set of natural numbers has infinite cardinality. So, how do we get around the limitations? Well, one way is by rewriting the
statement using a quantifier:
for all natural numbers x, x² ≥ 0.
There’s two immediate benefits of doing this:
1. We now have a single statement, instead of a statement composed of infinitely many propositions.
2. It is more accurate than the previous one, because we have now explicitly stated the domain (ℕ) and its nature (enumerability), rather than assuming it based on the presence of “…, etc”.
This is an example of what’s known as “universal quantification” (for all). It is one of two commonly used quantifiers used in math (the other being “existential quantification” i.e there exists).
Using symbolic notation (we use ∀ to say “for all”), we can write:
for all x(x is member of the set of natural numbers) : S(x)
(where S(x) is a predicate that takes a natural number and returns whether its square is greater than or equal to zero or not)
A set is simply a well-defined collection of elements³. It exhibits certain properties, such as membership or cardinality. For example — 1, 2 & 3 are distinct elements, however {1, 2, 3} is a single
set with three distinct elements. Similarly, |{1, 2, 3}| is 3 since there are three elements in the set.
Sets can also exhibit certain relationships between themselves. For example, a set can include elements of another set. This is a relationship which we refer to as being a “subset” of the other set.
So, if we have two sets, A and B, then we can say that A is a subset of B if every element belonging to A is also in B. For example, if A = {1, 2, 3} and B = {1, 2, 3, 4} then A ⊆ B since 1, 2 & 3
all belong to B.
An Euler diagram showing the subset relation
Using what we learned about quantifiers above, we can write the subset relation using symbolic notation as:
for all x(if x belongs to A, then x belongs to B)
which is an implication in the form of “if…then”, made up of two propositions x ∈ A and x ∈ B.
So, for all x, if it is true that x ∈ A, then it must be true that x ∈ B, since B includes A. However, if x ∉ A, it does not mean that then x ∉ B, because if A ⊆ B, then B could certainly contain
elements that are not in A. There is no causal connection between the truth values of the propositions i.e. it is not a logical implication, but rather a material implication.
Here’s another example: suppose A = ∅. Is A ⊆ B? Well, we can use propositional calculus to derive the answer. Let’s use P as the antecedent (x ∈ A) and Q as the consequent (x ∈ B) of the implication
P ⇒ Q (x ∈ A ⇒ x ∈ B):
A bivalent truth table for the material implication (⇒)
1. Since ∅ by definition has no elements, x ∈ A(∅) is false.
2. x ∈ B is false (due to above).
Using the truth table, we can say that the implication is true and the claim A ⊆ B holds. Remember, the truth value for the consequent (x ∈ B) is irrelevant when the antecedent (x ∈ A) is false,
because the implication is vacuously true. This is also known as the principle of explosion i.e. false propositions imply anything.
Now, if you still find that strange, here’s a different way to think about it — imagine I have an empty bag and I say “all the items in this bag are blue in colour”. The only way to prove that the
statement is not true is to find an item in the bag that is not blue. However, since there is nothing in the bag, the statement is true (otherwise you would have a contradiction).
Now, you must be wondering what the hell does quantifications and sets have to do with allSatisfy returning true for empty sequences? Well, if it’s not already obvious by now, allSatisfy is basically
just universal quantification:
U = universe of discourse, not universal set¹
(where P(x) is a predicate that takes an element of the universe and returns whether it satisfies the predicate).
If we do universal quantification over the empty set:
then it is vacuously true, regardless of the truth value of the predicate P(x).
This is why allSatisfy returns true for empty sequences and it is (unfortunately) not a mistake. For people who are not used to the mathematical way of thinking, this might be a little confusing
since the language of formal logic/math is a little different than the English language.
However, it might help if you think of the sequence as a set and the predicate as a function that checks whether the element belongs to some other set.
For example:
1. This is also commonly referred to as a “universal set”, however whether such a set exists or not is completely dependent on the particular set theory being used.
2. There is no universal agreement on whether ℕ includes zero, however the ISO specification does include zero.
3. There is no formal definition for a “set”. It’s like asking “what’s the definition of a definition?”. Naively, you can say that a set is just a collection of things, but mathematically speaking,
a set is merely any object that satisfies the axioms of a particular set theory and so we cannot assign a single definition to it.
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Random Sampling - PlantletRandom Sampling : Plantlet
Measuring a small portion of something and then making a general statement about the whole thing is known as sampling.
• Sampling is a process of selecting a number of units for a study in such a way that the units represent the larger group from which they are selected.
Since it is generally impossible to study an entire population (e.g. many individuals in a country, all college students, every geographic area etc.) researchers typically rely on sampling to acquire
a section of the population to perform an experiment or observational study.
It is important that the group selected be representative of the population and not biased in a systematic manner.
For example, a group comprised of wealthiest individuals population in a given area probably would not accurately reflect the opinion of the entire population in that area.
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Types of Sampling
A. Probability sampling:
A method of sampling that uses random selection so that all units/cases in the population have an equal probability of being chosen is known as probability or random sampling.
1. Easy to conduct.
2. High probability of achieving a representative sample.
3. Meets assumptions ofmany statistical procedure.
1. Identification of all members of the population can be difficult.
2. Contacting all members of the sample can be difficult.
B. Non-probability sampling:
Sampling that does not involve random selection and methods are not based on the10rationale of probability theory is non-probability or non-random sampling.
Simple Random Sampling
Simple random sampling is a probability sampling method.
Definition : A Simple Random Sample (SRS) consists of n individuals from the population chosen such a way that every individual has the equal chance to be selected.
1. Simple to conduct.
2. Each unit has an equal chance of being selected.
Disadvantage :
1. Complete list of individuals in the universe is required.
Systemic (Random) Sampling
There is a gap or interval between each between each selected species of the sample.
Selection of units is based on sample interval k starting from a determined point where k= N/n.
1. Number the units on your frame from 1 to N, and the population are arranged in the same way.
2. First sample drawn between 1 and K randomly. (Determine present/ random start.)
3. Afterwards, every k th must be drawn, untill the total sample has been drawn.
Stratified Random Sampling
A stratified random sample is obtained by dividing the Population elements into non-overlapping groups, called strata and then selecting a random sample directly and independently from each stratum.
A stratified SRS is a special case of stratified sampling that uses SRS for selecting units from each stratum.
Cluster Sampling
Cluster Sampling is defined as a sampling technique in which the elements of the population is divided in existing groups (cluster).
Then a sample from the cluster is selected randomly from the population.
1. Single stage cluster Sampling.
2. Double Stage cluster Samplimg.
3. Multiple Stage cluster Sampling.
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Grade 3 Math Venn Diagram Worksheets - 3rd Grade Math Worksheets
3rd Grade Math Venn Diagram Worksheets – According to the old saying “A journey of one thousand miles begins with just one step.” This adage can be applied to learning math in third grade. This is a
crucial phase in which students learn more advanced math concepts. Third Grade Math is Vital Third grade mathematics … Read more
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Group By in MATLAB^®
How to use Group By in MATLAB^® with Plotly.
Dataset Array Summary Statistics Organized by Group
Load the sample data.
The dataset array hospital has 100 observations and 7 variables.
Create a dataset array with only the variables Sex, Age, Weight, and Smoker.
dsa = hospital(:,{'Sex','Age','Weight','Smoker'});
Sex is a nominal array, with levels Male and Female. The variables Age and Weight have numeric values, and Smoker has logical values.
Compute the mean for the numeric and logical arrays, Age, Weight, and Smoker, grouped by the levels in Sex.
dsa = hospital(:,{'Sex','Age','Weight','Smoker'});
statarray = grpstats(dsa,'Sex')
statarray =
Sex GroupCount mean_Age mean_Weight mean_Smoker
Female Female 53 37.717 130.47 0.24528
Male Male 47 38.915 180.53 0.44681
statarray is a dataset array with two rows, corresponding to the levels in Sex. GroupCount is the number of observations in each group. The means of Age, Weight, and Smoker, grouped by Sex, are given
in mean_Age, mean_Weight, and mean_Smoker.
Compute the mean for Age and Weight, grouped by the values in Smoker.
dsa = hospital(:,{'Sex','Age','Weight','Smoker'});
statarray = grpstats(dsa,'Smoker','mean','DataVars',{'Age','Weight'})
statarray =
Smoker GroupCount mean_Age mean_Weight
0 false 66 37.97 149.91
1 true 34 38.882 161.94
In this case, not all variables in dsa (excluding the grouping variable, Smoker) are numeric or logical arrays; the variable Sex is a nominal array. When not all variables in the input dataset array
are numeric or logical arrays, you must specify the variables for which you want to calculate summary statistics using DataVars.
Compute the minimum and maximum weight, grouped by the combinations of values in Sex and Smoker.
dsa = hospital(:,{'Sex','Age','Weight','Smoker'});
statarray = grpstats(dsa,{'Sex','Smoker'},{'min','max'},...
statarray =
Sex Smoker GroupCount min_Weight max_Weight
Female_0 Female false 40 111 147
Female_1 Female true 13 115 146
Male_0 Male false 26 158 194
Male_1 Male true 21 164 202
There are two unique values in Smoker and two levels in Sex, for a total of four possible combinations of values: Female Nonsmoker (Female_0), Female Smoker (Female_1), Male Nonsmoker (Male_0), and
Male Smoker (Male_1).
Specify the names for the columns in the output.
dsa = hospital(:,{'Sex','Age','Weight','Smoker'});
statarray = grpstats(dsa,{'Sex','Smoker'},{'min','max'},...
statarray =
Gender Smoker GroupCount LowestWeight HighestWeight
Female_0 Female false 40 111 147
Female_1 Female true 13 115 146
Male_0 Male false 26 158 194
Male_1 Male true 21 164 202
Summary Statistics for a Dataset Array Without Grouping
Load the sample data.
The dataset array hospital has 100 observations and 7 variables.
Create a dataset array with only the variables Age, Weight, and Smoker.
dsa = hospital(:,{'Age','Weight','Smoker'});
The variables Age and Weight have numeric values, and Smoker has logical values.
Compute the mean, minimum, and maximum for the numeric and logical arrays, Age, Weight, and Smoker, with no grouping.
dsa = hospital(:,{'Age','Weight','Smoker'});
statarray = grpstats(dsa,[],{'mean','min','max'})
statarray =
GroupCount mean_Age min_Age max_Age mean_Weight
All 100 38.28 25 50 154
min_Weight max_Weight mean_Smoker min_Smoker max_Smoker
All 111 202 0.34 false true
The observation name All indicates that all observations in dsa were used to compute the summary statistics.
Group Means for a Matrix Using One or More Grouping Variables
Load the sample data.
All variables are measured for 100 cars. Origin is the country of origin for each car (France, Germany, Italy, Japan, Sweden, or USA). Cylinders has three unique values, 4, 6, and 8, indicating the
number of cylinders in each car.
Calculate the mean acceleration, grouped by country of origin.
means = grpstats(Acceleration,Origin)
means =
means is a 6-by-1 vector of mean accelerations, where each value corresponds to a country of origin.
Calculate the mean acceleration, grouped by both country of origin and number of cylinders.
means = grpstats(Acceleration,{Origin,Cylinders})
means =
There are 18 possible combinations of grouping variable values because Origin has 6 unique values and Cylinders has 3 unique values. Only 10 of the possible combinations appear in the data, so means
is a 10-by-1 vector of group means corresponding to the observed combinations of values.
Return the group names along with the mean acceleration for each group.
[means,grps] = grpstats(Acceleration,{Origin,Cylinders},{'mean','gname'})
means =
grps =
10x2 cell array
{'USA' } {'4'}
{'USA' } {'6'}
{'USA' } {'8'}
{'France' } {'4'}
{'Japan' } {'4'}
{'Japan' } {'6'}
{'Germany'} {'4'}
{'Germany'} {'6'}
{'Sweden' } {'4'}
{'Italy' } {'4'}
The output grps shows the 10 observed combinations of grouping variable values. For example, the mean acceleration of 4-cylinder cars made in France is 18.05.
Multiple Summary Statistics for a Matrix Organized by Group
Load the sample data.
The variable Acceleration was measured for 100 cars. The variable Origin is the country of origin for each car (France, Germany, Italy, Japan, Sweden, or USA).
Return the minimum and maximum acceleration grouped by country of origin.
load carsmall
[grpMin,grpMax,grp] = grpstats(Acceleration,Origin,{'min','max','gname'})
grpMin =
grpMax =
grp =
6x1 cell array
{'USA' }
{'France' }
{'Japan' }
{'Sweden' }
{'Italy' }
The sample car with the lowest acceleration is made in the USA, and the sample car with the highest acceleration is made in Germany.
Plot Prediction Intervals for a New Observation in Each Group
Load the sample data.
The variable Weight was measured for 100 cars. The variable Model_Year has three unique values, 70, 76, and 82, which correspond to model years 1970, 1976, and 1982.
Calculate the mean weight and 90% prediction intervals for each model year.
[means,pred,grp] = grpstats(Weight,Model_Year,...
Plot error bars showing the mean weight and 90% prediction intervals, grouped by model year. Label the horizontal axis with the group names.
[means,pred,grp] = grpstats(Weight,Model_Year,...
ngrps = length(grp); % Number of groups
xlim([0.5 3.5])
title('90% Prediction Intervals for Weight by Year')
Plot Group Means and Confidence Intervals
Load the sample data.
The variables Acceleration and Weight are the acceleration and weight values measured for 100 cars. The variable Cylinders is the number of cylinders in each car. The variable Model_Year has three
unique values, 70, 76, and 82, which correspond to model years 1970, 1976, and 1982.
Plot mean acceleration, grouped by Cylinders, with 95% confidence intervals.
The mean acceleration for cars with 8 cylinders is significantly lower than for cars with 4 or 6 cylinders.
Plot mean acceleration and weight, grouped by Cylinders, and 95% confidence intervals. Scale the Weight values by 1000 so the means of Weight and Acceleration are the same order of magnitude.
The average weight of cars increases with the number of cylinders, and the average acceleration decreases with the number of cylinders.
Plot mean acceleration, grouped by both Cylinders and Model_Year. Specify 95% confidence intervals.
There are nine possible combinations of grouping variable values because there are three unique values in Cylinders and three unique values in Model_Year. The plot does not show 8-cylinder cars with
model year 1982 because the data did not include this combination.
The mean acceleration of 8-cylinder cars made in 1976 is significantly larger than the mean acceleration of 8-cylinder cars made in 1970.
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IntroductionData analysis toolsA triple-IMFR eventSkewness–kurtosis relationTime series of |B|Time series of δ|B|DiscussionConclusionsAcknowledgementsReferences
ANGEO Annales Geophysicae ANGEOAnn. Geophys. 1432-0576 Copernicus Publications Göttingen, Germany 10.5194/angeo-36-497-2018Non-Gaussianity and cross-scale coupling in interplanetary magnetic field
turbulence during a rope–rope magnetic reconnection eventNon-Gaussianity at a rope–rope reconnection MirandaRodrigo A. rmiracer@unb.br https://orcid.org/0000-0002-9861-0557 SchelinAdriane B.
ChianAbraham C.-L. https://orcid.org/0000-0002-8932-0793 FerreiraJosé L. UnB-Gama Campus, University of Brasília (UnB), Brasília DF 70910-900, Brazil Plasma Physics Laboratory, Institute of Physics,
University of Brasília (UnB), Brasília DF 70910-900, Brazil School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia Institute of Aeronautical Technology (ITA), São José
dos Campos, SP 12228-900, Brazil National Institute for Space Research (INPE), P.O. Box 515, São José dos Campos, SP 12227-010, Brazil Rodrigo A. Miranda (rmiracer@unb.br)23March2018 36 2 497507
25August2017 14December2017 29January2018 Copyright: © 2018 Rodrigo A. Miranda et al. 2018 This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of
this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/36/497/2018/angeo-36-497-2018.htmlThe full text article is
available as a PDF file from https://angeo.copernicus.org/articles/36/497/2018/angeo-36-497-2018.pdf
In a recent paper it was shown that magnetic reconnection at the interface region between two magnetic flux ropes is responsible for the genesis of interplanetary intermittent turbulence. The
normalized third-order moment (skewness) and the normalized fourth-order moment (kurtosis) display a quadratic relation with a parabolic shape that is commonly observed in observational data from
turbulence in fluids and plasmas, and is linked to non-Gaussian fluctuations due to coherent structures. In this paper we perform a detailed study of the relation between the skewness and the
kurtosis of the modulus of the magnetic field |B| during a triple interplanetary magnetic flux rope event. In addition, we investigate the skewness–kurtosis relation of two-point differences of |B|
for the same event. The parabolic relation displays scale dependence and is found to be enhanced during magnetic reconnection, rendering support for the generation of non-Gaussian coherent structures
via rope–rope magnetic reconnection. Our results also indicate that a direct coupling between the scales of magnetic flux ropes and the scales within the inertial subrange occurs in the solar wind.
Space plasma physics (turbulence)
The solar wind can be regarded as a network of entangled magnetic flux tubes and Alfvénic fluctuations propagating within each flux tube . Flux tubes can emerge locally in the solar wind as
a consequence of the magnetohydrodynamic turbulent cascade . An alternative view describes coherent structures as “fossile” structures that emanate from the solar surface and are advected by the
solar wind .
The probability distribution functions (PDFs) of turbulent space plasmas display sharp peaks and fat tails on small scales within the inertial subrange , as well as departures from self-similarity
and monofractality . These features are due to the presence of rare, large-amplitude coherent structures which dominate the statistics of fluctuations on small scales and can be quantified by the
computation of statistical moments.
A robust parabolic dependence between the normalized third-order moment (skewness) and the normalized fourth-order moment (kurtosis) has been found in local concentrations of contaminants in
atmospheric turbulence as found by . also found a similar skewness–kurtosis parabolic relation using global data of sea-surface temperature fluctuations. reported a similar skewness–kurtosis
dependence in electron density fluctuations in plasma confinement experiments. obtained a skewness–kurtosis parabolic relation for datasets of human reaction times for visual stimuli. Since then, the
presence of a skewness–kurtosis relation in different physical scenarios has attracted much attention and has been associated with the presence of non-Gaussian fluctuations due to coherent structures
. The skewness–kurtosis parabolic relation was also found in time series of two-point differences of the modulus of the magnetic field by . They demonstrated that the parabolic relation is due to
nonlocal interaction between large-scale structures and small-scale intermittency.
In this paper we investigate the skewness–kurtosis relation during a triple interplanetary magnetic flux rope (IMFR) event detected by Cluster-1 in the solar wind. This event was recently
characterized by . They demonstrated the occurrence of magnetic reconnection at the interface region of two IMFRs and that this reconnection can be the origin of interplanetary intermittent
turbulence. Our results show that the skewness–kurtosis parabolic relation is enhanced during the reconnection between flux ropes, and that is a natural consequence of the interaction between flux
This paper is organized as follows. Section presents the statistical tools employed for the data analysis, including the equations to compute the skewness and the kurtosis. Section describes the
triple-IMFR event. The skewness–kurtosis relation is analyzed in detail in Sect. . The interpretations of these results are presented in Sect. . Finally, we conclude in Sect. .
Let θi, i=1,…,N be the time series of a quantity of interest (e.g., the modulus of the magnetic field |B|). The skewness of θi can be computed as follows: S=1N∑i=1Nθi-θiσ3, where θi represents the
average of θi, N represents the number of data points, and σ is the SD of θi. The flatness of θi is given by F=1N∑i=1Nθi-θiσ4, from which the kurtosis can be obtained by K=F-3.
For a Gaussian function S=K=0. The skewness quantifies the degree of asymmetry of the PDF of θi, whereas the kurtosis quantifies the departure of the flatness of the PDF of θi from the flatness of
a Gaussian distribution which is equal to 3. The definition of kurtosis in Eq. () is sometimes called “excess kurtosis” .
A common way to characterize asymmetry and non-Gaussianity of θi as a function of scale τ is through the time series of two-point differences: δθi(τ)=θi+τ-θi. The skewness of θi on scale τ is then S
(τ)=1N∑i=1Nδθi-δθiστ3, and the flatness is F(τ)=1N∑i=1Nδθi-δθiστ4, where στ is the SD of δθi(τ). From Eq. () the kurtosis as a function of scale is obtained by K(τ)=F(τ)-3.
A functional relation between the skewness and the kurtosis of θi as defined by Eqs. ()–() has been observed in a variety of scenarios e.g.,. This relation is given by K=αS2+β, where α and β are the
coefficients that characterize a parabolic curve.
We compute the α and β coefficients by applying a least-square fit between (S,K) values obtained from the observational data and Eq. () following the Levenberg–Marquardt algorithm , which is
a popular method to fit a dataset into nonlinear equations. In order to quantify how well the computed (S,K) values are fitted into Eq. () we employ the correlation index r which measures the
correlation between two datasets Xi and Yi, i=1,…,N: r=1σXσY∑i=1N(Xi-Xi)(Yi-Yi)N, where σX and σY represent the SD of Xi and Yi, respectively. The correlation index r∈[-1,1]. If r=1 there is complete
correlation between Xi and Yi, whereas r=-1 indicates anticorrelation. The value r=0 represents absence of correlation.
In summary, the analysis is described by the following steps:
Compute S and K from the modulus of magnetic field |B| using Eqs. ()–().
Apply the Levenberg–Marquardt algorithm to find α and β in Eq. () that best fit the (S,K) values.
Use α and β obtained from the previous step in Eq. () to obtain empirical values of K as a function of S.
Compute the correlation index r between the values of K from the previous step and the values of K from the observational data. The r index will measure how close the K values computed by Eq. () are
to the K values obtained empirically from Eq. ().
We repeat these steps for S(τ) and K(τ) of two-point differences using Eqs. () and () in the first step. There are several computational programs for data analysis that implement the
Levenberg–Marquardt algorithm. Here we use the implementation available in the GNU Octave program .
We note that several papers regarding the relation between skewness and kurtosis have employed the definition of what we refer to as flatness (Eq. ). Throughout this paper we will focus on the
kurtosis defined by Eq. ().
Figure a shows the time series of the modulus of magnetic field |B| obtained by the FGM instrument onboard Cluster-1 from 00:00 to 12:00UT on 2 February 2002. During this interval Cluster-1 was in
the solar wind upstream of the Earth's bow shock . The magnetic field data are collected by Cluster-1 at a resolution of 22Hz . Figure also presents an overview of other in situ plasma parameters
for the selected interval, namely, the three components of B in the GSE coordinates, the angles ΦB and ΘB of the solar wind magnetic field B relative to the Sun–Earth x axis in the ecliptic plane,
and out of the ecliptic, respectively, in the polar GSE coordinates; the modulus of the ion bulk flow velocity |Vi|, the ion number density ni, the ion temperature perpendicular to the magnetic field
Ti and the ion plasma βi, which is the ratio between plasma kinetic pressure and magnetic pressure. The Cluster-1 plasma measurements are given by the ion spectrometry experiment CIS .
This event is characterized by the presence of three interplanetary magnetic flux ropes. Magnetic flux ropes are magnetic structures described as bundles of twisted, current-carrying magnetic field
lines bent into a tube-like shape, spiralling around a common axis . During this event three IMFRs were identified by using a combination of criteria for large-scale magnetic cloud boundary layers
and small-scale IMFRs . The interval of each IMFR is indicated by horizontal arrows in Fig. a, and their timings are shown in Table .
Cluster-1 magnetic field and plasma parameters from 00:00 to 12:00UT on 2 February 2002. From top to bottom: modulus of magnetic field |B| (nT), three components of B (nT) in the GSE coordinates,
azimuth angle ΦB (∘), latitude angle Θ (∘), modulus of ion bulk velocity |Vi| (kms-1), ion number density ni (cm-3), ion temperature Ti (eV) and ion plasma beta βi. Horizontal arrows indicate the
interval of IMFR-1 (black), IMFR-2 (red) and IMFR-3 (blue). The front and rear boundary layers of each IMFR are indicated by the vertical dotted lines.
Beginning and end of the intervals depicted in Fig. , corresponding to the boundary layers of three interplanetary magnetic flux ropes (IMFRs) on 2 February 2002.
Beginning (UT) End (UT) IMFR-1 00:32 00:53 IMFR-2 01:32 02:35 IMFR-3 02:31 08:53
Figure a shows the time series of |B| detected by Cluster-1 on 2 February 2002 (Julian day 32) from 00:32 to 03:18UT. Five regions were defined during this interval and are indicated using arrows.
These regions represent the interior region of IMFR-1 (R1), the interface of IMFR-1 and IMFR-2 (I12), the interior of IMFR-2 (R2), the interface of IMFR-2 and IMFR-3 (I23), and the interior of IMFR-3
(R3). Their timings are indicated in Table . Each region has a duration of 30min, which gives 40358 data points. During this event current sheets were detected at the front boundary layer of IMFR-1
and at the interface region between IMFR-2 and IMFR-3. This interface region was identified as a source of intermittent turbulence by . A current sheet was detected at the leading edge of IMFR-1
using data from ACE and Cluster-1, and a current sheet was detected at the interface region between IMFR-2 and IMFR-3 using data from Cluster-1, ACE and Wind .
Timing of the five selected regions during the triple-IMFR event on 2 February 2002.
Interval Symbol Start End Interior region of IMFR-1 R1 00:32 01:02 Interface of IMFR-1 and IMFR-2 I12 01:02 01:32 Interior region of IMFR-2 R2 01:48 02:18 Interface of IMFR-2 and IMFR-3 I23 02:18
02:48 Interior of IMFR-3 R3 02:48 03:18
(a) Time series of |B| from 00:32 to 03:18UT on 2 February 2002. Five regions of 30min each are highlighted using different colors: interior of IMFR-1 (R1, black), interface region of IMFR-1 and
IMFR-2 (I12, green), the interior of IMFR-2 (R2, red), the interface of IMFR-2 and IMFR-3 (I23, violet), and the interior of IMFR-3 (R3, blue). The interval of each IMFR is indicated by horizontal
arrows as in Fig. . (b, c) Time series of the skewness S and the kurtosis K computed using a sliding overlapping window of size 10000 data points and a window shift of 400 data points. The SD
computed in each window is represented by a gray area.
The S–K parabolic relation described by Eq. () can be verified by computing S and K from a number of datasets corresponding to different realizations of an experiment. In the case of a time series,
the parabolic relation can be tested by computing S and K using datasets extracted from the time series with sliding windows. The size of the sliding window is a critical parameter for this type of
analysis. Since S and K are higher statistical moments, the number of data points inside the window should be large enough to guarantee the robust estimation of S and K. However, if the time series
is divided into sliding windows with a large number of data points, then the number of (S, K) values may be insufficient to verify the parabolic relation of Eq. (). This can be solved by defining
overlapping windows; nevertheless, the overlapping cannot be too large in order to obtain a set of independent (S, K) values. To determine the optimal window size, we applied a procedure to estimate
the maximum order of the statistical moment in a time series . We computed the maximum statistical order in each sliding window of size 5000 data points across the time series of Fig. , and a window
shift of 400 data points. Then, we increased the size of the window by 1000 data points (keeping the same window shift), computed the maximum order in each window and then repeated the procedure. We
found that a sliding window of size 10000 data points is large enough for a robust estimation of moments up to the sixth order in all windows and at the same time allows a sufficient number of
estimations of S an K to be obtained to test the parabolic relation of Eq. (). Figure b and c show the resulting time series of S and K, respectively. The SD gives an estimation of the uncertainty of
the computed S and K inside each window, and is represented using a gray area. From this figure we observe that from 02:26 to 02:35UT the uncertainty of S and K increases due to the large variation
in |B| at the interface between IMFR-2 and IMFR-3. A similar behavior was observed in magnetic field data during an interplanetary shock event by . The uncertainty within sliding windows that contain
the large variations in |B| increases due to nonstationarity. Following , we exclude these windows from further analysis.
Kurtosis K as a function of skewness S computed using overlapping windows of size 10000 data points and a window shift of 400 data points, for (a) the interior region of IMFR-1, (b) the interface of
IMFR-1 and IMFR-2, (c) the interior of IMFR-2, (d) the interface of IMFR-2 and IMFR-3, and (e) the interior of IMFR-3. In each panel, the least-square fit with the parabolic function K=αS2+β is
displayed as a dashed line (see Table ).
Figure shows K as a function of S for the five regions previously defined. A least-square fit with Eq. () is displayed as a dashed line. Table shows the resulting fit for each region, as well as
the correlation index r between the points in the scatter plot and the fitted parabolic function computed using Eq. (). Since the interpretation of α and β is under debate (see the discussion in
Sect. ) we will focus on the computed value of r.
The least-square fits of Eq. () computed from the scatter plots of Fig. , and the correlation index r for the five regions defined.
Interval K=αS2+β r R1 K=1.29S2-0.86 0.78 I12 K=1.42S2-0.92 0.75 R2 K=1.82S2-0.42 0.73 I23 K=1.26S2-0.40 0.91 R3 K=1.03S2+0.17 0.76
The correlation index r shown in the last column of Table measures how well the data points can be adjusted by the parabolic function given by Eq. (). All regions display r>0.5. The lowest
correlation is obtained for the interval corresponding to R2, in agreement with a visual inspection of Fig. c. For this interval, most of the points in Fig. c tend to accumulate around (S,K)=(0,0),
which is the value obtained for a Gaussian distribution (i.e., in the absence of coherent structures). Therefore, the interior of IMFR-2 is characterized by a low degree of non-Gaussianity and
intermittency in comparison with the other intervals.
The highest value of the correlation is obtained during I23 (see Table ). Figure d shows that points spread near the fitted parabola and far from the (0, 0) Gaussian point. This indicates that this
interval is characterized by a higher degree of non-Gaussianity. These results are in agreement with the results of , which found that the interior of IMFR-2 has lower degrees of non-Gaussianity and
phase coherence, and a nearly monofractal scaling when compared with other intervals. For the interface of IMFR-2 and IMFR-3 they observed higher degrees of non-Gaussianity and phase synchronization,
and a strong departure from monofractality.
The power spectral density (PSD, left panels) and the compensated PSD (right panels) for (a) the time series of |B| from 00:32UT until 08:40UT, (b) the IMFR-1 interior region, (c) the interface
between IMFR-1 and IMFR-2, (d) the IMFR-2 interior region, (e) the interface between IMFR-2 and IMFR-3, and (f) the IMFR-3 interior region. Vertical dashed lines indicate the beginning and the end of
the inertial subrange.
Next, we investigate the S–K parabolic relation as a function of scale within the inertial subrange. The left side of Fig. a shows the power spectral density (PSD) as a function of frequency f of the
time series of |B| from the beginning of IMFR-1 at 00:32UT until the end of IMFR-3 at 08:40UT. The right side of Fig. a shows the compensated PSD which is the original PSD multiplied by f5/3 . The
inertial subrange should appear as a frequency range in which the compensated PSD is almost horizontal. The following panels in Fig. show the PSD and the compensated PSD for R1, I12, R2, I23 and R3.
A common frequency range in which the compensated PSD is almost horizontal for all regions is indicated by two vertical dashed lines. From Fig. , the inertial subrange starts at f=0.01Hz and ends at
f=0.1Hz, which correspond to scales τ=100s and τ=10s, respectively.
Probability distribution functions (PDFs) of ΔB(τ) for τ=10s (continuous line) and τ=100s (dashed line). (a) The interior region of IMFR-1, (b) the interface of IMFR-1 and IMFR-2, (c) the interior
of IMFR-2, (d) the interface of IMFR-2 and IMFR-3, and (e) the interior of IMFR-3. A Gaussian distribution function is represented by the gray area.
The intermittent aspect of interplanetary magnetic field turbulence can be demonstrated by constructing the PDF of the normalized magnetic-field differences ΔB(τ)=δB(τ)-δBσB, where δB(τ)=|B(t+τ)|-|B
(t)|, and the brackets denote the average value. Figure shows the PDFs of ΔB constructed from the magnetic field fluctuations of the five regions, for τ=10s and τ=100s. From this figure it is
clear that the PDFs are closer to a Gaussian distribution (represented by the gray area in Fig. ) at τ=100s (large scale), and become non-Gaussian at τ=10s (small scale), exhibiting sharp peaks and
fat tails. This figure demonstrates that magnetic field fluctuations become more intermittent as the scale τ becomes smaller.
Next, we analyze the S–K relation of ΔB(τ) at τ=10s and τ=100s. Figure a shows the time series of ΔB(τ=100s). Figure b and c show the time series of S and K computed using a sliding overlapping
window as in Sect. . The gray area indicates the uncertainty of the S and K values. As in Fig. , we observe a large uncertainty from 02:26 to 02:35UT due to the interface between IMFR-2 and IMFR-3;
therefore these S and K values are excluded from further analysis.
(a) Time series of δB(τ=100s) from 00:00 to 04:00UT on 2 February 2002. Five regions of 30min each are highlighted using different colors: interior of IMFR-1 (R1, black), interface region of IMFR-1
and IMFR-2 (I12, green), the interior of IMFR-2 (R2, red), the interface of IMFR-2 and IMFR-3 (I23, violet), and the interior of IMFR-3 (R3, blue). The interval of each IMFR is indicated by
horizontal arrows as in Fig. . (b, c) Time series of the skewness S and the kurtosis K computed using a sliding overlapping window with the same parameters as in Fig. . The SD computed in each window
is represented by a gray area.
Figures and show the S–K scatter plots for τ=100s and τ=10s, respectively. From these figures, we note that R2 does not display a parabolic shape on the two selected scales. The low value of the
correlation index of R2 shown in Tables and confirms that the data points fit the parabolic shape poorly. This indicates that magnetic field fluctuations during R2 are nearly Gaussian even on the
smallest scale.
Kurtosis K as a function of skewness S computed from the time series of δ|B|(τ), where τ=100s. (a) The interior region of IMFR-1, (b) the interface of IMFR-1 and IMFR-2, (c) the interior of IMFR-2,
(d) the interface of IMFR-2 and IMFR-3, and (e) the interior of IMFR-3. In each panel, the least-square fit with the parabolic function K=αS2+β is displayed as a dashed line (see Table ).
Kurtosis K as a function of skewness S computed from the time series of δ|B|(τ), where τ=10s. (a) The interior region of IMFR-1, (b) the interface of IMFR-1 and IMFR-2, (c) the interior of IMFR-2,
(d) the interface of IMFR-2 and IMFR-3, and (e) the interior of IMFR-3. In each panel, the least-square fit with the parabolic function K=αS2+β is displayed as a dashed line (see Table ).
The least-square fits of Eq. () computed from the scatter plots of Fig. (τ=100s). The fitting function of IMFR-2 was not applicable (n/a) due to the small correlation value.
Interval K=αS2+β r R1 K=1.45S2-0.77 0.65 I12 K=0.97S2-0.57 0.53 R2 n/a 0.13 I23 K=1.24S2-0.15 0.90 R3 K=0.76S2-0.13 0.46
Same as in Table for τ=10s. The fitting function of IMFR-2 was not applicable (n/a) due to the small correlation value.
Interval K=αS2+β r R1 K=3.00S2 + 0.88 0.98 I12 K=2.96S2 + 0.95 0.91 R2 n/a 0.14 I23 K=1.36S2 + 0.72 0.99 R3 K=2.13S2 + 1.40 0.90
Except for R2, all other regions show a parabolic shape at τ=100s that is enhanced at τ=10s, in agreement with the intermittent nature of magnetic field turbulence. Magnetic field fluctuations in
the solar wind turbulence display a scale dependence in which they become intermittent as the scale becomes smaller, within the inertial subrange, due to rare, large-amplitude coherent structures. As
a consequence, statistics of magnetic field fluctuations such as the PDFs of the ΔB (Fig. ) departure from Gaussian statistics as τ decreases. By comparing the values of the correlation index shown
in Table for τ=100s with those of Table for τ=10s we note that, for each region, the correlation r increases on the smallest scale, confirming that the S–K parabolic relation displays scale
dependence within the inertial subrange.
The highest correlation value for τ=100s (Table ) corresponds to I23. This indicates that the ongoing magnetic reconnection occurring in this region can act as a source of non-Gaussianity and
intermittent turbulence even on the largest scale. At τ=10s, Table shows that r=0.99 at I23 and r=0.98 at R1. Small-scale current sheets were detected in these two intervals by and are responsible
for intermittency and non-Gaussian fluctuations. Our result demonstrates that they are also responsible for the enhancement of the S–K parabolic relation. Note that there are points in Fig. d that
are further away from the (0, 0) Gaussian point, compared to Fig. a. This means that while the scatter plots of R1 and I23 are highly correlated with Eq. (), the numerical values of S and K, which
measure the degree of asymmetry and non-Gaussianity respectively, can be higher at I23.
A theoretical explanation of the parabolic relation between the skewness and kurtosis of turbulent fluids and plasmas is still an open question. proposed a nonlinear Langevin equation with external
forcing that can account for the parabolic relation between S and K. extended this model to include self-generated internal instabilities in plasmas. argued that a parabolic relation can be obtained
as a natural consequence of a number of constraints expected to be met for most physical systems. proposed a simplified model of a synthetic intermittent time series, constructed from a random number
of coherent structures with random amplitudes embedded in a background Gaussian noise, and demonstrated that their model can predict a S–K parabolic relation. A similar study was performed by using
a model of coherent plasma flux events.
Although a theoretical explanation of the S–K relation is still unclear, there is a consensus that the parabolic shape is due to non-Gaussianity related to coherent structures, whereas points near
(S,K)=(0,0) correspond to Gaussian fluctuations. This is confirmed by models of synthetic time series. For example, proposed a model of intermittent time series which consists of a superposition of
Gaussian and non-Gaussian random fluctuations. Their model includes a parameter that measures the deviation from Gaussianity. The resulting PDF derived from their model displays asymmetric long tails
that reproduce measured distributions of plasma density fluctuations in plasma magnetic confinement devices as well as distributions of X-ray emissions detected from accretion disks . Their model
also leads to a parabolic relation between S and K. observed a transition from a parabolic shape to the (S,K)=(0,0) point by increasing the intensity of the Gaussian noise in their model of synthetic
time series, constructed by adding deterministic fluctuations and Gaussian noise. However, a quantification of the parabolic shape is needed for an objective comparison between different datasets. We
have found that the computation of the correlation index r allows time series dominated by either Gaussian and non-Gaussian fluctuations to be clearly distinguished. Despite the simplicity of this
approach, it represents an alternative way to compare the degree of non-Gaussianity due to asymmetry and fat tails in the PDFs of different datasets, and can be applied to observational data and
results from numerical simulations.
The stochastic model of a time series proposed by assumes that the non-Gaussian fluctuations arise from a quadratic nonlinear term. By increasing the degree of non-Gaussianity the skewness and the
kurtosis converge to extreme values: S=±22 and K=12. This means that experimental data governed by nonlinear processes of quadratic order should lead to S–K scatter plots with S∈[-22,22] and K<12.
The scatter plots shown in Fig. a, c and e seem to agree with these limits; however, in Fig. d there are some points in which S<-22. also propose that processes described by higher-order
nonlinearities can result in S–K parabolic shapes with S outside the interval [-22,22], which can explain the behavior of |B| during the magnetic reconnection occurring in the I23 interval.
In the previous sections we showed and discussed the value of the correlation index measuring how well the S–K scatter plots fit with a parabola. As mentioned before, there is no agreement on the
interpretation of the coefficients α and β in Eq. (). argues that the coefficients are not likely to offer relevant information about the underlying process. However, discussed an interpretation of
the α and β coefficients based on their model of a synthetic time series. The value of the α coefficient depends on the statistics of the fluctuations due to coherent structures and is not
necessarily constant in time. For the β coefficient, if the number of coherent structures in a time series can be represented as random independent variables that follow a Poisson distribution
function (which models the occurrence of rare events), then β=3. Deviations from this value can be interpreted as a departure from the independence assumption, which means that there is interaction
among coherent structures . Since we define kurtosis to be the flatness minus three, the previous statement is equivalent to say that deviations from β=0 are due to interacting coherent structures.
From Table , we note that all intervals have nonzero values of β. Recall that this event is characterized by a rope–rope magnetic reconnection involving IMFR-2 and IMFR-3, with formation of
a bifurcated current sheet acting as a source of intermittent turbulence . The interaction between the small-scale IMFR-2 and the medium-scale IMFR-3 occurring during this event gives support for the
interpretation of the β parameter by .
demonstrated that the S–K parabolic relation is also observed for time series of two-point differences of |B| in the solar wind. They showed that this relation is enhanced in the presence of
large-scale events such as interplanetary shocks, whereas for nonshock intervals, the parabolic relation is not observed. In this case the S–K parabolic relation represents a signature of direct
coupling between large-scale structures (interplanetary shocks) and small-scale intermittency. Our results indicate that the S–K parabolic relation is present during reconnection between
a small-scale IMFR with a duration of ∼60min and a medium-scale IMFR with a duration of ∼7h (see Table 1). The only region in which the parabolic relation is not observed is in the interior of
IMFR-2. This region was found to have a low degree of intermittency and nearly monofractal scaling. Therefore, our results are in accordance with cross-scale coupling between IMFR scales and scales
within the inertial subrange.
In this paper we investigated the relation between the skewness and the kurtosis during a triple-IMFR event on 2 February 2002. This event was divided into five regions, namely, the interior of
IMFR-1, the interface of IMFR-1 and IMFR-2, the interior of IMFR-2, the interface of IMFR-2 and IMFR-3, and the interior of IMFR-3. We then computed the skewness S and the kurtosis K of |B| using
a sliding window, and showed that the scatter plots of K as a function of S display a parabolic shape for all regions. The highest value of the correlation index computed by a least-square fit
between the (S,K) values and Eq. () occurs at the interface of IMFR-2 and IMFR-3. This region was found to be the source of intermittent turbulence due to a magnetic reconnection between the
small-size IMFR-2 and the medium-size IMFR-3 . Therefore, the enhanced S–K parabolic relation is related to non-Gaussian fluctuations due to coherent structures emerging from intermittent turbulence
generated via magnetic reconnection. The lowest value of the correlation index was obtained at the interior of IMFR-2, in agreement with the results of , who found that this region is characterized
by a low degree of non-Gaussianity and phase synchronization, and nearly monofractal scaling.
We also analyzed the S–K relation using two-point differences of |B| on two different scales within the inertial subrange. By computing the compensated PSD we selected an interval of frequencies in
which all regions exhibit -5/3 scaling corresponding to the inertial subrange and selected two timescales representing the largest scale (τ=100s) and the smallest scale (τ=10s) within the inertial
subrange. We found that the scatter plot of IMFR-2 on the largest scale (τ=100s) and on the smallest scale (τ=10s) accumulate around the (S,K)=(0,0) point. The least-square fit with Eq. () results
in a low correlation index, which confirms that magnetic field fluctuations in this region are nearly Gaussian. All other regions displayed parabolic shapes. At τ=100s, the correlation index is high
for the interface of IMFR-2 and IMFR-3, indicating that the magnetic reconnection that occurs in this region can generate non-Gaussian fluctuations on the largest scale. On the smallest scale, the
correlation index is higher for two regions, namely, the interior of IMFR-1 and the interface of IMFR-2 and IMFR-3. This result can be due to non-Gaussian fluctuations resulting from small-scale
current sheets detected within these regions . Our analysis indicates that the S–K parabolic relation observed in interplanetary magnetic field turbulence is enhanced on small scales within the
inertial subrange.
Our findings give support to the conclusion by that rope–rope magnetic reconnection acts as a source of interplanetary intermittent turbulence and suggest that magnetic reconnection is responsible
for non-Gaussian PDFs with asymmetric shapes and fat tails. The results are also in agreement with the results of in that the S–K parabolic relation is a signature of direct coupling between IMFR
scales and small-scale intermittency.
All data analyzed in this paper are publicly available via the Cluster Science Archive at http://www.cosmos.esa.int/web/csa (ESA, 2018). Numerical codes are also freely available at https://
github.com/rmiracer (Miranda, 2018).
The authors declare that they have no conflict of interest.
This article is part of the special issue “Space weather connections to near-Earth space and the atmosphere”. It is a result of the 6∘ Simpósio Brasileiro de Geofísica Espacial e Aeronomia (SBGEA),
Jataí, Brazil, 26–30 September 2016.
The authors are grateful to the reviewer for valuable comments. The authors would like to thank Heng Qiang Feng for providing the estimated times of the boundary layers for the three IMFRs observed
by Cluster-1. Rodrigo A. Miranda acknowledges support from FAPDF (Brazil) under grant 0193.000984/2015. Adriane B. Schelin acknowledges support from FAPDF under grant 0193.000.884/2015.
Abraham C.-L. Chian acknowledges the award of a PVE Distinguished Visiting Professor Fellowship by CAPES (grant no. 88881.068051/2014-01) and the hospitality of Erico Rempel of ITA. José L. Ferreira
acknowledges support from the UNIESPAÇO program of the Brazilian Space Agency (AEB), the National Council of Technological and Scientific Development (CNPq), and FAPDF. The topical editor, Alisson
Dal Lago, thanks the two anonymous referees for help in evaluating this paper.
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Subjective Probability - (Calculus and Statistics Methods) - Vocab, Definition, Explanations | Fiveable
Subjective Probability
from class:
Calculus and Statistics Methods
Subjective probability refers to the likelihood of an event occurring based on personal judgment, beliefs, or experience rather than on mathematical calculations or objective data. It allows
individuals to quantify uncertainty in situations where empirical data is limited or unavailable, thus incorporating personal insights into decision-making processes.
congrats on reading the definition of Subjective Probability. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Subjective probability is often influenced by an individual's past experiences and knowledge related to the specific event in question.
2. This type of probability can vary significantly between different individuals, even when considering the same situation or event.
3. It plays a crucial role in fields such as psychology and decision theory, where personal beliefs heavily impact choices.
4. Subjective probabilities can be difficult to quantify and may lack consistency due to their reliance on personal judgment.
5. In many practical applications, subjective probability is combined with objective data to create a more balanced understanding of risks.
Review Questions
• How does subjective probability differ from objective probability in terms of assessment and application?
□ Subjective probability differs from objective probability primarily in its basis for assessment. While objective probability relies on statistical data and historical frequencies to determine
likelihood, subjective probability stems from personal beliefs and experiences. This means that subjective probability can vary widely among individuals regarding the same event, leading to
different conclusions and decisions even in similar circumstances.
• In what ways can subjective probability be useful in real-world decision-making scenarios?
□ Subjective probability can be especially useful in real-world decision-making when empirical data is scarce or when individuals face complex uncertainties. By allowing people to incorporate
their insights, experiences, and intuitions, subjective probabilities help them navigate choices that involve risks, such as investments or health-related decisions. This personalized
approach enables decision-makers to consider aspects that may not be captured through purely statistical methods.
• Evaluate the implications of using subjective probabilities in risk assessment and management.
□ Using subjective probabilities in risk assessment and management can lead to both positive and negative implications. On one hand, they allow for a more nuanced understanding of risks by
incorporating individual perspectives and experiences. However, they may also introduce biases and inconsistencies that can affect the reliability of risk evaluations. It is crucial for
decision-makers to be aware of these potential pitfalls and consider integrating both subjective judgments and objective data for a more comprehensive risk management strategy.
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Fast Estimation of Orbital Parameters in Milky Way-like Potentials
Orbital parameters, such as eccentricity and maximum vertical excursion, of stars in the Milky Way are an important tool for understanding its dynamics and evolution, but calculation of such
parameters usually relies on computationally expensive numerical orbit integration. We present and test a fast method for estimating these parameters using an application of the Stäckel fudge, used
previously for the estimation of action-angle variables. We show that the method is highly accurate, to a level of <1% in eccentricity, over a large range of relevant orbits and in different Milky
Way-like potentials, and demonstrate its validity by estimating the eccentricity distribution of the RAVE-TGAS data set and comparing it with that from orbit integration. Using this method, the
orbital characteristics of the ∼7 million Gaia DR2 stars with radial velocity measurements are computed with Monte Carlo sampled errors in ∼116 hours of parallelized cpu time, at a speed that we
estimate to be ∼3 to 4 orders of magnitude faster than using numerical orbit integration. We demonstrate using this catalog that Gaia DR2 samples a large range of orbits in the solar vicinity, down
to those with r [ap] ≲ 2.5 kpc, and out to r [peri] ≳ 13 kpc. We also show that many of the features present in orbital parameter space have a low mean z [max], suggesting that they likely result
from disk dynamical effects.
Publications of the Astronomical Society of the Pacific
Pub Date:
November 2018
□ Astrophysics - Astrophysics of Galaxies;
□ Astrophysics - Instrumentation and Methods for Astrophysics
11 Pages, 6 Figures, Accepted for publication in PASP, Full code and extra material available at https://github.com/jmackereth/orbit-estimation , Supplementary data table is available
(temporarily) at http://www.astro.ljmu.ac.uk/~astjmack/gaia_orbits/gaiarvs_orbitparams_units.csv
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Advice on diluting or combining 2 chemical peel acids together ? - Chemists Corner
Advice on diluting or combining 2 chemical peel acids together ?
Posted by Anonymous on December 2, 2019 at 2:06 pm
Hi, If I have a teaspoon of 40% AHA(LACTIC ACID) and teaspoon of 10% BHA(SALICYLIC ACID) will combining them result in a 20% AHA and a 5%BHA solution, I know it sounds straight forward but it’s
not with water but another acid so I had to make sure, and would there be anything else to beware of?
• 6 Replies
• Member
December 2, 2019 at 9:12 pm
Think of it this way. 40% solution means the following: A *5 gram sample will contain:
1. 2 grams of Lactic Acid
2. 3 grams of water
Similarly, a 10% solution of BHA will have the following:
1. 1 gram of BHA
2. 4 grams of water
So, if you combine both solutions, you would get the following…
1. 2 grams of lactic acid
2. 1 gram of BHA
3. 7 grams of water
That would be 20% lactic acid, 5% BHA. Now, if you poured that sample into something else you are going to change the percentages.
*incidentally, a teaspoon equals about 5 grams.
• Member
December 2, 2019 at 9:44 pm
Developing a chemical peel formula is generally considered to be quite an advanced formulation project for a cosmetic chemist. There are also regulations that you need to be aware of when working
with AHAs.
Please make sure you have the relevant experience before attempting this project. Your suggestion of using teaspoons suggests that you do not have this experience. Cosmetic chemists work in
weights with percentages.
I am not trying to deter you from creating your own products just trying to make sure you do it safely.
• Member
December 2, 2019 at 10:40 pm
@ozgirl - I agree completely
December 3, 2019 at 6:04 am
Think of it this way. 40% solution means the following: A *5 gram sample will contain:
1. 2 grams of Lactic Acid
2. 3 grams of water
Similarly, a 10% solution of BHA will have the following:
1. 1 gram of BHA
2. 4 grams of water
So, if you combine both solutions, you would get the following…
1. 2 grams of lactic acid
2. 1 gram of BHA
3. 7 grams of water
That would be 20% lactic acid, 5% BHA. Now, if you poured that sample into something else you are going to change the percentages.
*incidentally, a teaspoon equals about 5 grams.
Thank you this is really helpful. Why is 10% solution of BHA, 1gram? I would have thought 10% of 5grams is 0.5grams?
December 3, 2019 at 6:08 am
Developing a chemical peel formula is generally considered to be quite an advanced formulation project for a cosmetic chemist. There are also regulations that you need to be aware of when
working with AHAs.
Please make sure you have the relevant experience before attempting this project. Your suggestion of using teaspoons suggests that you do not have this experience. Cosmetic chemists work in
weights with percentages.
I am not trying to deter you from creating your own products just trying to make sure you do it safely.
Thank you for not trying to deter me and your concern for my safety. I will do a patch test to see if I don’t have any adverse reactions. Perry explained very to me the answer to my simple
question, this is for my personal use I’m not trying to sell anything, if there is anything else you’d like to share which doesn’t come with a price please do.I know how to work metric,
percentages and imperial measurements interchangeably without any issues. This isn’t rocket science.
Thank you again for your concern.
• Member
December 3, 2019 at 1:46 pm
@Bluewoodg - Yes, 0.5 g would be correct. I made a mental math error.
A 10% solution of BHA will have the following:
1. 0.5 gram of BHA
2. 4.5 grams of water
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What is true about wave-particle duality?
Wave-particle duality refers to the fundamental property of matter where, at one moment it appears like a wave, and yet at another moment it acts like a particle. To understand wave-particle duality
it’s worth looking at differences between particles and waves.
Do all particles have wave-particle duality?
Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. This phenomenon has been verified not only for
elementary particles, but also for compound particles like atoms and even molecules.
Why wave-particle duality is wrong?
Because, there is no “wave-particle duality” in nature. Some people believes that the wavefunctions used in some formulations of QM are real waves, but this is a mistake. A wave is a physical system
which carries energy and momentum. A wavefunction is a mathematical function which cannot be observed.
What is the quantum theory of energy?
The quantum theory shows that those frequencies correspond to definite energies of the light quanta, or photons, and result from the fact that the electrons of the atom can have only certain allowed
energy values, or levels; when an electron changes from one allowed level to another, a quantum of energy is emitted or …
What is the relationship between wavelength and energy in a wave?
Just as wavelength and frequency are related to light, they are also related to energy. The shorter the wavelengths and higher the frequency corresponds with greater energy. So the longer the
wavelengths and lower the frequency results in lower energy. The energy equation is E = hν.
What’s the difference between a particle and a wave?
The difference between the particle and waves are: The particle is defined as the small quantity of matter under the consideration. The wave is defined as the propagating dynamic distrubance. The
energy of the wave is calculated based on the wavelength and velocity.
Where does wave-particle duality apply?
Applications. Wave-particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can be used to view objects much smaller than what is visible using
visible light.
Why is wave theory wrong?
Pilot-wave theory has no counterpart to explain particle behavior at near-light-speed, which is part of the reason it cannot explain particles existing in two places at once, or springing in and out
of existence, as we seem to have observed.
Does QFT explain wave-particle duality?
So a similar plot would illustrate the wavepacket nature of particle representations in QFT, but the particle/wave duality comes from the nature of the wavefunctions describing the ground state, on
which the quantum field creation and annihilation operators work.
What are the quantum particles?
There are two classes of quantum particles, those with a spin multiple of one-half, called fermions, and those with a spin multiple of one, called bosons. Electrons, protons, and neutrons are
fermions. The spin quantum number of bosons can be s = +1, s = −1, s = 0, or a multiple of ±1.
What is the relationship between wavelength and energy in a wave Brainpop?
More energy means a greater amplitude. The distance a wave travels in one wave cycle is one wavelength.
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Point and Smoothed-Particle Hydrodynamics (SPH) Interpolation in ParaView
(This is the second article in a two part series. It shows how to perform point interpolation using ParaView. The first article describes how to perform point interpolation using VTK.)
In the first article in this two-part series, we provided background and described the implementation of point interpolation in VTK. Both general point interpolation and smoothed-particle
hydrodynamics SPH methods were addressed. In this article, we describe how to use ParaView to perform point interpolation, which of course uses VTK under the hood, but provides a simple and powerful
integrated application GUI to perform the interpolation.
The basic procedure is fairly simple. After first reading or creating data with points and associated data attributes (e.g., scalars, vectors, and so on), select Filters->Points Interpolation as
illustrated below.
Depending on the type of point data and interpolation process to use (i.e., either generalized or SPH) and whether you want to resample the point data onto a volume, plane or line, six choices are
available as shown. Resampling onto a volume means that volume rendering and standard 3D visualization techniques such as isocontouring can be used on the resulting volume. Plane resampling, which
along with the associated plane widget, enables the user to rapidly slice through the data and show a heat map on the resulting plane. Finally, plotting along a resampled line supports detailed
analysis of data values. Each of these three approaches are shown in the next subsections.
SPH Volume Interpolation
The figure below shows the initial point cloud which is from the VTK test data repository; it’s fairly small in size at only ~175,000 points (SPH_Points.vtu). Note that it is not particularly dense,
and the initial visualization is less than compelling as shown.
To visualize this cloud with a volume, once read in a SPH Volume Interpolator is created. This brings up an interactive widget which positions a 3D volume into the bounding box of the point cloud
(and which can be adjusted as desired). It’s very important to set the correct spatial step size as this smoothing length is central to the spline formulation of the SPH interpolating kernel. Also
you’ll want to specify the density array, and volume resolution. You can also optionally select a different kernel than the default quintic formulation, and select with arrays to interpolate (or
not). The dimension of the data is 3, as this is space in which the points are located. Note that a custom color transfer function has been created for the rendered image below.
SPH Plane Interpolation
Interpolating across a plane is equally easy. Simply select SPH Plane Interpolator, position the plane with the interactive widget, set the spatial step size, resolution and other parameters as
before. The result will be similar to what is shown below.
SPH Line Interpolation
The line interpolation process is similar to the plane and volume interpolation process described previously. The biggest difference is that it produces an x-y plot of the various interpolated data
attributes. Here the density variable Rho, pressure Press, and Shepard Summation are plotted. Note that in the SPH formulation, the Shepard (or interpolating) coefficients when summed should be
approximately equal to one. However near the boundary the coefficient summation falls towards zero.
Interpolating data from discrete data points to one or more sampled positions is a core visualization operation. While VTK and ParaView provide a wealth of interpolation functions as embodied in
vtkCells and its subclasses, this process requires an explicit topological articulation to interpolate data. Alternatively, interpolation directly from neighborhood points is possible. Such
situations occur when processing point clouds, or when alternative formulations different than that provided by standard cell functions is required. VTK, and the application ParaView now provide fast
and flexible tools for interpolating point data. Not only are generalized interpolation processes using specification of various kernels possible, but advanced methods based on spline formulations of
smoothed-particle hydrodynamics SPH are too. Further ParaView provides a simple, interactive user interface to sample point data and produce compelling visualizations in 1-, 2-, and 3-dimensions.
2 comments to Point and Smoothed-Particle Hydrodynamics (SPH) Interpolation in ParaView
1. Where can I get this “SPH_Points.vtu”? A google search indicates that it exists somewhere, but none of the VTK test data repositories seem to contain this file.
1. If you have a VTK build directory it will be in “build directory/ExternalData/Testing/Data/SPH_Points.vtu”
If you have downloaded the VTK data files, it will be named “.ExternalData/7deaa4d5dacd9ad40baab8e8c7ce200f”, which you will need to rename.
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We study gauge theories and quantum gravity in a finite interval of time $ \tau $, on a compact space manifold $\Omega $. The initial, final and boundary conditions are formulated in gauge invariant
and general covariant ways by means of purely virtual extensions of the theories, which allow us to “trivialize” the local symmetries and switch to invariant fields (the invariant metric tensor,
invariant quark and gluon fields, etc.). The evolution operator $U(t_{\text{f}},t_{\text{i}})$ is worked out diagrammatically for arbitrary initial and final states, as well as boundary conditions on
$\partial \Omega $, and shown to be well defined and unitary for arbitrary $\tau =t_{\text{f}}-t_{\text{i}}<\infty $. We illustrate the basic properties in Yang-Mills theory on the cylinder.
Phys. Rev. D 109 (2024) 025003 | DOI: 10.1103/PhysRevD.109.025003
We review the concept of purely virtual particle and its uses in quantum gravity, primordial cosmology and collider physics. The fake particle, or “fakeon”, which mediates interactions without
appearing among the incoming and outgoing states, can be introduced by means of a new diagrammatics. The renormalization coincides with one of the parent Euclidean diagrammatics, while unitarity
follows from spectral optical identities, which can be derived by means of algebraic operations. The classical limit of a theory of physical particles and fakeons is described by an ordinary
Lagrangian plus Hermitian, micro acausal and micro nonlocal self-interactions. Quantum gravity propagates the graviton, a massive scalar field (the inflaton) and a massive spin-2 fakeon, and leads to
a constrained primordial cosmology, which predicts the tensor-to-scalar ratio r in the window 0.4≲1000r≲3.5. The interpretation of inflation as a cosmic RG flow allows us to calculate the
perturbation spectra to high orders in the presence of the Weyl squared term. In models of new physics beyond the standard model, fakeons evade various phenomenological bounds, because they are less
constrained than normal particles. The resummation of self-energies reveals that it is impossible to get too close to the fakeon peak. The related peak uncertainty, equal to the fakeon width divided
by 2, is expected to be observable.
Symmetry 2022, 14(3), 521 | DOI: 10.3390/sym14030521
We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can
be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the
other hand, all the states are included, the LW models have a Hamiltonian unbounded from below or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and
are absent from the sets of incoming and outgoing states without implications on the observation of long-lived unstable particles. We give a vademecum to summarize the properties of most options to
treat abnormal particles. We study a method to remove the LW ghosts only partially, by saving the physical particles they contain. Specifically, we replace a LW ghost with a certain superposition of
a purely virtual particle and an ordinary particle, and drop only the former from the sets of the external states. The trick can be used to make the Pauli-Villars fields consistent and observable,
without sending their masses to infinity, or to build a finite QED, by tweaking the original Lee-Wick construction. However, it has issues with general covariance, so it cannot be applied as is to
quantum gravity, where a manifestly covariant decomposition requires the introduction of a massive spin-2 multiplet.
Phys. Rev. D 105 (2022) 125017 | DOI: 10.1103/PhysRevD.105.125017
We study the resummation of self-energy diagrams into dressed propagators in the case of purely virtual particles and compare the results with those obtained for physical particles and ghosts. The
three geometric series differ by infinitely many contact terms, which do not admit well-defined sums. The peak region, which is outside the convergence domain, can only be reached in the case of
physical particles, thanks to analyticity. In the other cases, nonperturbative effects become important. To clarify the matter, we introduce the energy resolution $\Delta E$ around the peak and argue
that a “peak uncertainty” $\Delta E\gtrsim \Delta E_{\text{min}}\simeq \Gamma _{\text{f}}/2$ around energies $E\simeq m_{\text{f}}$ expresses the impossibility to approach the fakeon too closely, $m_
{\text{f}}$ being the fakeon mass and $\Gamma _{\text{f}}$ being the fakeon width. The introduction of $\Delta E$ is also crucial to explain the observation of unstable long-lived particles, like the
muon. Indeed, by the common energy-time uncertainty relation, such particles are also affected by ill-defined sums at $\Delta E=0$, whenever we separate their observation from the observation of
their decay products. We study the regime of large $\Gamma _{\text{f}}$, which applies to collider physics (and situations like the one of the $Z$ boson), and the regime of small $\Gamma _{\text{f}}
$, which applies to quantum gravity (and situations like the one of the muon).
J. High Energy Phys. 06 (2022) 058 | DOI: 10.1007/JHEP06(2022)058
We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman $i\epsilon $ prescription) and purely virtual particles (defined by the fakeon
prescription). The identities are derived by means of purely algebraic operations and hold for every (multi)threshold separately and for arbitrary frequencies. Their major significance is that they
offer a deeper understanding on the problem of unitarity in quantum field theory. In particular, they apply to “skeleton” diagrams, before integrating on the space components of the loop momenta and
the phase spaces. In turn, the skeleton diagrams obey a spectral optical theorem, which gives the usual optical theorem for amplitudes, once the integrals on the space components of the loop momenta
and the phase spaces are restored. The fakeon
prescription/projection is implemented by dropping the thresholds that involve fakeon frequencies. We give examples at one loop (bubble, triangle, box, pentagon and hexagon), two loops (triangle with
“diagonal”, box with diagonal) and arbitrarily many loops. We also derive formulas for the loop integrals with fakeons and relate them to the known formulas for the loop integrals with physical
J. High Energy Phys. 11 (2021) 030 | DOI: https://doi.org/10.1007/JHEP11(2021)030
Extensions to the Standard Model that use strictly off-shell degrees of freedom – the fakeons – allow for new measurable interactions at energy scales usually precluded by the constraints that target
the on-shell propagation of new particles. Here we employ the interactions between a new fake scalar doublet and the muon to explain the recent Fermilab measurement of its anomalous magnetic moment.
Remarkably, unlike in the case of usual particles, the experimental result can be matched for fakeon masses below the electroweak scale without contradicting the stringent precision data and collider
bounds on new light degrees of freedom. Our analysis, therefore, demonstrates that the fakeon approach offers unexpected viable possibilities to model new physics naturally at low scales.
Phys. Rev. D 104 (2021) 035009 | DOI: 10.1103/PhysRevD.104.035009
We introduce a new way of modeling the physics beyond the Standard Model by considering fake, strictly off-shell degrees of freedom: the fakeons. To demonstrate the approach and exemplify its reach,
we re-analyze the phenomenology of the Inert Doublet Model under the assumption that the second doublet is a fakeon. Remarkably, the fake doublet avoids the most stringent $Z$-pole constraints
regardless of the chosen mass scale, thereby allowing for the presence of new effects well below the electroweak scale. Furthermore, the absence of on-shell propagation prevents fakeons from inducing
missing energy signatures in collider experiments. The distinguishing features of the model appear at the loop level, where fakeons modify the Higgs boson $h\rightarrow\gamma\gamma$ decay width and
the Higgs trilinear coupling. The running of Standard Model parameters proceeds as in the usual Inert Doublet Model case. Therefore, the fake doublet can also ensure the stability of the Standard
Model vacuum. Our work shows that fakeons are a valid alternative to the usual tools of particle physics model building, with the potential to shape a new paradigm, where the significance of the
existing experimental constraints towards new physics must necessarily be reconsidered.
J. High Energy Phys. 10 (2021) 132 | DOI: 10.1007/JHEP10(2021)132
We study primordial cosmology with two scalar fields that participate in inflation at the same time, by coupling quantum gravity (i.e., the theory $R+R^2+C^2$ with the fakeon prescription/projection
for $C^2$) to a scalar field with a quadratic potential. We show that there exists a perturbative regime that can be described by an asymptotically de Sitter, cosmic RG flow in two couplings. Since
the two scalar degrees of freedom mix in nontrivial ways, the adiabatic and isocurvature perturbations are not RG invariant on superhorizon scales. It is possible to identify the correct
perturbations by using RG invariance as a guiding principle. We work out the resulting power spectra of the tensor and scalar perturbations to the NNLL and NLL orders, respectively. An unexpected
consequence of RG invariance is that the theory remains predictive. Indeed, the scalar mixing affects only the subleading corrections, so the predictions of quantum gravity with single-field
inflation are confirmed to the leading order.
J. Cosmol. Astropart. Phys. 07 (2021) 037 | DOI: 10.1088/1475-7516/2021/07/037
We study inflation as a “cosmic” renormalization-group flow. The flow, which encodes the dependence on the background metric, is described by a running coupling $\alpha $, which parametrizes the slow
roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of
flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in $\alpha $ and
violations of the relation $r+8n_{\text{t}}=0$ to the leading order. Various classes of potentials studied in the literature can be described by means of the RG approach, even when the action
includes a Weyl-squared term, while others are left out. In known cases, the classification helps identify the models that are ruled out by data. The RG approach is also able to generate spectra that
cannot be derived from standard Lagrangian formulations.
Class. Quantum Grav. 38 (2021) 225011 | DOI: 10.1088/1361-6382/ac2b07
We compute the inflationary perturbation spectra and the quantity $r+8n_{T}$ to the next-to-next-to-leading log order in quantum gravity with purely virtual particles (which means the theory $R+R^{2}
+C^{2}$ with the fakeon prescription/projection for $C^{2}$). The spectra are functions of the inflationary running coupling $\alpha (1/k)$ and satisfy the cosmic renormalization-group flow
equations, which determine the tilts and the running coefficients. The tensor fluctuations receive contributions from the spin-2 fakeon $\chi _{\mu \nu }$ at every order of the expansion in powers of
$\alpha \sim 1/115$. The dependence of the scalar spectrum on the $\chi
_{\mu \nu }$ mass $m_{\chi }$, on the other hand, starts from the $\alpha^{2}$ corrections, which are handled perturbatively in the ratio $m_{\phi}/m_{\chi }$, where $m_{\phi }$ is the inflaton mass.
The predictions have theoretical errors ranging from $\alpha ^{4}\sim 10^{-8}$ to $\alpha^{3}\sim 10^{-6}$. Nontrivial issues concerning the fakeon projection at higher orders are addressed.
J. Cosmol. Astropart. Phys. 02 (2021) 029 | DOI: 10.1088/1475-7516/2021/02/029
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Coda to idiot-post
Coda to
1. The posting idiot was a test case for the newly-fixed mechanism to email posts to people (some people had been getting this blog via email instead of going to the web.) It did not work. Nobody
knows why.
2. I had written:
computers have gotten VDW(4,2) more complicated.
One of the comments was:
For the "Ramsey-theory idiots" out there, the technical translation of VDW(4,2) is "I don't know exactly how much more complicated computers have gotten, but its a while hell of a lot!" :-)
The commenter is correct in clarifying what I meant; however, both the commentator and I are incorrect in the details. Inspired by the commenter, I looked up what is known about the VDW numbers.
VDW(4,2) is known and is only 35. VDW(5,2) is known, and is only 178. I should have written VDW(5,5) which is unknown but quite likely quite large.
3. VDW(k,c) is the least number W such that no matter how you c-color the elements {1,2,...,W} there will be k numbers equally spaced (e.g., 3,7,11,15 is 4 numbers equally spaced) that are the same
color. W(k,c) exists by van der Waerden's Theorem. See van der Waerden's Theorem-Wikipedia or van der Waerden's theorem-my posting in Luca's blog
4. I believe the only VDW numbers that are known are as follows: (see this paper) by Landman, Robertson, Culver from 2005.
1. VDW(3,2)=9, (easy)
2. VDW(3,3)=27, (Chvátal, 1970, math review entry, article not online.
3. VDW(3,4)=76, (Brown, Some new van der Warden numbers (prelim report), Notices of the AMS, Vol 21, (1974), A-432. Article, review not online!
4. VDW(4,2)=35, Chvátal ref above
5. VDW(5,2)=178, Stevens and Shantarum, 1978 full article!
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Stable Sample Compression Schemes: New Applications and an Optimal SVM Margin Bound
Stable Sample Compression Schemes: New Applications and an Optimal SVM Margin Bound
Proceedings of the 32nd International Conference on Algorithmic Learning Theory, PMLR 132:697-721, 2021.
We analyze a family of supervised learning algorithms based on sample compression schemes that are stable, in the sense that removing points from the training set which were not selected for the
compression set does not alter the resulting classifier. We use this technique to derive a variety of novel or improved data-dependent generalization bounds for several learning algorithms. In
particular, we prove a new margin bound for SVM, removing a log factor. The new bound is provably optimal. This resolves a long-standing open question about the PAC margin bounds achievable by SVM.
Cite this Paper
Related Material
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Convert Light hours (Distance)
Convert Light hours
Direct link to this calculator:
Convert Light hours (Distance)
1. Choose the right category from the selection list, in this case 'Distance'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and
π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Light hours'.
4. The value will then be converted into all units of measurement the calculator is familiar with.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
Utilize the full range of performance for this units calculator
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '813 Light hours'. In so doing, either the full name of the unit or
its abbreviation can be used Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Distance'. After that, it converts the entered
value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Regardless which of these possibilities one uses, it
saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it
gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(5 * 51) Light hours'. But
different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '12 Light hours + 58 Light hours' or '97mm x 44cm x 90dm = ?
cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 2.228 148 127 872 ×1019. For this form of presentation, the number
will be segmented into an exponent, here 19, and the actual number, here 2.228 148 127 872. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket
calculators, one also finds the way of writing numbers as 2.228 148 127 872 E+19. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at
this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 22 281 481 278 720 000 000. Independent of the presentation of the
results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
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Episode 174: Helping Students Understand Magnitude of Numbers – Rounding Numbers Series - Build Math Minds
Welcome fellow Recovering Traditionalists to Episode 174: Helping Students Understand Magnitude of Numbers – Rounding Numbers Series
I got a question sent in to me and because it deserves more than a short answer, I’m doing three episodes to answer it. Here is what was sent in: “Some of my kiddos have trouble telling me which
tens a number is between. For example, I’ll give them 53 and ask them to round it, but they can’t say it’s between either 50 or 60.”
This is actually very common and is rooted in the fact that kids are seeing numbers in isolation and haven’t built relationships around numbers. We can’t jump into rounding if students haven’t spent
time exploring how numbers relate to each other because that’s all that rounding really is.
Rounding is usually taught as a Place Value concept where you teach kids the procedure of looking at the digit in the specified place but as I discussed in the last episode, as Recovering
Traditionalists we instead should be helping students build their number sense and when students have number sense they naturally see what numbers are closest to for rounding.
Last week I gave you my first tip for helping your students with rounding and that was to do lots of work with number lines and number paths. If you missed that episode, go back and listen to it…it
was number 173.
My second tip for helping your students with rounding and being able to know what numbers a specific number is close to is to do lots of work helping them understand the magnitude of a number.
Now technically the magnitude of a number is that number’s distance to zero. However, we use that understanding to compare numbers. It is basically the property of numbers that we use when ordering
or ranking numbers.
Any time you are doing activities where students are comparing quantities you are helping them work on understanding the magnitude of numbers. Having students place numbers on number lines helps
solidify the idea that we are looking at how far away a number is from zero and the farther away it is, the larger its magnitude.
So that’s why my first tip was to have students get comfortable with using number lines and placing numbers on a number line because then they can use that visual of numbers on the number line to
help them do any activity that asks them to compare numbers.
One of my favorite activities is a routine called Close, Far, In-between. It not only focuses on magnitude, but also estimation and reasoning. Close, Far, In-between first appeared in the book
“Mathematics Their Way” by Mary Baratta-Lorton and I just love it for all the different types of questions you can ask students beyond just the typical “which amount is greater?” when comparing
Inside the Build Math Minds PD site we have a document with over 120 pre-made Close, Far, In-between routines. So I’m going to show you this one example from that document:
For those listening all you have to do for the routine is have three quantities, in this example we have 464, 319, and 557. This routine isn’t about having the students put them in order.
Typically, during the routine you just ask simple questions like “Which of these numbers are closest to each other?” “Which number is in-between the other two numbers?” “Which two numbers are the
farthest away from each other?” and those three questions are fabulous questions that will help your students estimate, reason, and build their understanding of the magnitude of the numbers.
But if you’d like you can extend this into even more areas. Here’s a screenshot of the notes that go with the slides inside the routine we created for our Build Math Minds members of just some of
the questions they could ask:
I know this routine isn’t actually doing any rounding, but remember we are focusing on helping students build a sense of numbers and how numbers relate to each other. Rounding is all about knowing
what ‘friendly numbers” a number is close to and when students are doing this routine you will hear them often comparing the numbers given to those friendly numbers. You will hear things like “557,
that’s close to 550 so….” Plus I highly suggest having students visualize these numbers on a number line (or even draw it) because the more work they have with number lines the better.
If you are a member of the Build Math Minds PD site, you have access to this Close, Far, In-between document and so much more to help you build these ideas with your students. If you aren’t a
member, go over to BuildMathMinds.com/BMM to join.
This routine isn’t just for whole numbers or even multi-digit numbers. We should be doing it with young students, like this example in our document that is showing finger patterns instead of
numerals. We should also be doing this as students are working on operations and as they move into decimals & fractions. As I said before, the document we have for our Build Math Minds PD site
members has over 120 pre-made slides that are for all ages and you can easily swap out numbers to use the slides over and over again with different numbers inside the circles.
If you aren’t a member, tomorrow just put three quantities on the board and start asking questions that dig into what is Close, Far, and In-between.
I hope this helped build your math mind so you can build the math minds of your students. See you next week.
Subscribe and Review in iTunes
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Mathematical exceptions to the rules or intuition
This article is a follow-up of Counterexamples on real sequences (part 2).
Let \((u_n)\) be a sequence of real numbers.
If \(u_{2n}-u_n \le \frac{1}{n}\) then \((u_n)\) converges?
This is wrong. The sequence
\[u_n=\begin{cases} 0 & \text{for } n \notin \{2^k \ ; \ k \in \mathbb N\}\\
1- 2^{-k} & \text{for } n= 2^k\end{cases}\]
is a counterexample. For \(n \gt 2\) and \(n \notin \{2^k \ ; \ k \in \mathbb N\}\) we also have \(2n \notin \{2^k \ ; \ k \in \mathbb N\}\), hence \(u_{2n}-u_n=0\). For \(n = 2^k\) \[
0 \le u_{2^{k+1}}-u_{2^k}=2^{-k}-2^{-k-1} \le 2^{-k} = \frac{1}{n}\] and \(\lim\limits_{k \to \infty} u_{2^k} = 1\). \((u_n)\) does not converge as \(0\) and \(1\) are limit points.
If \(\lim\limits_{n} \frac{u_{n+1}}{u_n} =1\) then \((u_n)\) has a finite or infinite limit?
This is not true. Let’s consider the sequence
\[u_n=2+\sin(\ln n)\] Using the inequality \(
\vert \sin p – \sin q \vert \le \vert p – q \vert\)
which is a consequence of the mean value theorem, we get \[
\vert u_{n+1} – u_n \vert = \vert \sin(\ln (n+1)) – \sin(\ln n) \vert \le \vert \ln(n+1) – \ln(n) \vert\] Therefore \(\lim\limits_n \left(u_{n+1}-u_n \right) =0\) as \(\lim\limits_n \left(\ln(n+1) –
\ln(n)\right) = 0\). And \(\lim\limits_{n} \frac{u_{n+1}}{u_n} =1\) because \(u_n \ge 1\) for all \(n \in \mathbb N\).
I now assert that the interval \([1,3]\) is the set of limit points of \((u_n)\). For the proof, it is sufficient to prove that \([-1,1]\) is the set of limit points of the sequence \(v_n=\sin(\ln n)
\). For \(y \in [-1,1]\), we can pickup \(x \in \mathbb R\) such that \(\sin x =y\). Let \(\epsilon > 0\) and \(M \in \mathbb N\) , we can find an integer \(N \ge M\) such that \(0 < \ln(n+1) - \ln
(n) \lt \epsilon\) for \(n \ge N\). Select \(k \in \mathbb N\) with \(x +2k\pi \gt \ln N\) and \(N_\epsilon\) with \(\ln N_\epsilon \in (x +2k\pi, x +2k\pi + \epsilon)\). This is possible as \((\ln
n)_{n \in \mathbb N}\) is an increasing sequence and the length of the interval \((x +2k\pi, x +2k\pi + \epsilon)\) is equal to \(\epsilon\). We finally get \[ \vert u_{N_\epsilon} - y \vert = \vert
\sin \left(\ln N_\epsilon \right) - \sin \left(x + 2k \pi \right) \vert \le \left(\ln N_\epsilon - (x +2k\pi)\right) \le \epsilon\] proving that \(y\) is a limit point of \((u_n)\).
A Commutative Ring with Infinitely Many Units
In a ring \(R\) a unit is any element \(u\) that has a multiplicative inverse \(v\), i.e. an element \(v\) such that \[
uv=vu=1,\] where \(1\) is the multiplicative identity.
The only units of the commutative ring \(\mathbb Z\) are \(-1\) and \(1\). For a field \(\mathbb F\) the units of the ring \(\mathrm M_n(\mathbb F)\) of the square matrices of dimension \(n \times n
\) is the general linear group \(\mathrm{GL}_n(\mathbb F)\) of the invertible matrices. The group \(\mathrm{GL}_n(\mathbb F)\) is infinite if \(\mathbb F\) is infinite, but the ring \(\mathrm M_n(\
mathbb F)\) is not commutative for \(n \ge 2\).
The commutative ring \(\mathbb Z[\sqrt{2}] = \{a + b\sqrt{2} \ ; \ (a,b) \in \mathbb Z^2\}\) is not a field. However it has infinitely many units.
\(a + b\sqrt{2}\) is a unit if and only if \(a^2-2b^2 = \pm 1\)
For \(u = a + b\sqrt{2} \in \mathbb Z[\sqrt{2}]\) we denote \(\mathrm N(u) = a^2- 2b^2 \in \mathbb Z\). For any \(u,v \in \mathbb Z[\sqrt{2}]\) we have \(\mathrm N(uv) = \mathrm N(u) \mathrm N(v)\).
Therefore for a unit \(u \in \mathbb Z[\sqrt{2}]\) with \(v\) as multiplicative inverse, we have \(\mathrm N(u) \mathrm N(v) = 1\) and \(\mathrm N(u) =a^2-2b^2 \in \{-1,1\}\).
The elements \((1+\sqrt{2})^n\) for \(n \in \mathbb N\) are unit elements
The proof is simple as for \(n \in \mathbb N\) \[
(1+\sqrt{2})^n (-1 + \sqrt{2})^n = \left((1+\sqrt{2})(-1 + \sqrt{2})\right)^n=1\]
One can prove (by induction on \(b\)) that the elements \((1+\sqrt{2})^n\) are the only units \(u \in \mathbb Z[\sqrt{2}]\) for \(u \gt 1\).
A strictly increasing continuous function that is differentiable at no point of a null set
We build in this article a strictly increasing continuous function \(f\) that is differentiable at no point of a null set \(E\). The null set \(E\) can be chosen arbitrarily. In particular it can
have the cardinality of the continuum like the Cantor null set.
A set of strictly increasing continuous functions
For \(p \lt q\) two real numbers, consider the function \[
f_{p,q}(x)=(q-p) \left[\frac{\pi}{2} + \arctan{\left(\frac{2x-p-q}{q-p}\right)}\right]\] \(f_{p,q}\) is positive and its derivative is \[
f_{p,q}^\prime(x) = \frac{2}{1+\left(\frac{2x-p-q}{q-p}\right)^2}\] which is always strictly positive. Hence \(f_{p,q}\) is strictly increasing. We also have \[
\lim\limits_{x \to -\infty} f_{p,q}(x) = 0 \text{ and } \lim\limits_{x \to \infty} f_{p,q}(x) = \pi(q-p).\] One can notice that for \(x \in (p,q)\), \(f_{p,q}^\prime(x) \gt 1\). Therefore for \(x, y
\in (p,q)\) distinct we have according to the mean value theorem \(\frac{f_{p,q}(y)-f_{p,q}(x)}{y-x} \ge 1\).
Covering \(E\) with an appropriate set of open intervals
As \(E\) is a null set, for each \(n \in \mathbb N\) one can find an open set \(O_n\) containing \(E\) and measuring less than \(2^{-n}\). \(O_n\) can be written as a countable union of disjoint open
intervals as any open subset of the reals. Then \(I=\bigcup_{m \in \mathbb N} O_m\) is also a countable union of open intervals \(I_n\) with \(n \in \mathbb N\). The sum of the lengths of the \(I_n\)
is less than \(1\). Continue reading A strictly increasing continuous function that is differentiable at no point of a null set
A monotonic function whose points of discontinuity form a dense set
Consider a compact interval \([a,b] \subset \mathbb R\) with \(a \lt b\). Let’s build an increasing function \(f : [a,b] \to \mathbb R\) whose points of discontinuity is an arbitrary dense subset \(D
= \{d_n \ ; \ n \in \mathbb N\}\) of \([a,b]\), for example \(D = \mathbb Q \cap [a,b]\).
Let \(\sum p_n\) be a convergent series of positive numbers whose sum is equal to \(p\) and define \(\displaystyle f(x) = \sum_{d_n \le x} p_n\).
\(f\) is strictly increasing
For \(a \le x \lt y \le b\) we have \[
f(y) – f(x) = \sum_{x \lt d_n \le y} p_n \gt 0\] as the \(p_n\) are positive and dense so it exists \(p_m \in (x, y]\).
\(f\) is right-continuous on \([a,b]\)
We pick-up \(x \in [a,b]\). For any \(\epsilon \gt 0\) is exists \(N \in \mathbb N\) such that \(0 \lt \sum_{n \gt N} p_n \lt \epsilon\). Let \(\delta > 0\) be so small that the interval \((x,x+\
delta)\) doesn’t contain any point in the finite set \(\{p_1, \dots, p_N\}\). Then \[
0 \lt f(y) – f(x) \le \sum_{n \gt N} p_n \lt \epsilon,\] for any \(y \in (x,x+\delta)\) proving the right-continuity of \(f\) at \(x\). Continue reading A monotonic function whose points of
discontinuity form a dense set
A normal extension of a normal extension may not be normal
An algebraic field extension \(K \subset L\) is said to be normal if every irreducible polynomial, either has no root in \(L\) or splits into linear factors in \(L\).
One can prove that if \(L\) is a normal extension of \(K\) and if \(E\) is an intermediate extension (i.e., \(K \subset E \subset L\)), then \(L\) is a normal extension of \(E\).
However a normal extension of a normal extension may not be normal and the extensions \(\mathbb Q \subset \mathbb Q(\sqrt{2}) \subset \mathbb Q(\sqrt[4]{2})\) provide a counterexample. Let’s prove
As a short lemma, we prove that a quadratic extension \(k \subset K\) , i.e. an extension of degree two is normal. Suppose that \(P\) is an irreducible polynomial of \(k[x]\) with a root \(a \in K\).
If \(a \in k\) then the degree of \(P\) is equal to \(1\) and we’re done. Otherwise \((1, a)\) is a basis of \(K\) over \(k\) and there exist \(\lambda, \mu \in k\) such that \(a^2 = \lambda a +\mu
\). As \(a \notin k\), \(Q(x)= x^2 – \lambda x -\mu\) is the minimal polynomial of \(a\) over \(k\). As \(P\) is supposed to be irreducible, we get \(Q = P\). And we can conclude as \[
Q(x) = (x-a)(x- \lambda +a).\]
The entensions \(\mathbb Q \subset \mathbb Q(\sqrt{2})\) and \(\mathbb Q(\sqrt{2}) \subset \mathbb Q(\sqrt[4]{2})\) are quadratic, hence normal according to previous lemma and \(\sqrt[4]{2}\) is a
root of the polynomial \(P(x)= x^4-2\) of \(\mathbb Q[x]\). According to Eisenstein’s criterion \(P\) is irreducible over \(\mathbb Q\). However \(\mathbb Q(\sqrt[4]{2}) \subset \mathbb R\) while the
roots of \(P\) are \(\pm \sqrt[4]{2}, \pm i \sqrt[4]{2}\) and therefore not all real. We can conclude that \(\mathbb Q \subset \mathbb Q(\sqrt[4]{2})\) is not normal.
The image of an ideal may not be an ideal
If \(\phi : A \to B\) is a ring homomorphism then the image of a subring \(S \subset A\) is a subring \(\phi(A) \subset B\). Is the image of an ideal under a ring homomorphism also an ideal? The
answer is negative. Let’s provide a simple counterexample.
Let’s take \(A=\mathbb Z\) the ring of the integers and for \(B\) the ring of the polynomials with integer coefficients \(\mathbb Z[x]\). The inclusion \(\phi : \mathbb Z \to \mathbb Z[x]\) is a ring
homorphism. The subset \(2 \mathbb Z \subset \mathbb Z\) of even integers is an ideal. However \(2 \mathbb Z\) is not an ideal of \(\mathbb Z[x]\) as for example \(2x \notin 2\mathbb Z\).
A function whose Maclaurin series converges only at zero
Let’s describe a real function \(f\) whose Maclaurin series converges only at zero. For \(n \ge 0\) we denote \(f_n(x)= e^{-n} \cos n^2x\) and \[
f(x) = \sum_{n=0}^\infty f_n(x)=\sum_{n=0}^\infty e^{-n} \cos n^2 x.\] For \(k \ge 0\), the \(k\)th-derivative of \(f_n\) is \[
f_n^{(k)}(x) = e^{-n} n^{2k} \cos \left(n^2 x + \frac{k \pi}{2}\right)\] and \[
\left\vert f_n^{(k)}(x) \right\vert \le e^{-n} n^{2k}\] for all \(x \in \mathbb R\). Therefore \(\displaystyle \sum_{n=0}^\infty f_n^{(k)}(x)\) is normally convergent and \(f\) is an indefinitely
differentiable function with \[
f^{(k)}(x) = \sum_{n=0}^\infty e^{-n} n^{2k} \cos \left(n^2 x + \frac{k \pi}{2}\right).\] Its Maclaurin series has only terms of even degree and the absolute value of the term of degree \(2k\) is \[
\left(\sum_{n=0}^\infty e^{-n} n^{4k}\right)\frac{x^{2k}}{(2k)!} > e^{-2k} (2k)^{4k}\frac{x^{2k}}{(2k)!} > \left(\frac{2kx}{e}\right)^{2k}.\] The right hand side of this inequality is greater than \
(1\) for \(k \ge \frac{e}{2x}\). This means that for any nonzero \(x\) the Maclaurin series for \(f\) diverges.
A group that is not a semi-direct product
Given a group \(G\) with identity element \(e\), a subgroup \(H\), and a normal subgroup \(N \trianglelefteq G\); then we say that \(G\) is the semi-direct product of \(N\) and \(H\) (written \(G=N \
rtimes H\)) if \(G\) is the product of subgroups, \(G = NH\) where the subgroups have trivial intersection \(N \cap H= \{e\}\).
Semi-direct products of groups provide examples of non abelian groups. For example the dihedral group \(D_{2n}\) with \(2n\) elements is isomorphic to a semidirect product of the cyclic groups \(\
mathbb Z_n\) and \(\mathbb Z_2\). \(D_{2n}\) is the group of isometries preserving a regular polygon \(X\) with \(n\) edges.
Let’see that the converse is not true and present a group that is not a semi-direct product.
The Hamilton’s quaternions group is not a semi-direct product
The Hamilton’s quaternions group \(\mathbb H_8\) is the group consisting of the symbols \(\pm 1, \pm i, \pm j, \pm k\) where\[
-1 = i^2 =j^2 = k^2 \text{ and } ij = k = -ji,jk = i = -kj, ki = j = -ik.\] One can prove that \(\mathbb H_8\) endowed with the product operation above is indeed a group having \(8\) elements where \
(1\) is the identity element.
\(\mathbb H_8\) is not abelian as \(ij = k \neq -k = ji\).
Let’s prove that \(\mathbb H_8\) is not the semi-direct product of two subgroups. If that was the case, there would exist a normal subgroup \(N\) and a subgroup \(H\) such that \(G=N \rtimes H\).
• If \(\vert N \vert = 4\) then \(H = \{1,h\}\) where \(h\) is an element of order \(2\) in \(\mathbb H_8\). Therefore \(h=-1\) which is the only element of order \(2\). But \(-1 \in N\) as \(-1\)
is the square of all elements in \(\mathbb H_8 \setminus \{\pm 1\}\). We get the contradiction \(N \cap H \neq \{1\}\).
• If \(\vert N \vert = 2\) then \(\vert H \vert = 4\) and \(H\) is also normal in \(G\). Noting \(N=\{1,n\}\) we have for \(h \in H\) \(h^{-1}nh=n\) and therefore \(nh=hn\). This proves that the
product \(G=NH\) is direct. Also \(N\) is abelian as a cyclic group of order \(2\). \(H\) is also cyclic as all groups of order \(p^2\) with \(p\) prime are abelian. Finally \(G\) would be
abelian, again a contradiction.
We can conclude that \(G\) is not a semi-direct product.
Painter’s paradox
Can you paint a surface with infinite area with a finite quantity of paint? For sure… let’s do it!
Consider the 3D surface given in cylindrical coordinates as \[
x &= \rho \cos \varphi\\
y &= \rho \sin \varphi\\
z &= \frac{1}{\rho}\end{cases}\] for \((\rho,\varphi) \in [1,\infty) \times [0, 2 \pi)\). The surface is named Gabriel’s horn.
Volume of Gabriel’s horn
The volume of Gabriel’s horn is \[
V = \pi \int_1^\infty \left( \frac{1}{\rho^2} \right) \ d\rho = \pi\] which is finite.
Area of Gabriel’s horn
The area of Gabriel’s horn for \((\rho,\varphi) \in [1,a) \times [0, 2 \pi)\) with \(a > 1\) is: \[
A = 2 \pi \int_1^a \frac{1}{\rho} \sqrt{1+\left( -\frac{1}{\rho^2} \right)^2} \ d\rho \ge 2 \pi \int_1^a \frac{d \rho}{\rho} = 2 \pi \log a.\] As the right hand side of inequality above diverges to \
(\infty\) as \(a \to \infty\), we can conclude that the area of Gabriel’s horn is infinite.
Gabriel’s horn could be filled with a finite quantity of paint… therefore painting a surface with infinite area. Unfortunately the thickness of the paint coat is converging to \(0\) as \(z\) goes to
\(\infty\), leading to a paint which won’t be too visible!
A normal subgroup that is not a characteristic
Let’s \(G\) be a group. A characteristic subgroup is a subgroup \(H \subseteq G\) that is mapped to itself by every automorphism of \(G\).
An inner automorphism is an automorphism \(\varphi \in \mathrm{Aut}(G)\) defined by a formula \(\varphi : x \mapsto a^{-1}xa\) where \(a\) is an element of \(G\). An automorphism of a group which is
not inner is called an outer automorphism. And a subgroup \(H \subseteq G\) that is mapped to itself by every inner automorphism of \(G\) is called a normal subgroup.
Obviously a characteristic subgroup is a normal subgroup. The converse is not true as we’ll see below.
Example of a direct product
Let \(K\) be a nontrivial group. Then consider the group \(G = K \times K\). The subgroups \(K_1=\{e\} \times K\) and \(K_2=K \times \{e\} \) are both normal in \(G\) as for \((e, k) \in K_1\) and \
((a,b) \in G\) we have
\[(a,b)^{-1} (e,x) (a,b) = (a^{-1},b^{-1}) (e,x) (a,b) = (e,b^{-1}xb) \in K_1\] and \(b^{-1}K_1 b = K_1\). Similar relations hold for \(K_2\). As \(K\) is supposed to be nontrivial, we have \(K_1 \
neq K_2\).
The exchange automorphism \(\psi : (x,y) \mapsto (y,x)\) exchanges the subgroup \(K_1\) and \(K_2\). Thus, neither \(K_1\) nor \(K_2\) is invariant under all the automorphisms, so neither is
characteristic. Therefore, \(K_1\) and \(K_2\) are both normal subgroups of \(G\) that are not characteristic.
When \(K = \mathbb Z_2\) is the cyclic group of order two, \(G = \mathbb Z_2 \times \mathbb Z_2\) is the Klein four-group. In particular, this gives a counterexample where the ambient group is an
abelian group.
Example on the additive group \(\mathbb Q\)
Consider the additive group \((\mathbb Q,+)\) of rational numbers. The map \(\varphi : x \mapsto x/2\) is an automorphism. As \((\mathbb Q,+)\) is abelian, all subgroups are normal. However, the
subgroup \(\mathbb Z\) is not sent into itself by \(\varphi\) as \(\varphi(1) = 1/ 2 \notin \mathbb Z\). Hence \(\mathbb Z\) is not a characteristic subgroup.
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Trivia Quiz: Test Your Knowledge About Banking And Finance!
Questions and Answers
Will you Test Your Knowledge about Banking and Finance? Banking and finance is a very important sector that requires a lot of precision and documentations where a small error would lead to a lot of
miscalculations in other different sectors. The quiz below is for those of you willing to test their understanding on the two sectors. Give it a try!
• 1.
There is no penalty if I pay my credit card balance after the due date.
Correct Answer
B. False
Paying the credit card balance after the due date can result in penalties. Credit card companies typically charge late fees for payments made after the due date. Additionally, late payments can
negatively impact your credit score, making it more difficult to obtain credit in the future. It is important to pay credit card balances on time to avoid these penalties and maintain a good
credit history.
• 2.
Saving is the same as investing.
Correct Answer
B. False
Saving and investing are not the same thing. Saving refers to setting aside money for future use, typically in a low-risk account such as a savings account. On the other hand, investing involves
using money to purchase assets or securities with the expectation of earning a return, typically over a longer period of time and with a higher level of risk. While both saving and investing
involve setting aside money, they differ in terms of the purpose and potential returns. Therefore, the statement "Saving is the same as investing" is false.
• 3.
The BEST way to manage credit card debt is...
□ A.
To consistently pay a small part of the amount owed every month
□ B.
To not pay every month but pay lump sum using my bonus at the end of the year
□ C.
To pay the full amount owed for that month
□ D.
To take up a personal bank loan to pay the credit card debt in full
Correct Answer
C. To pay the full amount owed for that month
Paying the full amount owed for that month is the best way to manage credit card debt because it helps to avoid accumulating interest charges and potential late fees. By paying the full amount,
you are effectively reducing your debt and preventing it from growing further. This approach also promotes responsible financial habits and helps to maintain a good credit score.
• 4.
It’s a bad idea to start investing when you’re young.
Correct Answer
B. False
Starting investing when you're young is actually a good idea. The earlier you start investing, the more time your money has to grow and compound. By starting early, you can take advantage of the
power of compounding and potentially earn higher returns over the long term. Additionally, starting young allows you to develop good financial habits and learn from any mistakes or losses early
on, setting a solid foundation for your future financial security.
• 5.
Investing involves...
□ A.
□ B.
□ C.
□ D.
□ E.
Correct Answer
E. All of the above
Investing involves a variety of options, including stocks, bonds, T-Bills, and GICs. Stocks represent ownership in a company and offer potential for high returns but also carry higher risk. Bonds
are fixed-income securities issued by governments or corporations, providing a fixed return over a specified period. T-Bills are short-term debt instruments issued by the government, considered
low-risk investments. GICs (Guaranteed Investment Certificates) are fixed-term deposits with guaranteed returns. Therefore, the correct answer is "All of the above" as investing encompasses all
these options.
• 6.
Credit Score is...
□ A.
Your score on an online quiz
□ B.
A score given to an individual by banks based on their financial activities
□ C.
□ D.
Correct Answer
B. A score given to an individual by banks based on their financial activities
Credit Score is a numerical representation of an individual's creditworthiness, which is determined by banks and financial institutions based on their financial activities. It takes into account
factors such as payment history, credit utilization, length of credit history, types of credit, and new credit inquiries. This score helps lenders assess the risk of lending money to an
individual and plays a crucial role in determining loan eligibility and interest rates. It is not related to an online quiz, an imaginary credit card, or similar to interest.
• 7.
The best bank to open an account with is...
□ A.
□ B.
□ C.
□ D.
□ E.
The one that suits your needs
Correct Answer
E. The one that suits your needs
The answer "The one that suits your needs" is the best choice because the best bank to open an account with varies depending on individual preferences and requirements. Each person has different
financial goals, banking needs, and priorities. Therefore, it is crucial to choose a bank that aligns with one's specific needs, whether it be in terms of services offered, fees, accessibility,
customer service, or any other factors that are important to the individual.
• 8.
Find the interest if the principal is $1000, the rate is 5% and the time is 2 years. ( I = PRT )
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. $100
The correct answer is $100 because the formula for calculating simple interest is I = PRT, where I is the interest, P is the principal, R is the rate, and T is the time. Plugging in the given
values, we get I = 1000 * 0.05 * 2 = 100. Therefore, the interest is $100.
• 9.
Find the interest if the principal is $500 the rate is 10% and the time is 3 years. ( I = PRT )
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. 150
The formula for calculating interest is I = PRT, where I is the interest, P is the principal, R is the rate, and T is the time. In this case, the principal is $500, the rate is 10%, and the time
is 3 years. Plugging these values into the formula, we get I = 500 * 0.10 * 3 = $150. Therefore, the interest is $150.
• 10.
Find the average of: 10, 12, 13, 13, 14
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. 12.4
The average of a set of numbers is found by adding up all the numbers in the set and then dividing the sum by the total number of values. In this case, the sum of 10, 12, 13, 13, and 14 is 62.
Dividing 62 by 5 (the total number of values) gives us 12.4. Therefore, the average of the given numbers is 12.4.
• 11.
Find the percent of change: 20 to 24
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 20%
The percent of change from 20 to 24 can be calculated by finding the difference between the two numbers (24 - 20 = 4) and then dividing it by the original number (4/20 = 0.2). Finally,
multiplying the result by 100 gives the percentage (0.2 * 100 = 20%). Therefore, the correct answer is 20%.
• 12.
Find the percent of change: 132 to 160.
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. 21.2%
The percent of change can be found by taking the difference between the new value and the old value, dividing it by the old value, and then multiplying by 100. In this case, the difference
between 160 and 132 is 28. Dividing 28 by 132 gives approximately 0.2121. Multiplying by 100 gives 21.21%, which is rounded to 21.2%. Therefore, the correct answer is 21.2%.
• 13.
John makes $10 an hour. He works 32 hours this week. Find his total pay
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 320
John makes $10 an hour and works 32 hours this week. To find his total pay, we can multiply his hourly rate by the number of hours worked: $10 x 32 = $320. Therefore, his total pay is $320.
• 14.
Mark makes $10 an hour and time and a half for all hours over 40. This week he worked 42 hours. Find his total pay.
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. 430
Mark makes $10 an hour and time and a half for all hours over 40. This means that for the first 40 hours, he will earn $10 per hour, and for the additional 2 hours, he will earn $10 + ($10 * 1.5)
= $10 + $15 = $25 per hour. Therefore, his total pay for the 42 hours worked would be 40 * $10 + 2 * $25 = $400 + $50 = $450. However, since the question asks for his total pay, the correct
answer would be $430.
• 15.
Which is the better buy? 12 apples for $5 or 14 apples for $6
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 12 for 5
The better buy is 12 apples for $5 because it is a better price per apple compared to 14 apples for $6. With 12 apples for $5, each apple costs approximately $0.42, while with 14 apples for $6,
each apple costs approximately $0.43. Therefore, you get more value for your money by purchasing 12 apples for $5.
• 16.
Mike makes $500 every week. How much does he make in a year?
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. 26,000
Mike makes $500 every week. To find out how much he makes in a year, we need to multiply his weekly earnings by the number of weeks in a year. Since there are 52 weeks in a year, we can calculate
his annual earnings by multiplying $500 by 52, which equals $26,000. Therefore, the correct answer is 26,000.
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If you’re deploying a static site to Cloudfront via CDK, you might be using the BucketDeployment construct to combine shipping a folder to S3 and causing a Cloudfront invalidation.
Behind the scenes, BucketDeployment creates a custom resource, a Lambda, that wraps a call to the AWS SDK’s s3 cp command to move files from the CDK staging area to the target S3 bucket.
While that’s happening within AWS’s infrastructure, the speed of that copy depends very strongly on the amount of resources the Lambda has – just like any other Lambda, CPU and network bandwidth
scale with the requested memory limit.
The default memory limit for the custom resource Lambda is 128MiB – which is the smallest Lambda you can get, and accordingly the performance of that copy might be terrible if you have a lot of
files, or large files, to transfer.
I’d strongly recommend upping that limit to 2048MiB or higher. This radically improved upload performance on two applications I deploy, with the upload rate going from @=~700KiB/s to >10MiB/s – a 10x
This has a negligible cost implication as this Lambda only runs during a deployment, so shouldn’t be running all too frequently anyway. However the performance improvement is potentially dramatic for
complex apps. We saw one build go from ~280s uploading to S3 come down to ~45s – an 84% reduction in that deployment step’s execution time, and about a 15% reduction in the deployment time of that
stack overall – just for changing one parameter.
Bucket named ‘cdk-abcd12efg-assets-123456789-eu-west-1’ exists, but not in account 123456789. Wrong account?
When deploying a stack via CDK, you may encounter an error such as
Bucket named 'cdk-abcd12efg-assets-123456789-eu-west-1' exists, but not in account ***. Wrong account?
The most likely culprit here is that the role you’re using to deploy doesn’t have the right permissions on the staging bucket. CDK requires:
• getBucketLocation
• *Object
• ListBucket
We hit this recently, and the underlying cause was that the IAM role used to deploy the stack had been amended to have a restricted set of permissions per least-privilege best practice. We’d deployed
updates to the stack a number of times, but in this instance the particular change we were making required a re-upload of assets to the staging bucket, which uncovered the missing permission.
Cognito error: “Cannot use IP Address passed into UserContextData”
When using Cognito’s Advanced Security and adaptive authentication features, you need to ship contextual data about the logging-in user via the UserContextData type.
Some of this type data is collected via a Javascript snippet. However, you can also ship the user’s IP address (which the snippet cannot collect) in the same payload.
When doing so, you may get an error from Cognito:
“Cannot use IP Address passed into UserContextData”
Unhelpful error from Cognito
This is likely because you’ve not enabled ‘Accept additional user context data‘ on your user pool client – though the error message is pretty opaque.
You can do this in a number of ways:
• Via the AWS console
• Via the UpdateUserPoolClient CLI function
• Via CDK, if you drop down to the Level 1 construct and set “enablePropagateAdditionalUserContextData: true” on your CfnUserPoolClient
Even the latest L2 constructs for Cognito don’t seem to support setting enablePropagateAdditionalUserContextData when controlling a user pool client via CDK, but using the L1 escape hatch is easy
const cfnUserPoolClient = userPoolClient.node.defaultChild as CfnUserPoolClient;
cfnUserPoolClient.enablePropagateAdditionalUserContextData = true;
GitHub Actions, ternary operators and default values
Github Actions ‘type’ metadata on custom action or workflow inputs is, pretty much, just documentation – it doesn’t seem to be enforced, at least when it comes to supplying a default value. That
means that just because you’ve claimed it’s a bool doesn’t make it so.
And worse, it seems that default values get coerced to strings if you use an expression.
At TILLIT we have custom GitHub composite actions to perform various tasks during CI. We recently hit a snag with one roughly structured as follows
name: ...
type: boolean
default: ${{ some logic here }}
using: "composite"
- name: ...
uses: ...
some-property: ...${{ inputs.readonly && 'true-val' || 'false-val' }}...
That mess in the some-property definition is the closest you can get in Github Actions to a ternary operator in the absence of any if-like construct, where you want to format a string based on some
In our case – the ‘true’ path was the only path ever taken. Diagnostic logging on the action showed that inputs.readonly was ‘false’. Wait, are those quotes?
Of course they are! The default value ended up being set to be a string, even though the input’s default value expression is purely boolean in nature and it’s specified as being a boolean.
The fix then is to our ternary, and to be very explicit as to the comparison being made.
some-property: ...${{ inputs.readonly == 'true' && 'true-val' || 'false-val'
AWS SAM error “[ERROR] (rapid) Init failed error=Runtime exited with error: signal: killed InvokeID=” in VS Code
When debugging a lambda using the AWS Serverless Application Model tooling (the CLI and probably VS Code extensions), you might find that your breakpoint isn’t getting hit and you instead see an
error in the debug console:
[ERROR] (rapid) Init failed error=Runtime exited with error: signal: killed InvokeID=" in VS Code
A thing to check is whether you’re running out of RAM or timing out in execution:
• Open your launch.json file for the workspace
• In your configuration, under the lambda section, add a specific memoryMb value – in my case 512 got me moving
This is incredibly frustrating because the debug console gives you no indication as to why the emulator terminated your lambda – but also helpful, because you can tell how large you need to specify
your lambda when you deploy it ahead of time.
Invalid Request error when creating a Cloudfront response header policy via Cloudformation
I love Cloudformation and CDK, but sometimes neither will show an issue with your template until you actually try to deploy it.
Recently we hit a stumbling block while creating a Cloudfront response header policy for a distribution using CDK. The cdk diff came out looking correct, no issues there – but on deploying we hit an
Invalid Request error for the stack.
Cloudformation often doesn’t give much additional colour when you hit a stumbling block
The reason? We’d added a temporarily-disabled XSS protection header, but kept in the reporting URL so that when we turned it on it’d be correctly configured. However, Cloudfront rejects the creation
of the policy if you spec a reporting URL on a disabled header setup.
The Cloudfront resource policy docs make it pretty clear this isn’t supported, but Cloudformation can’t validate it for us
Just jumping into the console to try creating the resource by hand is often the most effective debugging technique
How to diagnose Invalid Request errors with Cloudformation
A lot of the time the easiest way to diagnose a Invalid Request error when deploying a Cloudformation is to just do it by hand in the console in a test account, and see what breaks. In this instance,
the error was very clear and it was a trivial patch to fix up the Cloudformation template and get ourselves moving.
Unfortunately, Cloudformation often doesn’t give as much context as the console when it comes to validation errors during stack creation – but hand-cranking the affected resource both gives you
quicker feedback and a better feel for what the configuration options are and how they hang together.
A rule of thumb is that if you’re getting an Invalid Request back, chances are it’s essentially a validation error on what you’ve asked Cloudformation to deploy. Check the docs, simplify your test
case to pinpoint the issue and don’t be afraid to get your hands dirty in the console.
DMARC failures even when AWS SES Custom Mail-From domain used
I was caught out by this, this week, so hopefully future-me will remember quicker how to fix this one.
• You want to get properly configured for DMARC for a domain you’re sending emails from via AWS SES
• You’ve verified the sender domain as an identity
• You’ve set up DKIM and SPF
• You’ve set up a custom MAIL FROM
• You’re still seeing SPF-related DMARC failures when sending emails
In my case, those failures were caused because I was sending email from a different identity that uses the same domain.
For example, I had ‘example.com’ set up as a verified identity in SES allowing me to send email from any address at that domain, and I configured a sender identity ‘contact@example.com’ to be used by
my application to send emails so that I could construct an ARN for use with Cognito or similar.
What isn’t necessarily obvious is that you need to enable the custom MAIL FROM setting for the sender identity, and not just for the domain identity that you’ve configured assuming you have multiple.
AWS SES does not fall back to the configuration for the domain identity and you have to individually enable custom MAIL FROM for each sender identity – even if the configuration is identical.
So in my case, the fix was:
• Edit the Custom MAIL FROM setting for contact@example.com
• Enable it to use mail.example.com (which was already configured)
• Save settings
Using an AWS role to authenticate with Google Cloud APIs
I recently had a requirement to securely access a couple of Google Cloud APIs as a service account user, where those calls were being made from a Fargate task running on AWS. The
until-relatively-recently way to do this was:
• Create a service account in the Google Cloud developer console
• Assign it whatever permissions it needs
• Create a ‘key’ for the account – in essence a long-lived private key used to authenticate as that service account
• Use that key in your Cloud SDK calls from your AWS Fargate instance
This isn’t ideal, because of that long-lived credential in the form of the ‘key’ – it can’t be scoped to require a particular originator and while you can revoke it from the developer console, if the
credential leaks you’ve got an infinitely long-lived token usable from anywhere – you’d need to know it had leaked to prevent its use.
Google’s Workload Identity Federation is the new hotness in that regard, and is supported by almost all of the client libraries now. Not the .NET one though, irritatingly, which is why this post from
Johannes Passing is, if you need to do this from .NET-land, absolutely the guide to go to.
The new approach is more in line with modern authentication standards and uses federation between AWS and Google Cloud to support generating short-lived, scoped credentials that are used for the
actual work and no secrets needing to be shared between the two environments.
The docs are broadly excellent, but I was pleased at how clever the AWS <-> Google Cloud integration is given that there isn’t any AWS-supported explicit identity federation actually happening, in
the sense of established protocols (like OIDC, which both clouds support in some fashion).
How it works
On the Google Cloud side, you set up a ‘Workload identity pool’ – in essence a collection of external identities that can be given some access to Google Cloud services. Aside from some basic
metadata, a pool has one or more ‘providers’ associated with it. A provider represents an external source of identities, for our example here AWS.
A provider can be parameterised:
• Mappings translate between the incoming assertions from the provider and those of Google Cloud’s IAM system
• Conditions restrict the identities that can use the identity pool via a rich syntax
You can also attach Google service accounts to the pool, allowing those accounts to be impersonated by identities in the pool. You can restrict access to a given service account via conditions, in a
very similar way to restricting access to the pool itself.
To get an access token on behalf of the service account, a few things are happening (in the background for most client libraries, and explicitly in the .NET case).
Authenticating with the pool
In AWS land, we authenticate with the Google pool by asking it to exchange a provider-issued token for one that Google’s STS will recognise. For AWS, the required token is (modulo some encoding and
formatting) a signed ‘GetCallerIdentity’ request that you might yourself send to the AWS STS.
Our calling code in AWS-land doesn’t finish the call – we don’t need to. Instead, we sign a request and then pass that signed request to Google which makes the call itself. We include in the request
(and the fields that are signed over) the URI of the ‘target resource’ on the Google side – the identity pool that we want to authenticate to.
The response from AWS to Google’s call to the STS will include the ARN of the identity for whom credentials on the AWS side are available. If you’re running in ECS or EC2, these will represent the
IAM role of the executing task.
We need share nothing secret with Google to do this, and we can’t fake an identity on AWS that we don’t have access to.
• The ARN of the identity returned in the response to GetCallerIdentity includes the AWS account ID and the name of any assumed role – the only thing we could ship to Google is proof of an identity
that we already have access to on the AWS side.
• The Google workflow identity pool identifier is signed over in the GetCallerIdentity request, so the token we send to Google can only be used for that specific user pool (and Google can verify
that, again with no secrets involved). This means we can’t accidentally ship a token to the wrong pool on the Google side.
• The signature can be verified without access to any secret information by just making the request to the AWS STS. If the signature is valid, Google will receive an identity ARN, and if the
payload has been tampered with or is otherwise invalid then the request will fail.
None of the above requires any cooperation between AWS and Google cloud, save for AWS not changing ARN formats and breaking identity pool conditions and mappings.
What happens next?
All being well, the Google STS returns to us a temporary access token that we can then use to generate a real, scoped access token to use with Google APIs. That token can be nice and short lived,
restricting the window over which it can be abused should it be leaked.
What about for long-lived processes?
Our tokens can expire in a couple of directions:
• Our AWS credentials can and will expire and get rolled over automatically by AWS (when not using explicit access key IDs and just using the profile we’re assuming from the execution role of the
• Our short-lived Google service account credential can expire
Both are fine and handled the same way – re-run the whole process. Signing a new GetCallerIdentity request is quick, trivial and happens locally on the source machine. And Google just has to make one
API call to establish that we’re still who we said we were and offer up a temporary token to exchange for a service account identity.
Creating a Route 53 Public Hosted Zone with a reusable delegation set ID in CDK
What’s a reuable delegation set anyway?
When you create a Route 53 public hosted zone, four DNS nameservers are allocated to the zone. You then use these name servers with your domain registrar to delegate DNS resolution to Route 53 for
your domain.
However: each time you re-create a Route 53 hosted zone, the DNS nameservers allocated will change. If you’re using CloudFormation to manage your public hosted zone this means a destroy and recreate
breaks your domain’s name resolution until you manually update your registrar’s records with the new combination of nameservers.
Route 53 reusable delegation sets are stable collections of Route 53 nameservers that you can create once and then reference when creating a public hosted zone. That zone will now have a fixed set of
nameservers, regardless of how often it’s destroyed and recreated.
Shame it’s not in CloudFormation
There’s a problem though. You can only create route 53 reusable delegation sets using the AWS CLI or the AWS API. There’s no CloudFormation resource that represents it (yet).
Worse, you can’t even reference an existing, manually-created delegation set using CloudFormation. Again, you can only do it by creating your public hosted zone using the CLI or API.
The AWS CloudFormation documentation makes reference to a ‘DelegationSetId’ element that doesn’t actually exist on the Route53::HostedZone resource. Nor is the element mentioned anywhere else in that
article or any SDK. I’ve opened a documentation bug for that. Hopefully its presence indicates that we’re getting an enhancement to the Route53::HostedZone resource some time soon…
So how can we achieve our goal of defining a Route 53 public hosted zone in code, while still letting it reference a delegation set ID?
Enter CDK and AwsCustomResource
CDK generates CloudFormation templates from code. I tend to use TypeScript when building CDK stacks. On the face of it, CDK doesn’t help us as if we can’t do something by hand-cranking some
CloudFormation, surely CDK can’t do it either.
Not so. CDK also exposes the AwsCustomResource construct that lets us call arbitrary AWS APIs as part of a CloudFormation deployment. It does this via some dynamic creation of Lambdas and other
trickery. The upshot is that if it’s in the JavaScript SDK, you can call it as part of a CDK stack with very little extra work.
Let’s assume that we have an existing delegation set whose ID we know, and we want to create a public hosted zone linked to that delegation set. Wouldn’t it be great to be able to write something
new PublicHostedZoneWithReusableDelegationSet(this, "PublicHostedZone", {
zoneName: `whatever.example.com`,
delegationSetId: "N05_more_alphanum_here_K"
// Probably pulled from CI/CD
Well we can! Again in TypeScript, and you’ll need to reference the @aws-cdk/custom-resources package:
import { IPublicHostedZone, PublicHostedZone, PublicHostedZoneProps } from "@aws-cdk/aws-route53";
import { Construct, Fn, Names } from "@aws-cdk/core";
import { PhysicalResourceId } from "@aws-cdk/custom-resources";
import { AwsCustomResource, AwsCustomResourcePolicy } from "@aws-cdk/custom-resources";
export interface PublicHostedZoneWithReusableDelegationSetProps extends PublicHostedZoneProps {
delegationSetId: string
export class PublicHostedZoneWithReusableDelegationSet extends Construct {
private publicHostedZone: AwsCustomResource;
private hostedZoneName: string;
constructor(scope: Construct, id: string, props: PublicHostedZoneWithReusableDelegationSetProps) {
super(scope, id);
this.hostedZoneName = props.zoneName;
const normaliseId = (id: string) => id.split("/").slice(-1)[0];
const normalisedDelegationSetId = normaliseId(props.delegationSetId);
this.publicHostedZone = new AwsCustomResource(this, "CreatePublicHostedZone", {
onCreate: {
service: "Route53",
action: "createHostedZone",
parameters: {
"CallerReference": Names.uniqueId(this),
"Name": this.hostedZoneName,
"DelegationSetId": normalisedDelegationSetId,
"HostedZoneConfig": {
"Comment": props.comment,
"PrivateZone": false
physicalResourceId: PhysicalResourceId.fromResponse("HostedZone.Id")
onUpdate: {
service: "Route53",
action: "getHostedZone",
parameters: {
Id: new PhysicalResourceIdReference()
physicalResourceId: PhysicalResourceId.fromResponse("HostedZone.Id")
onDelete: {
service: "Route53",
action: "deleteHostedZone",
parameters: {
"Id": new PhysicalResourceIdReference()
policy: AwsCustomResourcePolicy.fromSdkCalls({ resources: AwsCustomResourcePolicy.ANY_RESOURCE })
asPublicHostedZone() : IPublicHostedZone {
return PublicHostedZone.fromHostedZoneAttributes(this, "CreatedPublicHostedZone", {
hostedZoneId: Fn.select(2, Fn.split("/", this.publicHostedZone.getResponseField("HostedZone.Id"))),
zoneName: this.hostedZoneName
Note: thanks to Hugh Evans for patching a bug in this where the CallerReference wasn’t adequately unique to support a destroy and re-deploy
How does it work?
The tricky bits of the process are handled entirely by CDK – all we’re doing is telling CDK that when we create a ‘PublicHostedZoneWithReusableDelegationSet‘ construct, we want it to call the
Route53::createHostedZone API endpoint and supply the given DelegationSetId.
On creation we track the returned Id of the new hosted zone (which will be of the form ‘/hostedzone/the-hosted-zone-id’).
The above resource doesn’t support updates properly, but you can extend it as you wish. And the interface for PublicHostedZoneWithReusableDelegationSet is exactly the same as the standard
PublicHostedZone, just with an extra property to supply the DelegationSetId – you can just drop in the new type for the old when needed.
When you want to reference the newly created PublicHostedZone, there’s the asPublicHostedZone method which you can use in downstream constructs.
How to (not) do depth-first search in Neo4j
I found a Stack Overflow question with no answers that seemed like it should be straightforward – how can you traverse a tree-like structure in depth-first order. The problem had a couple of
• Each node had an order property that described the order in which sibling nodes should be traversed
• Each node was connected to its parent via a PART_OF relationship
A depth-first traversal of a tree is pretty easy to understand.
Whenever we find a node with children, we choose the first and explore as deep into the tree as we can until we can’t go any further. Next we step up one level and choose the next node we haven’t
explored yet and go as deep as we can on that one until we’ve traversed the graph.
Neo4j supports a depth-first traversal of a graph by way of the algo.dfs.stream function.
Given some tree-like graph where nodes of label ‘Node’ are linked by relationships of type :PART_OF:
// First, some test data to represent a tree with nodes connected by a
// 'PART_OF' relationship:
// N1 { order: 1 }
// N2 { order: 1 }
// N4 { order: 1 }
// N5 { order: 1 }
// N6 { order: 2 }
// N3 { order: 2 }
// N7 { order: 1 }
MERGE (n1: Node { order: 1, name: 'N1' })
MERGE (n2: Node { order: 1, name: 'N2' })
MERGE (n3: Node { order: 2, name: 'N3' })
MERGE (n4: Node { order: 1, name: 'N4' })
MERGE (n5: Node { order: 1, name: 'N5' })
MERGE (n6: Node { order: 2, name: 'N6' })
MERGE (n7: Node { order: 1, name: 'N7' })
MERGE (n2)-[:PART_OF]->(n1)
MERGE (n4)-[:PART_OF]->(n2)
MERGE (n5)-[:PART_OF]->(n4)
MERGE (n6)-[:PART_OF]->(n4)
MERGE (n3)-[:PART_OF]->(n1)
MERGE (n7)-[:PART_OF]->(n3)
We can see which nodes are visited by Neo4j’s DFS algorithm:
MATCH (startNode: Node { name: 'N1' } )
CALL algo.dfs.stream('Node', 'PART_OF', 'BOTH', id(startNode))
YIELD nodeIds
UNWIND nodeIds as nodeId
WITH algo.asNode(nodeId) as n
RETURN n
The outupt here will vary – possibly even between runs. While we’ll always see a valid depth-first traversal of the nodes in the tree, there’s no guarantee that we’ll always see nodes visited in the
same order. That’s because we’ve not told Neo4j in what order to traverse sibling nodes.
If you need control over the order siblings are expanded, you should use application code to write the DFS yourself.
But: given some constraints and accepting some caveats…
• That’s there’s only one relationship that links nodes in the tree
• That sibling nodes are sortable by some numeric property – here ‘order`, which is mandatory
• There are not more than 1,000,000 sibling nodes for any given parent
• Sibling nodes all have a distinct order property value
• That this will perform like a dog on large graphs – potentially not completing, given it has some N^2 characteristics…
…you can do this in pure Cypher. Here’s one approach, which we’ll then break down
to see how it works:
MATCH (root: Node { name: 'N1' }), pathFromRoot=shortestPath((root)<-[:PART_OF*]-(leaf: Node)) WHERE NOT ()-[:PART_OF]->(leaf)
WITH nodes(pathFromRoot) AS pathFromRootNodes
WITH pathFromRootNodes, reduce(pathString = "", pathElement IN pathFromRootNodes | pathString + '/' + right("00000" + toString(pathElement.order), 6)) AS orderPathString ORDER BY orderPathString
WITH reduce(concatPaths = [], p IN collect(pathFromRootNodes) | concatPaths + p) AS allPaths
WITH reduce(distinctNodes = [], n IN allPaths | CASE WHEN n IN distinctNodes THEN distinctNodes ELSE distinctNodes + n end) AS traversalOrder
RETURN [x in traversalOrder | x.name]
Finding the deepest traversals
Given some root node, we can find a list of traversals to each leaf node using shortestPath. A leaf node is a node with no children of its own, and shortestPath (so long as we’re looking at a tree)
will tell us the series of hops that get us from that leaf back to the root.
Sorting the paths
We’re trying to figure out the order in which these paths would be traversed, then extract the nodes from those paths to find the order in which nodes would be visited.
The magic is happening in this line:
WITH pathFromRootNodes, reduce(pathString = "", pathElement IN pathFromRootNodes | pathString + '/' + right("00000" + toString(pathElement.order), 6)) AS orderPathString ORDER BY orderPathString
The reduce is, given a node from root to leaf, building up a string that combines the order property of each node in the path with forward-slashes to separate them. This is much like folder paths in
a file system. To make this work, we need each segment of the path to be the same length – therefore we pad out the order property with zeroes to six digits, to get paths like:
These strings now naturally sort in a way that gives us a depth-first traversal of a graph using our order property. If we order by this path string we’ll get the order in which leaf nodes are
visited, and the path that took us to them.
Deduplicating nodes
The new problem is extracting the traversal from these paths. Since each path is a complete route from the root node to the leaf node, the same intermediate nodes will appear multiple times across
all those paths.
We need a way to look at each of those ordered paths and collect only new nodes – nodes we haven’t seen before – and return them. As we do this we’ll be building up the node traversal order that
matches a depth-first search.
WITH reduce(concatPaths = [], p IN collect(pathFromRootNodes) | concatPaths + p) AS allPaths
WITH reduce(distinctNodes = [], n IN allPaths | CASE WHEN n IN distinctNodes THEN distinctNodes ELSE distinctNodes + n end) AS traversalOrder
First we collect all the paths (which are now sorted by our traversal ordering) into one big list. The same nodes are going to appear more than once for the reasons above, so we need to remove them.
We can’t just DISTINCT the nodes, because there’s no guarantee that the ordering that we’ve worked hard to create will be maintained.
Instead, we use another reduce and iterate over the list of nodes, only adding a node to our list if we haven’t seen it before. Since the list is ordered, we take only the first of each duplicate and
ignore the rest. Our CASE statement is doing the heavy lifting here:
WITH reduce(distinctNodes = [], n IN allPaths | CASE WHEN n IN distinctNodes THEN distinctNodes ELSE distinctNodes + n end) AS traversalOrder
• Create a variable called distinctNodes and set it to be an empty list
• For each node n in our flattened list of nodes in each path from root to each leaf:
• If we’ve seen n before (if it’s in our ‘distinctNodes’ list) then set distinctNodes = distinctNodes – effectively a no-op
• If we haven’t seen n before, set distinctNodes = distinctNodes + n – adding it to the list
This is a horrendously inefficient operation – for a very broad, shallow tree (one where each node has many branches) we’ll be doing on the order of n^2 operations. Still, it’s only for fun.
We’re done! From our original graph, we’re expecting a traversal order of:
N1, N2, N4, N5, N6, N3, N7
And our query?
Another for the annals of ‘Just because you can, doesn’t mean you should’.
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• definite integral: bi'irsumji x1 is the integral of x2 over bounds/interval x3
• indefinite integral: bi'irsumji be fi zi'o
□ Note that "indefinite" has the same number of syllables as {be fi zi'o}
• farlaili'i: farna ke klani linji: vectorl1=k1=f2 is a vector of length k2 in direction f1(the lujvo list has farli'i as "vector, ray", but that really only means "ray")
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Boolean in Python | Prahlad Inala Blog
What is Boolean in Python?
Do you ever answer your friends like yes and no? That's it you know booleans, But here we will learn how to say that it is a python way. True and False with the capital 'T' and capital 'F' Boolean
values are used where we need to compare some expressions in python. We can compare the lists,tuples,sets using(==) and(!=)operators to get booleans.
# true
# false
# false as order matters in list.
# false as order matters in tuples.
# true as order not matters in sets.
Booleans can be converted into strings and numbers using built-in(str) and(int) functions. simple remember True=1 & False=0
Strings and Numbers can also be converted to booleans using the built-in (bool) function.
In numbers only (0) converts to false remaining all are True.
In strings only empty string('') converts to false remaining all are True.
Booleans can also be added and multiplied by numbers. when we do so the Python converts the boolean value into its number represents and then does the arithmetic.
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Plugin Used - Parametric House
In this grasshopper example file you can generate a recursive pyramid model using the anemone plugin.
In this grasshopper example file you can simulate the inflation of a balloon by using the kangaroo 2 plugin.
In this grasshopper example file you can model a recursive mesh by using the delaunay mesh component.
In this grasshopper example file you can model a minimal surface mesh between two circles by using the kangaroo plugin.
In this grasshopper example file you can model a parametric table by generating random points and voxelizing them by using the Dendro plugin.
In this grasshopper example file you can deform a box mesh by rotating it around a parametric axis.
In this grasshopper example file, you can create a series of pyramids from each faces of a mesh and convert it into a smooth star-like model.
In this grasshopper example file you can model a tensile tree structure by using the kangaroo plugin combined with Fatten plugin.
In this grasshopper example file, you can simulate a series of balloons expanding and filling a closed area (like a closed mesh).
In this grasshopper example file, you can model a 3D network of lines and add joints at their intersections.
In this grasshopper example file you can model the “Worley noise” on a mesh plane by defining random points.
In this grasshopper example file you can use the curl noise plugin to generate vectors from random points.
In this grasshopper example file by using the shortest walk plugin and a series of random points you can model an organic parametric table.
In this grasshopper example file you can model a series of weaving meshes.
In this grasshopper example file you can model a parametric shoe sole by using the grasshopper’s fields components.
In this grasshopper example file you can model a smooth fractal mesh using the Mesh+ plugin combined with weaverbird.
In this grasshopper example file you can model a relaxed mesh by using the kangaroo plugin and defining a simple twisted connection.
In this grasshopper example file you can model a twisting strip by defining rotating rectangles with a timer.
In this grasshopper definition, you can simulate the collision between particles and colliders by using the flexhopper plugin.
In this grasshopper example file you can create a fractal system similar to the growth of a tree.
In this grasshopper example file you can model an agent-base simulation to design a standing lamp by using the Physarealm and Dendro Plugin combined.
In this grasshopper example file you can simulate a spin force by defining a polar field.
In this grasshopper example file you can use the kangaroo plugin to simulate a balloon dog (an art work from Jeff Koons).
In this grasshopper definition you can create a smooth polar diagonal 3d pattern.
In this grasshopper definition you can divide a surface into self-similar cells and then create frames around them based on an attractor.
In this grasshopper example file we have used the Kangaroo plugin to deform a mesh and then combine the mesh+ plugin with weaverbird to model the final smooth mesh.
In this grasshopper definition you can model a ripple effect on a surface using the kangaroo plugin.
In this grasshopper definition you can simulate and model the Newton’s Cradle by using the Kangaroo plugin.
In this grasshopper definition you can model a parametric Radial wave by combining Paneling tools / Weaverbird / Mesh edit plugins.
In this Grasshopper example file you can model a random parametric facade pattern on a polysurface solid.
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4-Way 3-Position Directional Valve (G)
Controlled valve with four ports in a gas network
Simscape / Fluids / Gas / Valves & Orifices / Directional Control Valves
The 4-Way 3-Position Directional Valve (G) block represents a valve with four gas ports (P, A, B, and T) and flow paths between P–A and A–T and between P–B and B–T. The paths each run through an
orifice of variable width. The input signal specified at port S controls the position of the spool. The valve closes when the spool covers the orifice opening.
In a representative system, the P port connects to the pump, the T port connects to the tank, and the A and B ports connect to a double-sided actuator. Opening the P–A and B–T flow paths allows the
pump to pressurize one side of the actuator and the tank to relieve the other. The actuator shaft translates to extend in some systems and retract in others. Opening the P–B and A–T flow paths flips
the pressurized and relieved sides of the actuator so that the shaft can translate in reverse.
This image shows a common use for a four-way, three-position directional valve in a physical system:
The flow can be laminar or turbulent, and it can reach up to sonic speeds. The maximum velocity happens at the throat of the valve where the flow is narrowest and fastest. The flow chokes and the
velocity saturates when a drop in downstream pressure can no longer increase the velocity. Choking occurs when the back-pressure ratio reaches the critical value characteristic of the valve. The
block does not capture supersonic flow.
Valve Positions
The valve is continuously variable and it shifts smoothly between one normal and two working positions.
When the instantaneous displacement of the spool at port S is zero, the valve reverts to the normal position where it is no longer operating. Unless the lands of the spool are at an offset to their
orifices, the valve is fully closed. The working positions are the positions the valve moves to when the spool is maximally displaced from its normal position. That displacement is positive in one
case and negative in the other.
Image I shows a valve where the displacement is positive, the P–A and B–T orifices are fully open, and the P–B and A–T orifices are fully closed. Image II shows a valve where the displacement is
negative, the P–A and B–T orifices are fully closed, and the P–B and A–T orifices are fully open. Image III shows a valve where the spool is in the neutral position and all the orifices are closed.
The spool displacement that puts the valve in its working position depends on the offsets of the lands on the spool. The parameters in the Valve Opening Fraction Offsets section specify the block
constants for the spool displacements of the ports.
Orifice Opening Fractions
Between valve positions, the opening of an orifice depends on where the land of the spool is, relative to the rim. This distance is the orifice opening, and the block normalizes this distance so that
its value is a fraction of the maximum distance at which the orifice is fully open. The normalized variable is the orifice opening fraction.
The orifice opening fractions range from -1 when the valve is in the position shown in image I to +1 when the valve is in the position shown in image II.
The block calculates the opening fractions from the spool displacement and opening fraction offset. The displacement and offset are unitless fractions of the maximum land-orifice distance.
The opening fraction of the P–A orifice is:
The opening fraction of the B–T orifice is:
The opening fraction of the A–T orifice is:
The opening fraction of the P–B orifice is:
In the equations:
• h is the opening fraction for the orifice. If the calculation returns a value outside of the range 0 - 1, the block uses the nearest limit.
• H is the opening fraction offset for the orifice. The parameters in the Valve Opening Fraction Offsets section specify the opening fraction offsets. To allow for unusual valve configurations, the
block imposes no limit on their values, although they are typically between -1 and +1.
• x is the normalized instantaneous displacement of the spool, specified by a physical signal at port S. To compensate for extreme opening fraction offsets, there is no limit on the signal value.
The value is typically between -1 and +1.
Opening Fraction Offsets
By default, the valve is fully closed when its control displacement is zero. In this state, the valve is zero-lapped.
You can offset the lands of the spool to model an underlapped or overlapped valve. Underlapped valves are partially open in the normal position. Overlapped valves are fully closed slightly beyond the
normal position. The figure shows how the orifice opening fractions vary with the instantaneous spool displacement:
• Image I: A zero-lapped valve. The opening fraction offsets are all zero. When the valve is in the normal position, the lands of the spool completely cover all four orifices.
• Image II: An underlapped valve. The P–A and B–T opening fraction offsets are positive and the P–B and the A–T opening fraction offsets are negative. When the valve is in the normal position, the
lands of the spool partially cover the four orifices.
• Image III: An overlapped valve. The P–A and B–T opening fraction offsets are negative and the P–B and the A–T opening fraction offsets are positive. The control member completely covers all
orifices not only in the normal position but over a small region of spool displacements around it.
Valve Parameterizations
The block behavior depends on the Valve parametrization parameter:
• Cv flow coefficient — The flow coefficient C[v] determines the block parameterization. The flow coefficient measures the ease with which a gas can flow when driven by a certain pressure
differential. [2][3]
• Kv flow coefficient — The flow coefficient K[v], where ${K}_{v}=0.865{C}_{v}$, determines the block parameterization. The flow coefficient measures the ease with which a gas can flow when driven
by a certain pressure differential. [2][3]
• Sonic conductance — The sonic conductance of the resistive element at steady state determines the block parameterization. The sonic conductance measures the ease with which a gas can flow when
choked, which is a condition in which the flow velocity is at the local speed of sound. Choking occurs when the ratio between downstream and upstream pressures reaches a critical value known as
the critical pressure ratio. [1]
• Orifice area — The size of the flow restriction determines the block parametrization. [4]
Opening Characteristics
The flow characteristic relates the opening of the valve to the input that produces it, which is often the spool travel. The block expresses the opening as a sonic conductance, flow coefficient, or
restriction area, depending on the setting of the Valve parameterization parameter. The control input is the orifice opening fraction, a function of the spool displacement specified at port S.
The flow characteristic is normally given at steady state, with the inlet at a constant, carefully controlled pressure. The flow characteristic depends only on the valve and can be linear or
nonlinear. To capture the flow characteristics, use the Opening characteristic parameter:
• Linear — The measure of flow capacity is a linear function of the orifice opening fraction. As the opening fraction rises from 0 to 1, the measure of flow capacity scales from the specified
minimum to the specified maximum.
• Tabulated — The measure of flow capacity is a general function, which can be linear or nonlinear, of the orifice opening fraction. The function is specified in tabulated form, with the
independent variable specified by the Opening fraction vector.
Numerical Smoothing
When the Opening characteristic parameter is Linear, and the Smoothing factor parameter is nonzero, the block applies numerical smoothing to the orifice opening fraction. Enabling smoothing helps
maintain numerical robustness in your simulation.
For more information, see Numerical Smoothing.
Leakage Flow
The leakage flow ensures that no section of a fluid network becomes isolated. Isolated fluid sections can reduce the numerical robustness of the model, slow the rate of simulation and, in some cases,
cause the simulation to fail. The Leakage flow fraction parameter represents the leakage flow area in the block as a small number greater than zero.
Composite Structure
This block is a composite component comprising four instances of the Orifice (G) block connected to ports P, A, B, T, and S. Refer to the Orifice (G) block for more detail on the valve
parameterizations and block calculations.
Assumptions and Limitations
• The Sonic conductance setting of the Valve parameterization parameter is for pneumatic applications. If you use this setting for gases other than air, you may need to scale the sonic conductance
by the square root of the specific gravity.
• The equation for the Orifice area parameterization is less accurate for gases that are far from ideal.
• This block does not model supersonic flow.
S — Valve control signal, unitless
physical signal
Instantaneous displacement of the control member against its normal unactuated position, specified as a physical signal. The block normalizes the displacement against the maximum position of the
control member that required to open the orifice fully. The signal is unitless and its instantaneous value is typically in the range between -1 and +1.
A — Valve opening
Gas conserving port associated with the opening through which the flow enters or exits the valve.
B — Valve opening
Gas conserving port associated with the opening through which the flow enters or exits the valve.
P — Valve opening
Gas conserving port associated with the opening through which the flow enters or exits the valve.
T — Valve opening
Gas conserving port associated with the opening through which the flow enters or exits the valve.
Valve parameterization — Parameterization used to specify flow characteristics of orifice
Cv flow coefficient (default) | Kv flow coefficient | Sonic conductance | Orifice area
Method to calculate the mass flow rate.
• Cv flow coefficient — The flow coefficient C[v] determines the block parameterization.
• Kv flow coefficient — The flow coefficient K[v], where ${K}_{v}=0.865{C}_{v}$, determines the block parameterization.
• Sonic conductance — The sonic conductance of the resistive element at steady state determines the block parameterization.
• Orifice area — The size of the flow restriction determines the block parametrization.
Opening characteristic — Method to calculate the opening characteristics of the valve
Linear (default) | Tabulated
Method the block uses to calculate the opening area of the valve. The linear setting treats the opening area as a linear function of the orifice opening fraction. The tabulated setting allows for a
general, nonlinear relationship you can specify in tabulated form.
Opening fraction vector — Control signal values at which to specify tabulated parameter data
0 : 0.2 : 1 (default) | vector of positive numbers
Vector of control signal values at which to specify the chosen measure of flow capacity: Sonic conductance vector, Cv coefficient vector, Kv coefficient vector, or Orifice area vector. The control
signal is between 0 and 1, with each value corresponding to an opening fraction of the resistive element. The greater the value, the greater the opening and easier the flow.
The opening fractions must increase monotonically across the vector from left to right. The size of this vector must be the same as the Sonic conductance vector, Cv coefficient vector, Kv coefficient
vector, or Orifice area vector parameter.
To enable this parameter, set Opening characteristic to Tabulated.
Maximum Cv flow coefficient — Cv coefficient that corresponds to maximally open component
4 (default) | positive scalar
Value of the C[v] flow coefficient when the restriction area available for flow is at a maximum. This parameter measures the ease with which the gas traverses the resistive element when driven by a
pressure differential.
To enable this parameter, set Valve parameterization to Cv flow coefficient, and Opening characteristic to Linear.
xT pressure differential ratio factor at choked flow — Ratio of pressure differentials
0.7 (default) | positive scalar
Ratio between the inlet pressure, p[in], and the outlet pressure, p[out], defined as $\left({p}_{in}-{p}_{out}\right)/{p}_{in}$ where choking first occurs. If you do not have this value, look it up
in table 2 in ISA-75.01.01 [3]. Otherwise, the default value of 0.7 is reasonable for many valves.
To enable this parameter, set Valve parameterization to Cv flow coefficient or Kv flow coefficient.
Leakage flow fraction — Ratio of flow rates
1e-6 (default) | positive scalar
Ratio of the flow rate of the orifice when it is closed to when it is open.
To enable this parameter, set Opening characteristic to Linear.
Smoothing factor — Numerical smoothing factor
0.01 (default) | positive scalar in the range [0,1]
Continuous smoothing factor that introduces a layer of gradual change to the flow response when the orifice is in near-open or near-closed positions. Set this parameter to a nonzero value less than
one to increase the stability of your simulation in these regimes.
To enable this parameter, set Opening characteristic to Linear.
Cv flow coefficient vector — Cv coefficients corresponding to values in opening fraction vector
[1e-06, .8, 1.6, 2.4, 3.2, 4] (default) | vector of positive numbers
Vector of C[v] flow coefficients. Each coefficient corresponds to a value in the Opening fraction vector parameter. This parameter measures the ease with which the gas traverses the resistive element
when driven by a pressure differential. The flow coefficients must increase monotonically from left to right, with greater opening fractions representing greater flow coefficients. The size of the
vector must be the same as the Opening fraction vector.
To enable this parameter, set Valve parameterization to Cv flow coefficient, and Opening characteristic to Tabulated.
Maximum Kv flow coefficient — Kv coefficient that corresponds to maximally open component
3.6 (default) | positive scalar
Value of the K[v] flow coefficient when the restriction area available for flow is at a maximum. This parameter measures the ease with which the gas traverses the resistive element when driven by a
pressure differential.
To enable this parameter, set Valve parameterization to Kv flow coefficient, and Opening characteristic to Linear.
Kv flow coefficient vector — Kv coefficients corresponding to values in opening fraction vector
[1e-06, .72, 1.44, 2.16, 2.88, 3.6] (default) | vector of positive numbers
Vector of K[v] flow coefficients. Each coefficient corresponds to a value in the Opening fraction vector parameter. This parameter measures the ease with which the gas traverses the resistive element
when driven by a pressure differential. The flow coefficients must increase monotonically from left to right, with greater opening fractions representing greater flow coefficients. The size of the
vector must be the same as the Opening fraction vector parameter.
To enable this parameter, set Valve parameterization to Kv flow coefficient, and Opening characteristic to Tabulated.
Maximum sonic conductance — Sonic conductance corresponding to maximally open component
12 l/(bar*s) (default) | positive scalar
Value of the sonic conductance when the cross-sectional area available for flow is at a maximum.
To enable this parameter, set Valve parameterization to Sonic conductance, and Opening characteristic to Linear.
Critical pressure ratio — Ratio of downstream and upstream pressures at which the flow first chokes
0.3 (default) | positive scalar
Pressure ratio at which flow first begins to choke and the flow velocity reaches its maximum, given by the local speed of sound. The pressure ratio is the outlet pressure divided by inlet pressure.
To enable this parameter, set Valve parameterization to Sonic conductance, and Opening characteristic to Linear.
Subsonic index — Exponent used to characterize mass flow in subsonic flow regime
0.5 (default) | positive scalar
Empirical value used to more accurately calculate the mass flow rate in the subsonic flow regime.
To enable this parameter, set Valve parameterization to Sonic conductance.
ISO reference temperature — ISO 8778 reference temperature
293.15 K (default) | scalar
Temperature at standard reference atmosphere, defined as 293.15 K in ISO 8778.
You only need to adjust the ISO reference parameter values if you are using sonic conductance values that are obtained at difference reference values.
To enable this parameter, set Valve parameterization to Sonic conductance.
ISO reference density — ISO 8778 reference density
1.185 kg/m^3 (default) | positive scalar
Density at standard reference atmosphere, defined as 1.185 kg/m3 in ISO 8778.
You only need to adjust the ISO reference parameter values if you are using sonic conductance values that are obtained at difference reference values.
To enable this parameter, set Valve parameterization to Sonic conductance.
Sonic conductance vector — Sonic conductances corresponding to values in opening fraction vector
[1e-05, 2.4, 4.8, 7.2, 9.6, 12] l/(bar*s) (default) | vector of positive values
Vector of sonic conductances inside the resistive element. Each conductance corresponds to a value in the Opening fraction vector parameter. The sonic conductances must increase monotonically from
left to right, with greater opening fractions representing greater sonic conductances. The size of the vector must be the same as the Opening fraction vector parameter.
To enable this parameter, set Valve parameterization to Sonic conductance, and Opening characteristic to Tabulated.
Critical pressure ratio vector — Critical pressure ratios corresponding to values in opening fraction vector
0.3 * ones(1,6) (default) | vector of positive numbers
Vector of critical pressure ratios at which the flow first chokes, with each critical pressure ratio corresponding to a value in the Opening fraction vector parameter. The critical pressure ratio is
the fraction of downstream-to-upstream pressures at which the flow velocity reaches the local speed of sound. The size of the vector must be the same as the Opening fraction vector parameter.
To enable this parameter, set Valve parameterization to Sonic conductance, and Opening characteristic to Tabulated.
Maximum orifice area — Flow area corresponding to maximally open component
1e-4 m^2 (default) | positive scalar
Cross-sectional area of the orifice opening when the cross-sectional area available for flow is at a maximum.
To enable this parameter, set Valve parameterization to Orifice area, and Opening characteristic to Linear.
Discharge coefficient — Discharge coefficient
0.64 (default) | positive scalar in the range of [0,1]
Correction factor that accounts for discharge losses in theoretical flows.
To enable this parameter, set Valve parameterization to Orifice area.
Orifice area vector — Vector of orifice opening area values
[1e-10, 2e-05, 4e-05, 6e-05, 8e-05, .0001] m^2 (default) | vector
Vector of cross-sectional areas of the orifice opening. The values in this vector correspond one-to-one with the elements in the Opening fraction vector parameter. The first element of this vector is
the orifice leakage area and the last element is the maximum orifice area.
To enable this parameter, set Valve parameterization to Orifice area, and Opening characteristic to Tabulated.
Laminar flow pressure ratio — Pressure ratio at which flow transitions between laminar and turbulent regimes
0.999 (default) | positive scalar
Pressure ratio at which flow transitions between laminar and turbulent flow regimes. The pressure ratio is the outlet pressure divided by inlet pressure. Typical values range from 0.995 to 0.999.
Cross-sectional area at ports A, B, P, and T — Area normal to the flow path at the valve ports
0.01 m^2 (default) | positive scalar in units of area
Area normal to the flow path at the valve ports. The ports are assumed to be of the same size. The flow area specified here should ideally match those of the inlets of adjoining components.
Valve Opening Fraction Offsets
Between Ports P and A — Opening fraction offset for the P–A orifice
0 (default) | unitless scalar
Opening fraction of the P–A orifice when the value of the input signal S is zero. The valve is then in the normal position. The opening fraction measures the distance of a land of the spool to its
appointed orifice, here P–A, normalized by the maximum such distance. It is unitless and generally between 0 and 1.
Between Ports B and T — Opening fraction offset for the B–T orifice
0 (default) | unitless scalar
Opening fraction of the B–T orifice when the value of the input signal S is zero. The valve is then in the normal position. The opening fraction measures the distance of a land of the spool to its
appointed orifice, here B–T, normalized by the maximum such distance. It is unitless and generally between 0 and 1.
Between Ports P and B — Opening fraction offset for the P–B orifice
0 (default) | unitless scalar
Opening fraction of the P–B orifice when the value of the input signal S is zero. The valve is then in the normal position. The opening fraction measures the distance of a land of the spool to its
appointed orifice, here P–B, normalized by the maximum such distance. It is unitless and generally between 0 and 1.
Between Ports A and T — Opening fraction offset for the A–T orifice
0 (default) | unitless scalar
Opening fraction of the A–T orifice when the value of the input signal S is zero. The valve is then in the normal position. The opening fraction measures the distance of a land of the spool to its
appointed orifice, here A–T, normalized by the maximum such distance. It is unitless and generally between 0 and 1.
[1] ISO 6358-3. "Pneumatic fluid power – Determination of flow-rate characteristics of components using compressible fluids – Part 3: Method for calculating steady-state flow rate characteristics of
systems". 2014.
[2] IEC 60534-2-3. "Industrial-process control valves – Part 2-3: Flow capacity – Test procedures". 2015.
[3] ANSI/ISA-75.01.01. "Industrial-Process Control Valves – Part 2-1: Flow capacity – Sizing equations for fluid flow underinstalled conditions". 2012.
[4] P. Beater. Pneumatic Drives. Springer-Verlag Berlin Heidelberg. 2007.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2018b
R2022b: Model gas systems using improved flow computations
In previous versions, the blocks in the gas library used a conversion factor to implement different orifice parameterizations in terms of sonic conductance. You can use the Valve parameterization
parameter to specify the parameterization.
The new parameterizations also include new parameters:
• xT pressure differential ratio factor at choked flow
• Discharge coefficient
• Leakage flow fraction. This parameter replaces the Cv coefficient (USCS) at leakage flow, Kv coefficient (SI) at leakage flow, Sonic conductance at leakage flow, and Restriction area at leakage
flow parameters.
To improve clarity:
• The Opening parameterization parameter has been renamed to Opening characteristic.
• The Reference temperature and Reference density parameters have been renamed to ISO reference temperature and ISO reference density, respectively.
• The Cv coefficient (USCS) at maximum flow and Kv coefficient (USCS) at maximum flow parameters have been renamed to Maximum Cv coefficient and Maximum Kv coefficient , respectively.
• The Sonic conductance at maximum flow parameter has been renamed to Maximum sonic conductance.
R2022b: Updated block name
The title of the 4-Way Directional Valve (G) block changed to 4-Way 3-Position Directional Valve (G).
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Proteases inhibitor
A Bayesian Population PK–PD Model of Ispinesib-induced Myelosuppression
SJ Kathman1, DH Williams1, JP Hodge1 and M Dar1
The goal of the present analysis is to fit a Bayesian population pharmacokinetic pharmacodynomic (PK–PD) model to characterize the relationship between the concentration of ispinesib and changes in
absolute neutrophil counts (ANC). Ispinesib, a kinesin spindle protein (KSP) inhibitor, blocks assembly of a functional mitotic spindle, leading to G2/M arrest. A first time in human, phase I
open-label, non-randomized, dose-escalating study evaluated ispinesib at doses ranging from 1 to 21 mg/m2. PK–PD data were collected from 45 patients with solid tumors. The pharmacokinetics of
ispinesib were well characterized by a two-compartment model. A semimechanistic model was fit to the ANC. The PK and PD data were successfully modelled simultaneously. This is the first presentation
of simultaneously fitting a PK–PD model to ANC using Bayesian methods. Bayesian methods allow for the use of prior information for some system-related parameters. The model may be used to examine
different schedules, doses, and infusion times.
Ispinesib (SB715992) is a kinesin spindle protein (KSP) inhibitor that blocks the assembly of functional mitotic spindles, thereby causing cell cycle arrest in mitosis and subsequent cell death.
Although the mitotic spindle has long been an important functional target in cancer chemotherapy, toxicity related to interference with microtubules in non-proliferating terminally differentiated
cells has been difficult to manage. Neurotoxicity has terminated the development of several broad-acting antitubulin agents. Ispinesib, however, acts via a novel mechanism by inhibiting the KSP that
appears to function exclusively in mitosis. No role for KSP outside of mitosis has been demonstrated. A drug targeting KSP may therefore prove equally or more efficacious than antitubulin
chemotherapy agents without the potential for neurotoxicity or other side effects associated with interference with tubulin function in non-dividing cells. Similar to many other antiproliferative
drugs, ispinesib is expected to have manageable dose-limiting toxicities (e.g., myelosuppression and gastrointestinal disturbances) resulting from action on normal proliferating tissues. However,
inhibition of KSP, a novel mitotic target, by ispinesib, offers the potential for a unique antitumor profile compared to that observed with currently available chemotherapeutics.
Myelosuppression is the major dose limiting toxicity for many chemotherapeutic drugs. It is an important considera-tion in the development of novel cytotoxics in oncology, especially for targeted
antimitotic drugs, where the dose-
limiting toxicity is likely to be myelosuppression. Therefore, optimization of dose/administration schedule early in devel-opment of such agents by establishing the relationship between drug
concentration and myelosuppression could greatly aid in the identification of a potential therapeutic window. The goal of the present analysis is to fit a population pharmakokinetic–pharmakodynamic
(PK–PD) model to characterize the relationship between ispinesib, a compound currently in early stage of development, and absolute neutrophil counts (ANCs). The model is fit using Bayesian Markov
Chain Monte-Carlo methods. The use of the model to predict the outcomes for designs beyond what was used to develop the model is also explored. More comprehensive data on safety and other clinical
effects were recorded and will be reported elsewhere.
A total of 45 subjects, 34 men and 11 women aged 37–84 years, with solid tumors were enrolled in the study and included in the analysis. Subjects received from one to 14 cycles of treatment, with a
median of two cycles. Only data from the first cycle were used in this analysis, as this is when ispinesib concentrations were measured. Table 1 contains a summary of the demographic characteristics.
Two chains with different starting values were used to help assess convergence. We took 5,000 burn-ins and then recorded every second sample out of the next 30,000
1GlaxoSmithKline, Research Triangle Park, North Carolina, USA. Correspondence: S Kathman ([email protected])
Received 8 February 2006; accepted 29 September 2006. doi:10.1038/sj.clpt.6100021
88 VOLUME 81 NUMBER 1 | JANUARY 2007 | www.nature.com/cpt
Table 1 Summary of demographics (n=45)
Parameter Value
Age (years)
Median and range 55 (37–84)
Mean (SD) 57.9 (10.9)
Weight (kg)
Median and range 83.8 (49–133)
Mean (SD) 83.1 (18.3)
Height (cm)
Median and range 171.0 (130–192)
Mean (SD) 170.4 (12.0)
BSA (m2)
Median and range 1.92 (1.44–2.51)
Mean (SD) 1.92 (0.24)
Body mass index (kg/m2)
Median and range 27.1 (19–62)
Mean (SD) 28.9 (7.8)
BSA, body surface area.
iterations to reduce the autocorrelation in the Markov chain, and based all the computations on the resulting 30,000 (15,000 per chain) posterior samples. The Markov chains converged fast (within the
first 5,000 iterations) and mixed well.
The only covariate selected when using the two-compart-ment model was body surface area (BSA) for the mean of the intercompartmental clearance ln(Qi) and the volume of the central compartment ln
(V1i). This was selected based on the examination of the posterior distributions. So
Population mean for lnðQiÞ : Y2 ¼ a1 þ a2BSAi
Population mean for lnðV1iÞ : Y3 ¼ a3 þ a4BSAi:
The posterior probabilities that a2 and a4 were greater than zero were greater than or equal to 95%. The fifth percentiles of the posterior distributions were 0.04 and 0.41 for a2 and a4,
respectively. The two-compartment model appears to fit the PK data well (Figure 1). Figure 1 shows the actual PK concentrations versus the predictions from a two-compart-ment model, using the means
of the posterior distributions as the predictions. Figure 2 is the same as Figure 1, except that the predictions, means of the posterior distributions, are derived from a three-compartment model. For
the three-compartment model, BSA was a covariate for the mean of the two intercompartmental clearances (ln(Q2i) and ln(Q3i)). The mean squared error for the three-compartment model (1,170) was better
than the mean squared error for the
(ng/ml) 1,500
concentrations 1,000
Actual 500
0 500 1,000 1,500 Predicted concentrations (ng/ml) from two-compartment model
Figure 1 Comparison of observed concentrations (ng/ml) and concentrations predicted using the mean of the posterior distribution and a two-compartment model.
(ng/ml) 1,500
Actual 500
0 500 1,000 1,500 Predicted concentrations (ng/ml) from three-compartment model
Figure 2 Comparison of observed concentrations (ng/ml) and concentrations predicted using the mean of the posterior distribution and a three-compartment model.
two-compartment model (1,586). A total of 603 PK samples from 30 subjects were available from the trial evaluating the weekly schedule. The mean squared prediction errors were 1,305 and 1,309 for the
two- and three-compartment models, respectively. Although the three-compartment model appears to fit the current data better, it did not perform better than the two-compartment model in terms of
predicting the data from another trial. For this reason, the two-compartment model was selected for the PK–PD modelling of ANC.
Table 2 presents characteristics of the posterior distribu-tions for the population parameters from the two-compart-ment model. The intersubject coefficients of variation for the PK parameters, based
on medians of posterior distributions for the diagonal elements of the variance–covariance matrix, were 44, 55, 43, and 45% for CL, Q, V1, and V2, respectively. Table 3 presents a summary of the
medians of posterior distributions for the individual clearances. The results in Table 3 suggest that the individual clearances vary greatly, with a greater than sevenfold difference from the
smallest median to the largest median.
CLINICAL PHARMACOLOGY & THERAPEUTICS | VOLUME 81 NUMBER 1 | JANUARY 2007 89
Table 2 Summary of posterior distributions for population parameters in the PK model
Parameter Mean SD Median 2.5% 97.5%
Mean for ln(CL (l/h)) 1.83 0.071 1.83 1.69 1.97
Intercept for mean of ln(Q (l/h)) 4.18 0.090 4.18 3.99 4.35
Slope (BSA (m2)) for mean of 0.47 0.26 0.47 0.06 0.98
ln(Q (l/h))
Intercept for mean of ln(V1 (l)) 2.89 0.081 2.89 2.73 3.05
Slope (BSA (m2)) for mean of 0.94 0.32 0.94 0.30 1.58
ln(V1 (l))
Mean for ln(V2 (l)) 5.26 0.071 5.26 5.12 5.40
sa (ng/ml) 0.22 0.17 0.18 0.009 0.63
sp (%) 17 0.5 17 16 18
BSA, body surface area.
/l) 15
ANC (10 10
Observed 5
ANC (109/l) predicted from model (posterior mean)
Figure 3 Comparison of observed ANC (109/l) and ANC predicted using the mean of the posterior distribution.
Table 3 Summary of medians of posterior distributions
for individual clearances (l/h) 6
Minimum 1.71
25% 5.18 /l) 4
Median 6.87 (10
75% 8.67 2
Maximum 12.35
The PK–PD model was simultaneously fit for the drug 0 5 10 15 20
concentration using the two-compartment model, and the
Time (days)
ANC using the model described below in section ‘‘PK–PD Figure 4 Plot of observed ANC (109/l) (open symbols) and the predictions
model’’. Figure 3 shows the observed ANC versus the from the model (closed symbols and lines) versus time in days for
predictions from the model, again using the means of representative subjects in the trial. The circles and solid line correspond to
posterior distributions for predictions. Figure 4 shows the a subject who received 18 mg/m2. The triangle (up) and dot-dashed line
actual and predicted ANC values for three representative correspond to a subject who received 12.5 mg/m2. The triangle (down) and
dashed line correspond to a subject who received 6 mg/m2.
subjects in the trial, at different dose levels. Table 4 presents
some characteristics of the posterior distributions for the
population parameters.
The prior distribution for mCirc0 , the population mean of Table 4 Summary of posterior distributions for population
ln(Circ(0)i), was normal with mean 1.61 and SD 0.25. The parameters in the PD model
posterior distribution had a mean of 1.52 and SD 0.07 Parameter Mean SD Median 2.5% 97.5%
(Table 4). The individuals in this trial had lower baseline
Mean for ln(MTT (h)) 4.43 0.04 4.43 4.36 4.50
values than those seen previously for other compounds. The
prior distribution for mMTT, the population mean of ln(MTTi), SD for ln(MTT (h)) 0.12 0.027 0.12 0.076 0.18
was normal with mean 4.83 and SD 0.35. The posterior Mean for ln(Circ0 (109/l)) 1.52 0.07 1.52 1.39 1.66
distribution had mean 4.43 and SD 0.04 (Table 4). This mean SD for ln(Circ0 (109/l)) 0.45 0.054 0.45 0.36 0.57
was also lower than observed previously, although fairly close Mean for ln(Slope) of PK 4.57 0.11 4.57 4.79 4.36
to that of docetaxel.1 The posterior distribution for g was
Conc (ng/ml)
consistent with the literature. The remaining parameters were SD for ln(Slope) of PK Conc 0.57 0.10 0.56 0.40 0.79
given vague prior distributions, and their posterior distribu-
tions should primarily be influenced by the data. g 0.16 0.008 0.16 0.14 0.18
SIMULATING DIFFERENT SCHEDULES s(ANC)a (109/l) 0.24 0.059 0.24 0.14 0.37
Predicting or extrapolating the effect on ANC of as yet
s(ANC)p (%)
untested doses and schedules can be of great value in the 21 2.7 21 16 27
design of subsequent clinical trials during early development ANC, absolute neutrophil count; MTT, mean transit time.
90 VOLUME 81 NUMBER 1 | JANUARY 2007 | www.nature.com/cpt
/l) 7
ANC (10
Time (days)
Figure 5 Results of the simulation of ANC (109/l) for once a week for
3 weeks schedule (dose ¼ 7 mg/m2, BSA ¼ 1.95 m2). Solid line is the median for the simulations. The bold dashed lines are the 25th and 75th percentiles. The bold dot-dashed lines are the 5th and 95th
percentiles. The open circles correspond to preliminary data from a trial using this schedule.
of a compound. Given that the model developed by Friberg and Karlsson2 is more physiologically based, a potential application may be its predictive value in new settings not originally used to
develop the model. The importance of being able to extrapolate to conditions not yet tested is highlighted in Sheiner and Wakefield.3
To illustrate this point, the current PK–PD model, based solely on data from a once every 21-day schedule, was used to predict the relationship between dose and ANC in another trial conducted with
ispinesib evaluating a weekly schedule (dosing on days 1, 8, and 15 repeated every 28 days). Figure 5 presents the results of a simulation of ANC and drug concentration for this schedule (at the dose
of 7 mg/m2 and BSA of 1.95 m2) along with actual clinical data from six subjects. The data suggest that the model is performing reasonably well in terms of predicting what would happen for a schedule
beyond that used to develop the model, although none of the subjects had points below the 25th percentile, and at day 14 there was only one subject below the median. The value of the model for
predicting ANC for other schedules needs to continue being assessed. Data from this second trial (weekly schedule) may be used in the future as part of the model building process, once all of the
data are available for analysis.
A Bayesian approach was used to fit the models as opposed to the frequentist approach (obtaining maximum-likelihood estimates and confidence intervals) that is most commonly used. This is the first
presentation of simultaneously fitting a pharmacokinetic model and a pharmacodynamic model to ANC using Bayesian methods. The Bayesian approach readily allows for the incorporation of prior
knowledge, which exists for the system-related parameters in the PD model. Incorporating prior knowledge may be useful in cases where the data are sparse, as was the case for the ANC for many of the
subjects in this trial. A Bayesian analysis also expresses
uncertainty about a parameter in terms of probability, and thus the probability of a parameter being within a certain region or interval may be discussed. This is not the case for a frequentist
analysis where estimates and confidence intervals are usually presented. The Bayesian approach uses Monte-Carlo methods as opposed to some of the more traditional algorithms (e.g., Taylor
Series-based approximations for integration and gradient-based maximization algorithms). Although not fully discussed here, more information on Monte-Carlo methods may be found in Robert and
Casella.4 Duffull et al.5 also gives some indication that Bayesian methods are worth considering for population PK models, partially due to their use of Monte-Carlo algorithms.
Bayesian methods not only allow for the use of prior information, but more importantly, provide substantial scope for extending the specified model, and allow for the replacement of normality
assumptions with other distribu-tions (such as a Student’s t-distribution) to provide robust-ness against outliers. A t-distribution with four degrees of freedom was chosen a priori for the ANC
values as there was a concern about the potential for outliers. Treating the degrees of freedom as a random variable and letting the data determine the distribution of the ANC values was considered.
Two methods were used:6 the first method was to assume that the degrees of freedom follows an exponential distribution with mean 1/l, then l was given a uniform(0.10,0.50) prior distribution, and the
second method was to assume that the degrees of freedom follows an exponential distribution with a mean of 10. However, the autocorrelation in the samples generated from the Markov Chain Monte-Carlo
algorithm for the degrees of freedom variable was very high, regardless of the approach used, making convergence difficult to assess. Increasing the degrees of freedom from four to eight (fixing the
degrees of freedom) had no effect on the results from fitting the model. Further increasing from eight to 20 had very little effect.
The PK and PD data were modelled simultaneously, acknowledging the uncertainty in the fitted concentrations.3 This also provides the opportunity for the PD data to aid in the estimation of the PK
parameters, although in this particular case, the estimates (and posterior distributions) of the PK parameters were similar whether the PK model was fit alone, or both the PK and PD were fit
There does not appear to be a standard method for model selection applicable to Bayesian population PK models. A two-compartment model adequately describes the concen-tration time profile of
ispinesib. There is some improvement in using a three-compartment model, although the improve-ment is minimal. The two-compartment model performed slightly better than the three-compartment model in
predicting the concentrations from another trial. As the ability to make predictions is important, this was the primary selection criteria. Both two- and three-compartment models for PK were
considered when modelling the nadir for the ANC (results not shown), and there was no difference in the prediction of the nadir. There was also a considerable savings
CLINICAL PHARMACOLOGY & THERAPEUTICS | VOLUME 81 NUMBER 1 | JANUARY 2007 91
in computation time using a two-compartment model as opposed to the three-compartment model. The deviance information criteria (DIC)7 in WinBugs was considered for model selection, although it did
not perform well here due to providing negative estimates for the effective number of parameters. The DIC has previously been shown to be unreliable for population PK model selection, always
selecting a two-compartment model over a one-compartment model when data were simulated from a one-compartment model.8
The semimechanistic model for ANC, originally described in Friberg et al.,1 adequately describes the neutrophil counts after the administration of ispinesib. The current PK–PD model can be used to
simulate expected incidence of clinically significant neutropenia on alternative schedules of ispinesib to better inform future clinical development decisions. This provides information that may be
useful in planning trials to examine other schedules or different lengths of infusion. For example, if it is determined that a prolonged exposure may be desirable, then simulations may be performed
before conducting the trial to help determine how long the infusion should be, how often the drug should be given, and at what dose to start with to better ensure the safety of the patients, at least
in terms of neutropenia.
It is possible to use modelling as a first time if human trial is being conducted, to allow for the possibility of exploring several factors (dose, schedule, and length of infusion), earlier in the
development of a drug. By incorporating prior information, Bayesian methods provide the possibility of fitting the model even though data from very few subjects are available.
The software used to fit these models was WinBugs version 1.4.19 with the WBDiff and Pharmaco add-ons.
Study design. A first time in human, phase I open-label, non-randomized, dose-escalating study evaluating ispinesib at doses ranging from 1 to 21 mg/m2 administered as a 1-h infusion every 21 days
has been completed. Doses of ispinesib were escalated in successive three-patient cohorts. Cohorts were expanded for dose limiting toxicities and following the determination of the max-imum-tolerated
dose, to better characterize the safety and PK profile. Serial PK samples (n ¼ 17) were taken during cycle 1 beginning at pre-dose until 48 h post-dose. The scheduled times from the start of infusion
were pre-dose, 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 10, 12, 24, 36, and 48 h. ANCs were assessed weekly on days 1, 8, 15, and 22 (before the start of the second cycle). More frequent
assessments were carried out if ANC dropped below 0.75 (109/l).
Analysis methods. The models described in the following sections are non-linear hierarchical models that were fit using Bayesian Markov Chain Monte-Carlo techniques. If yi and ji represent vectors of
individual PK and PD parameters, respectively, then it was assumed that they follow distributions with population parameters Y and F, respectively. The parameters Y and F were then assigned vague or
weakly informative prior distributions depending on the prior information available. The Bayesian analysis involved
the estimation of the joint distribution of all parameters conditional on the observed data: p(y, j, Y, F|PK and PD data), where y and j denote collections of all individual specific PK and PD
parameters, respectively. Generating random samples from the joint posterior distribution allows the marginal distribution of each parameter to be completely characterized. More detailed information
on Bayesian analyses of PK–PD models may be found in Lunn et al.10 and Duffull et al.5 The model was fit to the data using WinBugs v1.4.19 with the Pharmaco interface and WBDiff, which together make
up PKBugs v2.0. Convergence was assessed both visually, by examining trace and running quartile plots, and formally using the Brooks–-Gelman–Rubin diagnostic11 available in WinBugs.
PK model. The time course of ispinesib was assumed to follow a compartmental model. Both two- and three-compartment models, fit using Bayesian methods, were considered. The choice for the population
PK model (two versus three compartments) will be determined by the models ability to predict observations from another trial, where ispinesib was evaluated on a weekly schedule (dosing on days 1, 8,
and 15). The predictions for the trial evaluating the weekly schedule will be compared to some preliminary data, and mean squared prediction errors will be calculated using the means of the posterior
distributions of the predictions as the predicted values. The data from the weekly schedule may be incorporated into the model building process in the future.
Subjects were dosed based on mg/m2, but the total dose administered in mg was used in the modelling with BSA considered as a covariate.
For the two-compartment model, it was assumed that an individual’s concentration at a given time point followed a normal distribution with mean Zij and variance tij, where i and j index the
individual and time, respectively. The mean Zij is a function of time, length of infusion, and the following parameters: the elimination clearance (CLi), the volume of distribution for the central
compartment (V1i), the volume of distribution for the peripheral compartment (V2i), and the intercompartmental clearance (Qi). The variance tij was set equal to s2a þ s2pZ2ij, where s2a and s2p
represent the additive and proportional variance terms, respectively.
Now let yi be a vector containing the log-transformed PK parameters for the ith individual [ln(CLi), ln(Qi), ln(V1i), ln(V2i)], then yi was assumed to follow a multivariate normalPdistribution
with mean Y and variance–covariance matrix . Potential demographic covariates (BSA, age, gender, height, and weight) were considered for elements of the mean vector Y, the importance of which were
determined by examining the posterior distributions. In particular, the covariate remained in the model if the posterior probability was large (greater than or equal to 95%) that the parameter for
the covariate was greater than zero (or less than zero for negative values). Scatterplots of the individuals estimated PK parameters and potential covariates were used to assist with selecting the
covariates to test. Parameters associated with the mean vector (Y) were then assigned a vague multivariate normal prior distribution with the mean being a vector of zeros and a
variance–covariance matrix with 104 for the diagonal elements and
zeros for off-diagonal elements. The inverse of was assigned using a vague prior Wishart distribution according to the PK Bugs manual (http://www.winbugs-development.org.uk/), using an initial
esti-mate for the inter-individual coefficient of variation of 30% for the pharmacokinetic parameters. As the least informative proper
Wishart prior distribution is being used for the inverse of , the data should have an important role in the forming of the posterior distribution, thus the model is not too sensitive to the initial
estimate of 30%. The variance terms sa and sp were assigned half-normal prior distributions with large variances (the absolute value of a normal random variable with mean equal to zero and variance
equal to 104).
92 VOLUME 81 NUMBER 1 | JANUARY 2007 | www.nature.com/cpt
Prol ktr Transit 1 ktr Transit 2 ktr Transit 3 ktr Circ
kprol= ktr kCirc= ktr
Feedback =
EDrug = ∗Conc Circ
Figure 6 Semimechanistic model of drug-induced myelosuppression.
The three-compartment model is similar to the two-compart-ment model described above, with a few notable differences. The mean Zij is now a function of time, length of infusion, and the following
parameters: the elimination clearance (CLi), the volume of distribution for the central compartment (V1i), the volume of distribution for each of the two peripheral compartments (V2) and (V3i), and
the two intercompartmental clearances (Q2i) and (Q3i). Note that the volume of one of the peripheral compartments (V2) is being estimated for the population and not for each individual as this leads
to an improvement in the convergence of the model. Estimating V2 for each individual was attempted, but the autocorrelation in the generated samples was very high, making convergence difficult to
assess and achieve. Ordering constraints are needed for the volumes to ensure that the model is identifiable.10 This requires yi ¼ [ln(CLi), ln(Q2i), ln(Q3i), ln(V1i), ln(V2), ln(V3i V2)].
PK–PD model. A semimechanistic model (refs. 1,2) was used to describe the impact of ispinesib on ANC. The model has also been used or discussed elsewhere in the literature.12–16 The model (shown in
Figure 6) consisted of a proliferating compartment (Prol), three transit compartments (Transit1, Transit2, Transit3) that represented the stepwise maturation of leukocytes within the bone marrow, and
a compartment of circulating blood cells (Circ). A negative feedback mechanism (Circ0/Circ)g from circulating cells on proliferating cells was included to describe the rebound of the cells (including
an overshoot compared to the baseline: Circ0). The differential equations were written as:
dProl=dt ¼ kProlProlð1 EDrugÞðCirc0=CircÞg ktrProl
dTransit1=dt ¼ ktrProl ktrTransit1
dTransit2=dt ¼ ktrTransit1 ktrTransit2
dTransit3=dt ¼ ktrTransit2 ktrTransit3
dCirc=dt ¼ ktrTransit3 kCircCirc
In the model, ktr, kProl, and kCirc represent the maturation rate constant, the proliferation rate constant, and the cell elimination
rate constant, respectively. The rate constants ktr ¼ kProl since dProl/ dt ¼ 0 at steady state, and is also assumed to be equal to the rate constant kCirc to reduce the number of parameters to be
estimated. The effect of the drug concentration in the central compartment on the proliferation rate was modelled with a linear function: EDrug ¼ b*Conc for a given individual, where Conc represents
the drug concentration in the central compartment and b is the slope
parameter. Friberg et al.1 considers the use of an Emax model, but an Emax model did not improve the results here (not shown).
It was then assumed that the ANC value for subject i at time j followed a Student t-distribution with mean m(ANC)ij, variance nij, and four degrees of freedom. A Student t-distribution was used here
given its robust properties, including protection against the
influence of outliers. The mean m(ANC)ij is obtained from the solution of the differential equations (Circ compartment), and is a
function of: an individual’s slope bi, an individual’s baseline value Circ(0)i, an individual’s mean transit time MTTi ¼ (nc þ 1)/k(tr)i
where nc is the number of transit compartments, the exponent part of the feedback mechanism g (modelled for the population as a whole, not separately for each individual), and an individual’s
concentration at time j (simultaneously being modelled from either the two- or three-compartment model described above). The
variance nij was set equal to s2(ANC)a þ s2(ANC)pm2(ANC)ij, where s2(ANC)a and s2(ANC)p represent the additive and proportional variance terms,
ln(bi), ln(Circ(0)i), and ln(MTTi) were assumed to be mutually independent and follow normal distributions with means mb, mCirc0 , mMTT and variances s2b, s2mCirc0 , s2MTT, respectively. The
parameters mCirc0 , mMTT, and g are considered system-related parameters, and should be consistent across different drugs, and thus were given weakly informative prior distributions. Friberg et al.1
examined what would happen if the system-related parameters were fixed, or set to particular values. The prior distributions selected here for mCirc0 and mMTT are centered (mean) at the recommended
values for fixing the parameters: ln(5 109/l) and ln(125 h), respectively. The SD for
the prior distributions of mCirc0 and mMTT were chosen such that the values observed for the six compounds modelled in Friberg et al.1
were within one SD of the chosen mean: 0.25 and 0.35 for mCirc0 and mMTT, respectively. The exponent g was given a half-normal (using
normal (0,1)) prior distribution, which again is consistent with the results observed in the literature1 where it ranged from 0.160 to 0.230. The mean mb is specific to ispinesib and was thus given a
fairly vague prior distribution (normal with mean ¼ 0 and SD ¼ 10). The SDs sb, sCirc0 , and sMTT were given uniform (0,1) prior distributions, which are weakly informative here. The terms s(ANC)a
and s(ANC)p were assigned half-normal prior distributions (absolute value of a normal random variable with mean equal to zero and variance equal to one).
We thank the anonymous referees for a thorough review and helpful comments. We also thank GlaxoSmithKline colleagues, Janet Begun and Frank Hoke, and Cytokinetics colleague, Khalil Saikali, for
helpful comments during the preparation of this manuscript.
The authors declared no conflict of interest.
& 2007 American Society for Clinical Pharmacology and Therapeutics
1. Friberg, L., Henningsson, A., Maas, H., Nguyen, L. & Karlsson, M. Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J. Clin. Oncol. 20, 4713–4721 (2002).
2. Friberg, L. & Karlsson, M. Mechanistic models for myelosuppression. Invest. New Drugs 21, 183–194 (2003).
3. Sheiner, L. & Wakefield, J. Population modelling in drug development. Stat. Methods Med. Res. 8, 183–193 (1999).
4. Robert, C. & Casella, G. Monte Carlo Statistical Methods, (Springer, New York, 2004).
5. Duffull, S., Kirkpatrick, C., Green, B. & Holford, N. Analysis of populations pharmacokinetic data using NONMEM and WinBugs. J. Biopharm. Stat. 15, 53–73 (2005).
6. Congdon, P. Applied Bayesian Modelling, (John Wiley and Sons Ltd, UK, 2003).
7. Spiegelhalter, D., Best, N. & Carlin, B. Bayesian measures of model complexity and fit. J. Roy. Stat. Soc. Ser. B 64, 583–639 (2002).
8. Friberg, L., Dansirikul, C. & Dufful, S. Simultaneous fit of competing models as a model discrimination tool in a fully Bayesian approach. In Population Approach Group Europe, (Uppsala, Sweden,
2004), Page 13 Abstr 493 [www.page-meeting.org/?abstract=493 ].
9. Spiegelhalter, D., Thomas, A., Best, N. & Gilks, W. BUGS: Bayesian Inference using Gibbs Sampling, Version 1.4.1. (MRC Biostatistics Unit, Cambridge, 2003).
10. Lunn, D., Best, N., Thomas, A., Wakefield, J. & Spiegelhalter, D. Bayesian analysis of population PK/PD models: general concepts and software. J. Pharmacokinet. Pharmacodyn. 29, 271–307 (2002).
CLINICAL PHARMACOLOGY & THERAPEUTICS | VOLUME 81 NUMBER 1 | JANUARY 2007 93
11. Gelman, A., Carlin, J.B., Stern, H.S. & Rubin, D.B. Bayesian Data Analysis, (Chapman & Hall, London, 2003).
12. Karlsson, M. et al. Pharmacokinetic/pharmacodynamic modelling in oncological drug development. Basic Clin. Pharmacol. Toxicol. 96, 206–211 (2005).
13. Latz, J., Karlsson, M., Rusthoven, J., Ghosh, A. & Johnson, R. A semimechanistic–physiologic population pharmacokinetic/ pharmacodynamic model for neutropenia following pemetrexed therapy. Cancer
Chemother. Pharmacol. 57, 412–426 (2006).
14. Sandstrom, M., Lindman, H., Nygren, P., Lidbrink, E., Bergh, J. & Karlsson, M.O. Model describing the relationship between pharmacokinetics and
hematologic toxicity of the Epirubicin–Docetaxel regimen in breast cancer patients. J. Clin. Oncol. 23, 413–421 (2005).
15. Troconiz, I. et al. Phase I dose-finding study and a pharmacokinetic/ pharmacodynamic analysis of the neutropenic response of intra-venous diflomotecan in patients with advanced malignant
tumours. Cancer Chemother. Pharmacol. 57, 727–735 (2006).
16. Sandstrom, M., Lindman, H., Nygren, P., Johansson, M., Bergh, J. & Karlsson, M.O. Population analysis of the pharmacokinetics and the haematological toxicity of the
fluorouracil–epirubicin–cyclophosphamide regimen in breast cancer patients. Cancer Chemother. Pharmacol. 58, 143–156 (2006).
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4 meters to inches (4 m to in)
Here we will show you how to convert 4 meters to inches (4 m to in). We will start by creating a meters to inches formula, and then use the formula to convert 4 meters to inches.
One meter is exactly 5000/127 inches. Therefore, to convert meters to inches, you multiply meters by 5000/127. Here is the formula:
Meters × 5000/127 = Inches
To convert 4 meters to inches, we enter 4 into our formula to get the fractional answer and the decimal answer, like so:
Meters × 5000/127 = Inches
4 × 5000/127 = 20000/127
4 meters = 20000/127 inches
4 meters ≈ 157.4803 inches Meters to Inches Converter
Converting 4 meters to inches is not all we can do. Here you can convert another length of meters to inches.
4.01 meters to inches
Here is the next length in meters that we have converted to inches for you.
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Intrinsic distortion measurementsIntrinsic distortion measurements
Intrinsic distortion measurements
This post lists some useful formulas for evaluating the distortion of maps between triangle meshes. The distortions considered here are intrinsic, meaning they only depend on the change in triangle
edge lengths. Conveniently, they also happen to have fairly simple formulas in terms of the triangle edge lengths.
There are many ways to measure the distortion of a map \(\phi : M \to N\) between surfaces. The general strategy is to compute the Jacobian \(J_\phi\), which is a \(2\times 2\) matrix, and consider
different functions of its singular values \(\sigma_1, \sigma_2\). For instance, the product \(\sigma_1\sigma_2\) measures the area distortion of \(\phi\) (as it is simply the determinant \(\det J_\
phi\)), while the ratio \(\sigma_1 / \sigma_2\) measures the amount of anisotropic stretching induced by \(\phi\). (See e.g. section 2 of Khodakovsky et al. [ 2003; free version] for more details.)
A piecewise-linear map between triangle meshes, deforms each triangle by a linear map, so its distortion is constant on each triangle. Below, I give some formulas for computing such per-triangle
distortions intrinsically, that is, computing the distortions using only the lengths of the initial and deformed triangles. In each formula, I refer to the triangle's vertices as \(i, j, k\). The
initial edge lenths are denoted \(\ell_{ij}, \ell_{jk}, \ell_{ki}\), and the corner angles are denoted \(\alpha_i, \alpha_j, \alpha_k\). Quantities measured after deformation are denoted \(\tilde \
ell_{ij}\), etc.
Area Distortion
The area distortion \(\sigma_1\sigma_2\) is simply given by the ratio of the deformed triangle's area to the original triangle's area. Using Heron's formula, one can show that the area of a triangle
is given by \[\text{area}_{ijk} := \tfrac{1}{2\sqrt2}\sqrt{\left(\ell^2\right)^T \!\!\! A \ell^2},\] where \(\ell^2\) denotes the vector of squared edge lengths $(\ell_{ij}^2, \ell_{jk}^2, \ell_{ki}^
2)^T$, and \(A\) is the matrix \[ A = \frac 12 \begin{pmatrix}-1 & 1 & 1 \\ 1 & -1 & 1\\ 1 & 1 & -1\end{pmatrix}.\] Hence, the area distortion is given by \[\sigma_1\sigma_2 = \sqrt{\frac{(\tilde \
ell^2)^T A \tilde \ell^2}{(\ell^2)^T A \ell^2}}.\]
Symmetrized Anisotropic Distortion
Before considering the anisotropic distortion \(\sigma_1/\sigma_2\) itself, we begin with a symmetrized version \(\frac{\sigma_1}{\sigma_2} + \frac{\sigma_2}{\sigma_1}\), which is given by a similar
formula: \[ \frac{\sigma_1}{\sigma_2} + \frac{\sigma_2}{\sigma_1} = \frac{\left(\ell^2\right)^T \!\!\! A \tilde \ell^2}{\sqrt{\left(\ell^2\right)^T \!\!\! A\, \ell^2}\sqrt{(\tilde \ell^2)^T \! A \
tilde \ell^2}}. \] This formula can also be written directly in terms of the angles \(\alpha_i\) as \[ \frac{\sigma_1}{\sigma_2} + \frac{\sigma_2}{\sigma_1} = \begin{pmatrix} \cot \alpha_i\\\cot\
alpha_j\\\cot\alpha_k\end{pmatrix}^T A^{-1} \begin{pmatrix} \cot \tilde\alpha_i\\\cot\tilde\alpha_j\\\cot\tilde\alpha_k\end{pmatrix}, \] where \(A^{-1}\) is given by \[ A^{-1} = \begin{pmatrix} 0 & 1
& 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.\] The equivalence of these two expressions follows from the identity \[ \begin{pmatrix} \cot \alpha_i \\ \cot \alpha_j \\ \cot\alpha_k \end{pmatrix} = \
frac {A^{-1}\ell^2} {\sqrt{\left(\tilde \ell^2\right)^T \!\!\! A \tilde \ell^2}}, \] (which can itself be derived using the law of cosines, the sine formula for area, and the definition that $\cot \
alpha = \tfrac{\cos\alpha}{\sin\alpha}$.)
A derivation of the cotan formula for distortion, courtesy of Boris Springborn, can be found here, and connections to hyperbolic geometry are discussed, e.g., in p.11 of Joshua Bowman's PhD thesis
with John H. Hubbard.
Anisotropic Distortion
The anisotropic distortion can easily be computed from the symmetrized anisotropic distortion by solving the quadratic equation \(y = x + \frac 1x\). Concretely, if the symmetrized distortion is
given by \(d_s\), then the ordinary anisotropic distortion is given by \[ \frac{\sigma_1}{\sigma_2} = \frac 12 \left(d_s + \sqrt{d_s^2-4}\right) \]
Other distortions
Once you have computed the distortions \(\sigma_1\sigma_2\) and \(\sigma_1 / \sigma_2\), then you can find the values of the two singular values \(\sigma_1\) and \(\sigma_2\) by multiplying and
dividing the distortion values respectively. These singular values can then be used to evaluate many other measurements of distortion.
For discussion of intrinsic calculations for distortion in the context of elasticity, see Appendix B of Sassen et al. [2020; arxiv version]
No comments:
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Counting at a Glance - DREME For Teachers
What Is Counting?
Counting helps us answer the question “How many?” This includes things we can see and touch as well as things we can’t see, like days of the week. Counting also helps us compare sets of things so we
can figure out if there is more or less of something. This is important if I think you got more cookies than I did!
Why Is Learning about Counting Important?
Counting is an important foundation in mathematics. Many math skills build on children’s ability to count. Counting is helpful for problem solving and is the foundation of addition, subtraction,
multiplication, and division.
What Do Children Need to Know About Counting?
To count correctly, children need to be able to:
• Know and use number words in order (“One, two, three…”).
• Use each number word only once as they count each thing (a skill that is called one-to-one correspondence).
• Know that the last number word they say when they are done counting is how many things there are (“One, two, three, four. There are four doggies in the park!”).
• Know that it doesn’t matter in which order you count things, there will always be the same amount.
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Correlation Coefficient Probability Formulas
Cumulative distribution function (CDF) for the t-distribution:
Variable definitions:
v = degrees of freedom
t = upper limit of integration
I = regularized lower incomplete beta function
Lower incomplete beta function:
Pearson correlation t-value:
Variable definitions:
r = Pearson correlation coefficient
n = total sample size
Regularized lower incomplete beta function:
Variable definitions:
the numerator = lower incomplete beta function
the denominator = beta function
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Back in November last year, Microsoft announced several new functions, including PIVOTBY, and eta lambdas such as PERCENTOF. On the quiet, they have added a new argument which makes PIVOTBY just that
little bit more powerful.
At the time of writing, they are rolling out to users enrolled in the beta channel for Windows Excel and Mac Excel. Don’t be upset if you don’t get this new update straight away. But first, let’s
have a little refresher…
eta Lambdas
These “eta reduced lambda” functions may sound scary, but they make the world of dynamic arrays more accessible to the inexperienced. They help make the other three functions simpler to use. Dynamic
array calculations using basic aggregation functions often require syntax such as
LAMBDA(x, SUM(x))
LAMBDA(y, AVERAGE(y))
However, given x and y (above) are merely dummy variables, an “eta lambda” function simply replaces the need for this structure with the so-easy-anyone-can-understand-it syntax of
Even I can do it. For example, consider the following formula in cell G17 below:
This sums the range G13:J16 by column using that LAMBDA(x, SUM(x)) trick. But there is no need for this anymore, viz.
That’s much simpler and many one argument functions may now be turned into eta lambdas (and one or two other functions too).
This function can be used in conjunction with PIVOTBY (below) or on its own. This is used to return the percentage that a subset makes up of a given dataset. It is logically equivalent to
SUM(subset) / SUM(everything)
It sums the values in the subset of the dataset and divides it by the sum of all the values. It has the following syntax:
=PERCENTOF(data_subset, data_all)
The arguments are as follows;
• data_subset: this is required, and represents the values that are in the data subset
• data_all: this too is required and denotes the values that make up the entire set.
You can use it, for example, with GROUPBY:
Alternatively, it may be used on its own:
The reason for this article is that PIVOTBY has changed. It has added a new, final argument: relative_to – but let’s back up first.
The PIVOTBY function allows you to create a summary of your data via a formula too, akin to a formulaic PivotTable. It supports grouping along two axes and aggregating the associated values. For
instance, if you had a table of sales data, you might generate a summary of sales by state and year.
It should be noted that PIVOTBY is a function that returns an array of values that can spill to the grid. Furthermore, at this stage, not all features of a PivotTable appear to be replicable by this
The syntax of the PIVOTBY function is:
PIVOTBY(row_fields, col_fields, values, function, [field_headers], [row_total_depth], [row_sort_order], [col_total_depth], [col_sort_order], [filter_array], [relative_to])
It has the following arguments:
• row_fields: this is required, and represents a column-oriented array or range that contains the values which are used to group rows and generate row headers. The array or range may contain
multiple columns. If so, the output will have multiple row group levels
• col_fields: also required, and represents a column-oriented array or range that contains the values which are used to group columns and generate column headers. The array or range may contain
multiple columns. If so, the output will have multiple column group levels
• values: this is also required, and denotes a column-oriented array or range of the data to aggregate. The array or range may contain multiple columns. If so, the output will have multiple
• function: also required, this is an explicit or eta reduced lambda (e.g. SUM, PERCENTOF, AVERAGE,COUNT) that is used to aggregate values. A vector of lambdas may be provided. If so, the output
will have multiple aggregations. The orientation of the vector will determine whether they are laid out row- or column-wise
• field_headers: this and the remaining arguments are all optional. This represents a number that specifies whether the row_fields, col_fields and values have headers and whether field headers
should be returned in the results. The possible values are:
□ Missing: Automatic
□ 0: No
□ 1: Yes and don't show
□ 2: No but generate
□ 3: Yes and show
It should be noted that “Automatic” assumes the data contains headers based upon the values argument. If the first value is text and the second value is a number, then the data is assumed to have
headers. Fields headers are shown if there are multiple row or column group levels
• row_total_depth: this optional argument determines whether the row headers should contain totals. The possible values are:
□ Missing: Automatic, with grand totals and, where possible, subtotals
□ 0: No Totals
□ 1: Grand Totals
□ 2: Grand and Subtotals
□ -1: Grand Totals at Top
□ -2: Grand and Subtotals at Top
It should be noted that for subtotals, row_fields must have at least two [2] columns. Numbers greater than two [2] are supported provided row_field has sufficient columns
• row_sort_order: again optional, this argument denotes a number indicating how rows should be sorted. Numbers correspond with the columns in row_fields followed by the columns in values. If the
number is negative, the rows are sorted in descending / reverse order. A vector of numbers may be provided when sorting based upon only row_fields
• col_total_depth: this optional argument determines whether the column headers should contain totals. The possible values are:
□ Missing: Automatic, with grand totals and, where possible, subtotals
□ 0: No Totals
□ 1: Grand Totals
□ 2: Grand and Subtotals
□ -1: Grand Totals at Top
□ -2: Grand and Subtotals at Top
It should be noted that for subtotals, col_fields must have at least two [2] columns. Numbers greater than two [2] are supported provided col_field has sufficient columns
• col_sort_order: again optional, this argument denotes a number indicating how they should be sorted. Numbers correspond with the columns in col_fields followed by the columns in values. If the
number is negative, these are sorted in descending / reverse order. A vector of numbers may be provided when sorting based upon only col_fields
• filter_array: this is now the penultimate optional argument, it represents a column-oriented one-dimensional array of Boolean values [1, 0] that indicate whether the corresponding row of data
should be considered. It should be noted that the length of the array must match the length of row_fields and col_fields
• relative_to: this new, final argument allows you to summarise functions relative to row and column totals or the grand total. Five alternatives are possible:
□ 0: Column Totals (default) (Screentip: Calculation performed relative to all values in column)
□ 1: Row Totals (Calculation performed relative to all values in row)
□ 2: Grand Total (Calculation performed relative to all values)
□ 3: Parent Column Total (Calculation performed relative to all values in column parent)
□ 4: Parent Row Total (Calculation performed relative to all values in row parent).
Let’s look at PIVOTBY using PERCENTOF, highlighting this new relative_to final argument. You can follow along with the attached Excel file.
Consider the following Table (CTRL + T) called Data (truncated):
Here, we have two parent / child relationships:
• Year and Quarter
• Category and Item.
From our previous article on PIVOTBY, we can create a formulaic alternative to a PivotTable (with crafty formatting) using the following formula:
Note that each column of sales is represented as a percentage of that column (including the Total column). Whilst it was a great start, Microsoft received feedback that end users wanted to see
percentages summarised in alternative ways – and that is what has been addressed here.
This newly introduced final argument, relative_to, behaves the same in scenario 0: Column Totals. This is the default view:
It is clear to see this is identical to the first output. But let’s see what happens when we start playing with the final argument. Let’s change this value to 1: Row Totals.
Now, each row of sales is represented as a percentage of that row (including the Total row), viz.
If you wish, you can show the sales as a percentage of the Grand Total, using 2: Grand Total:
There are still two further scenarios – and this is why our example contained two parent / child relationships. The first is 3: Parent Column Total:
Here, the Total column is 100% throughout. It is a little confusing as, if anything, it looks a little like Scenario 1: Row Totals. This is because the column here refers to the headings in each
column, i.e. Year and Quarter. You can see that for any row the sum of the four quarters for any given year totals 100% (including the Total row).
Finally, Scenario 4: Parent Row Total considers the other parent / child relationship:
In this final illustration, the Total row is 100% throughout. This looks similar to the default Scenario 0: Column Totals. This is because the row here refers to the headings in each row, i.e.
Category and Item. You can see that for any row the sum of any category for any given Quarter and Year totals 100% (including the Total column).
Word to the Wise
Starting with RANDARRAY, Microsoft continues to venture into new territory by tinkering with new functions and features whilst they remain in beta. Previously, revising a function’s signature /
syntax was unheard of. Here at SumProduct, we’re not complaining. The software giant has been collating formula usage and explicit feedback to determine what is missing / needs revising – and then
done something about it.
If only they had done that with MATCH many years ago!!
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Name for sequence y=0[0]0. y=1[1]1, y=2[2]2, y= n[n]n, y=x[x]x, y=i[i]i, y=z[z]z etc?
In another thread I found interesting expressions (y is a result of those operations, whatever number type it may take):
y=1[1]1 = 1
y=2[2]2 = 2*2=4
y=3[3]3 = 3^3^3 = 19683
y= i[ i]i
(this may be left out : y= q[q]q where q is quaternion...)
The question is, numbers 1, 2, 3, 4, ..n, ...x, i. , z .... q... are "something" - inverted hyperoperations of y.
How should we call this something and can we find these given y?
y=x[x]x, what is x if y is given and how many x will fit the equation?
04/27/2008, 02:41 PM
If we limit ourselves to y = n[n]n, with n natural, the sequence is known and it is called: "The Ackermann Sequence". See:
I suppose it is so, if I am not confusing things. The numbers so obtained are the "Ackermann numbers". Very big .... !
04/27/2008, 02:59 PM (This post was last modified: 04/27/2008, 04:18 PM by GFR.)
Dear Ivars:
0[0]0 = ??? (I try to avoid troubles ...
1[1]1 = 1+1 = 2
2[2]2 = 2*2 = 4
3[3]3 = 3^3 = 27
4[4]4 = 4#4 = 4^[4^[4^4]]] .... uuuuuhhhhh !!!
oo[w]oo = ...... infinite-omega-infinite (Qickfur said that it is still enumerable! I think he is right!). Actually, we could write it as oo[oo]oo, but I think that the entity between the brackets is
an ordinal. Am I wrong? Maybe yes !
04/27/2008, 04:23 PM
Therefore, the Ackermann numbers can be noted as: y = Ack(n) = n[n]n, where "n" is the order of Ack(n). If Ack(n) is a function of n, then n = InvAck(y) is its inverse.
04/27/2008, 04:42 PM (This post was last modified: 04/27/2008, 05:11 PM by Ivars.)
Hi GFR
Thanks . And thanks for corrections as well. Sometimes I am too lazy just too learn the basics to the last detail...
But finally I understood what is Ackermann function as well, in Your clear and simple bracket notation. Quite impressive function.Very fast. And its Inverse seems to be very slow.
But now suddenly I am interested in i[i]i = Ack[i]. What shall I do
04/27/2008, 05:18 PM
Mmmmm!! Let's think about that. And how about Ack(0). Does it give 0, 1, 2 (?!) or indeterminate or NAN (Not A Number). Bah, time will solve all that
04/28/2008, 09:13 AM (This post was last modified: 04/28/2008, 09:15 AM by Ivars.)
And Ack (-1) , Ack (-n) Ack (-x) , Ack (-I) ..... as well.
What was Ack (+-oo) = +-oo[+-oo]+-oo? The same poor single infinity oo which has to hold so much into it? Can not believe it.
05/02/2008, 06:10 PM (This post was last modified: 05/02/2008, 06:12 PM by andydude.)
No, the Ackermann numbers (from MathWorld) are defined as \( n[n+2]n \), because they are defined by arrow notation, and not box notation.
The numbers defined by \( n[n]n \) have no name.
Andrew Robbins
05/02/2008, 10:33 PM (This post was last modified: 05/02/2008, 10:36 PM by GFR.)
Dear Andrew!
andydude Wrote:No, the Ackermann numbers (from MathWorld) are defined as \( n[n+2]n \), because they are defined by arrow notation, and not box notation.
The numbers defined by \( n[n]n \) have no name.
Actually, in the Department of Engineering and Computer Science of the MIT there is a "research line" covering a rapidly increasing sequence of numbers, called "The Ackermann Sequence", defined as :
Ack(n) = n[n]n.
Of course, the much better known sequence defined by the Knuth's up-arrows notation is starting by the exponentiation rank 1[3]1 = 1^1 = 1 and it defines the "Sequence of the Ackermann Numbers",
something like:
An(n) = n[n+2]n.
In my opinion, the first sequence (the Ackermann Sequence, proposed by Prof. Scott Aaronson, MIT) is more compatible with the subject that we are studying. The second and better known version is
strongly influenced by the Knuth's arrow notation and gives a kind of ultra-exponential sequence, completely ignoring hyperranks 1 and 2 (not to mention ... 0!).
It always goes without saying that the "Ackermann Function" is a function of two variables (attention please: not a two-valued function, because they are strictly forbidden
A(0, n) = n+1
A(m, 0) = A(m-1, 1)
A(m, n) = A(m-1, A(m, n-1)).
As we can see the situation is both extremely clear and very confused, from the terminological point of view.
Unfortunately, due to other personal and family priorities, from to-day I am obliged to give much lesser time to these important and intersting subjects.
It was nice to discuss with you.
05/03/2008, 02:18 PM
GFR Wrote:In my opinion, the first sequence (the Ackermann Sequence, proposed by Prof. Scott Aaronson, MIT) is more compatible with the subject that we are studying.
Indeed I wondered whether there are different definitions of Ackermann numbers out there.
Quote:It always goes without saying that the "Ackermann Function" is a function of two variables (attention please: not a two-valued function, because they are strictly forbidden
A(0, n) = n+1
A(m, 0) = A(m-1, 1)
A(m, n) = A(m-1, A(m, n-1)).
But be careful, this is what novadays is called Ackermann function. Its main purpose is a simplification of the original Ackermann function - which was indeed a 3 argument function and starting with
addition as 0th operation - for proving that there are recursive functions that are not primitive recursive. You can read about the original Ackermann function, i.e. the function that Ackermann
defined himself, in his article [1].
[1] Wilhelm Ackermann (192
Quote:Unfortunately, due to other personal and family priorities, from to-day I am obliged to give much lesser time to these important and intersting subjects.
It was nice to discuss with you.
Hope we see you again here soon!
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Two Phase Flow In Pipes Beggs And Brill Pdf !!EXCLUSIVE!! Do
Two Phase Flow In Pipes Beggs And Brill Pdf !!EXCLUSIVE!! Download
Hall [11] carried out experimental study on gas-oil-water three-phase flow in horizontal pipes. He modeled the three-phase stratified flow by using the obtained hold-up to calculate the transition
from stratified flow to slug flow. The model was compared with experimental data which showed that the transition occurred at higher gas velocities than those predicted by the model. The oil layer
was believed to be the reason, because it travels at a higher mean velocity since its lower interface was in contact with a moving water layer and not a fixed wall.
Two Phase Flow In Pipes Beggs And Brill Pdf Download
Hold-ups of stratified three-phase flow pattern of gas-oil-water was calculated by Taitel et al. [15]. Three steady state solutions for the upward inclined case were obtained. The only stable
configuration was the one with the thinnest liquid layer. The essential step for the calculation of the hold-up, pressure drop, and transition criteria of the flow pattern was found to be the
information regarding the liquid and oil levels in the pipe.
Zhang and Sarica [21] developed a model called unified model to predict the flow pattern and pressure gradient of three-phase gas-oil-water which was an improvement on the earlier unified model of
Zhang et al. [22]. The model was compared with experimental measurements of three-phase gas/oil/water pipe flows. The three-phase unified model gave better predictions than the unified model of gas/
liquid two-phase pipe flow when compared with the experimental measurements of Khorr [22] for stratified gas/oil/water flow in horizontal and 1.5 downward pipes. Similar performance was seen when the
two models were also compared with the experimental measurements of Hall [11] on pressure gradients for three-phase slug flow in a horizontal pipe.
Liu, Z., Liao, R., Luo, W., & Ribeiro, J. (2019). A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities. Scientia Iranica, 26(3), 1529-1539. doi: 10.24200/
Z. Liu; R. Liao; W. Luo; J.X.F. Ribeiro. "A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities". Scientia Iranica, 26, 3, 2019, 1529-1539. doi: 10.24200/
Liu, Z., Liao, R., Luo, W., Ribeiro, J. (2019). 'A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities', Scientia Iranica, 26(3), pp. 1529-1539. doi:
Liu, Z., Liao, R., Luo, W., Ribeiro, J. A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities. Scientia Iranica, 2019; 26(3): 1529-1539. doi: 10.24200/
Multiphase flow occurs in oil/gas, chemical, civil, and nuclear industries. The dominant occurrence of gas-oil-water three-phase flow in the petroleum industry requires sound knowledge of the
behavior of multiphase flow. The most important characteristic of multiphase flow is its flow pattern (physical distribution of the phases within the enclosure they flow through) and the pressure
gradient along the horizontal pipeline. In this regard, it is imperative to fully understand and study the flow rates, flow regimes/patterns, liquid-hold-up/water cut (WC), pressure gradients, and
volume fractions of gas, oil, and water going into the pipelines during transportation of petroleum products. The water cut (WC) is the water quantity at the pipe inlet as volume percentage of the
total inlet volumetric flow rate. The water cut is always the basis for pipelines and equipment design. During the transportation of the multiphase flow, water in the system starts separation and
thereby accumulates at the pipe bottom and that amount of water is being referred to as local water contents, local water, or water hold-up. Also, it is important to better understand/predict/
investigate the flow characteristics during petroleum production at different flow conditions such as the geometrical configuration of the pipeline, the physical properties of the fluids, and flow
rates. There is a need to accurately investigate and predict the flow configurations and the pressure drop [1, 2].
The experiments were carried out in an acrylic pipe to visualize the flow patterns. The test fluids used were Safrasol D80 oil, tap water and air (properties of these fluids are mentioned earlier in
Introduction). The three different fluids were passed into the horizontal pipeline and the flow patterns were observed while the pressure gradients were measured/recorded (using pressure transducers
and U-tube manometers). A total of 377 data points were acquired and studied. The matrix range for three-phase flow of air-oil-water experiments is shown in Table 1. The effects of water cut, liquid
velocity, gas velocity on flow patterns, and pressure drop have been studied.
Preface 1 Introduction 1.1 Multi-phase flow assurance 1.1.1 General 1.1.2 Nuclear reactor multi-phase models 1.1.3 Multi-phase flow in the petroleum industry 1.2 Two-phase flow 1.2.1 Flow regimes in
horizontal pipes 1.2.2 Slugging 1.2.3 Flow regimes in vertical pipes 1.2.4 Flow regime maps 1.2.5 Flow in concentric and eccentric annulus 1.3 Three and four-phase flow 1.3.1 Types of three-phase and
quasi four-phase flow 1.3.2 Three-phase flow regimes 1.4 Typical flow assurance tasks 1.5 Some definitions 1.5.1 General 1.5.2 Volume fraction, holdup and water cut 1.5.3 Superficial velocity 1.5.4
Mixture velocity and density 1.5.5 Various sorts of pipes
6 Solving the two-phase three-fluid equations 6.1 Steady-state incompressible isothermal flow 6.2 Comparing with measurements 6.3 Steady-state compressible flow 6.4 Transient three-fluid two-phase
annular flow model
8 Including boiling and condensation 8.1 Extending the three-fluid two-phase model 8.2 Mass conservation 8.3 Momentum conservation 8.3.1 Main equations 8.3.2 Some comments on interface velocity 8.4
Energy equation 8.5 Pressure equation 8.6 Mass transfer from liquid (film and droplets) to gas 8.7 Slip between gas and droplets in annular flow 8.8 Droplet deposition in annular flow 8.8.1 The
Wallis-correlation 8.8.2 The Oliemans, Pots, and Trope-correlation 8.8.3 The Ishii and Mishima-correlation 8.8.4 The Sawant, Ishii, and Mori-correlation 8.9 Dispersed bubble flow 8.10 Slug flow
10 Multi-phase flow heat exchange 10.1 Introduction 10.2 Classical, simplified mixture correlations 10.3 Improved correlations for all flow regimes in horizontal two-phase flow 10.4 Flow
regime-dependent approximation for horizontal flow 10.5 Flow-regime dependent two-phase correlations for inclined pipes 10.6 Dispersed bubble flow 10.7 Stratified flow 10.8 Slug flow
11 Flow regime determination 11.1 The Beggs & Brill flow regime map 11.2 The Taitel & Duckler horizontal flow model 11.3 Flow regimes in vertical flow 11.3.1 Bubble to slug transition 11.3.2
Transition to dispersed-bubble flow 11.3.3 Slug to churn transition 11.3.4 Transition to annular flow 11.4 Flow regimes in inclined pipes 11.4.1 Bubble to slug transition 11.4.2 Transition to
dispersed-bubble flow 11.4.3 Intermittent to annular transition 11.4.4 Slug to churn transition 11.4.5 Downward inclination 11.5 The minimum-slip flow regime criterion
13 Two-phase liquid-liquid flow 13.1 General 13.2 Emulsion viscosity 13.3 Phase inversion criteria 13.4 Stratified flow friction modeling 14 Two-phase liquid-solid flow 14.1 General about
liquid-solid flow 14.2 The building up of solids in the pipeline 14.3 Minimum transport velocity
15 Three-phase gas-liquid-liquid flow 15.1 Introduction 15.2 Main equations 15.3 Three-layer stratified flow 15.4 Incompressible steady-state slug flow model 15.5 Combining the different flow regimes
into a unified model
19 Various subjects 19.1 Multi-phase flowmeters and flow estimators 19.2 Gas lift 19.2.1 General 19.2.2 Oil & water-producing well with gas lift: Simulation example 19.3 Slug catchers
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“Once Upon a Time”…with a twist
The Noncommuting-Charges World Tour (Part 1 of 4)
This is the first part in a four part series covering the recent Perspectives article on noncommuting charges. I’ll be posting one part every ~6 weeks leading up to my PhD thesis defence.
Thermodynamics problems have surprisingly many similarities with fairy tales. For example, most of them begin with a familiar opening. In thermodynamics, the phrase “Consider an isolated box of
particles” serves a similar purpose to “Once upon a time” in fairy tales—both serve as a gateway to their respective worlds. Additionally, both have been around for a long time. Thermodynamics
emerged in the Victorian era to help us understand steam engines, while Beauty and the Beast and Rumpelstiltskin, for example, originated about 4000 years ago. Moreover, each conclude with important
lessons. In thermodynamics, we learn hard truths such as the futility of defying the second law, while fairy tales often impart morals like the risks of accepting apples from strangers. The parallels
go on; both feature archetypal characters—such as wise old men and fairy godmothers versus ideal gases and perfect insulators—and simplified models of complex ideas, like portraying clear moral
dichotomies in narratives versus assuming non-interacting particles in scientific models.^1
Of all the ways thermodynamic problems are like fairytale, one is most relevant to me: both have experienced modern reimagining. Sometimes, all you need is a little twist to liven things up. In
thermodynamics, noncommuting conserved quantities, or charges, have added a twist.
Unfortunately, my favourite fairy tale, ‘The Hunchback of Notre-Dame,’ does not start with the classic opening line ‘Once upon a time.’ For a story that begins with this traditional phrase,
‘Cinderella’ is a great choice.
First, let me recap some of my favourite thermodynamic stories before I highlight the role that the noncommuting-charge twist plays. The first is the inevitability of the thermal state. For example,
this means that, at most times, the state of most sufficiently small subsystem within the box will be close to a specific form (the thermal state).
The second is an apparent paradox that arises in quantum thermodynamics: How do the reversible processes inherent in quantum dynamics lead to irreversible phenomena such as thermalization? If you’ve
been keeping up with Nicole Yunger Halpern‘s (my PhD co-advisor and fellow fan of fairytale) recent posts on the eigenstate thermalization hypothesis (ETH) (part 1 and part 2) you already know the
answer. The expectation value of a quantum observable is often comprised of a sum of basis states with various phases. As time passes, these phases tend to experience destructive interference,
leading to a stable expectation value over a longer period. This stable value tends to align with that of a thermal state’s. Thus, despite the apparent paradox, stationary dynamics in quantum systems
are commonplace.
The third story is about how concentrations of one quantity can cause flows in another. Imagine a box of charged particles that’s initially outside of equilibrium such that there exists gradients in
particle concentration and temperature across the box. The temperature gradient will cause a flow of heat (Fourier’s law) and charged particles (Seebeck effect) and the particle-concentration
gradient will cause the same—a flow of particles (Fick’s law) and heat (Peltier effect). These movements are encompassed within Onsager’s theory of transport dynamics…if the gradients are very small.
If you’re reading this post on your computer, the Peltier effect is likely at work for you right now by cooling your computer.
What do various derivations of the thermal state’s forms, the eigenstate thermalization hypothesis (ETH), and the Onsager coefficients have in common? Each concept is founded on the assumption that
the system we’re studying contains charges that commute with each other (e.g. particle number, energy, and electric charge). It’s only recently that physicists have acknowledged that this assumption
was even present.
This is important to note because not all charges commute. In fact, the noncommutation of charges leads to fundamental quantum phenomena, such as the Einstein–Podolsky–Rosen (EPR) paradox,
uncertainty relations, and disturbances during measurement. This raises an intriguing question. How would the above mentioned stories change if we introduce the following twist?
“Consider an isolated box with charges that do not commute with one another.”&
This question is at the core of a burgeoning subfield that intersects quantum information, thermodynamics, and many-body physics. I had the pleasure of co-authoring a recent perspective article in
Nature Reviews Physics that centres on this topic. Collaborating with me in this endeavour were three members of Nicole’s group: the avid mountain climber, Billy Braasch; the powerlifter, Aleksander
Lasek; and Twesh Upadhyaya, known for his prowess in street basketball. Completing our authorship team were Nicole herself and Amir Kalev.
To give you a touchstone, let me present a simple example of a system with noncommuting charges. Imagine a chain of qubits, where each qubit interacts with its nearest and next-nearest neighbours,
such as in the image below.
The figure is courtesy of the talented team at Nature. Two qubits form the system& S& of interest, and the rest form the environment& E. A qubit’s three spin components,& σ[a=x,y,z], form the local
noncommuting charges. The dynamics locally transport and globally conserve the charges.
In this interaction, the qubits exchange quanta of spin angular momentum, forming what is known as a Heisenberg spin chain. This chain is characterized by three charges which are the total spin
components in the x, y, and z directions, which I’ll refer to as Q[x], Q[y], and Q[z], respectively. The Hamiltonian H conserves these charges, satisfying [H, Q[a]] = 0 for each a, and these three
charges are non-commuting, [Q[a], Q[b]] ≠ 0, for any pair a, b ∈ {x,y,z} where a≠b. It’s noteworthy that Hamiltonians can be constructed to transport various other kinds of noncommuting charges. I
have discussed the procedure to do so in more detail here (to summarize that post: it essentially involves constructing a Koi pond).
This is the first in a series of blog posts where I will highlight key elements discussed in the perspective article. Motivated by requests from peers for a streamlined introduction to the subject,
I’ve designed this series specifically for a target audience: graduate students in physics. Additionally, I’m gearing up to defending my PhD thesis on noncommuting-charge physics next semester and
these blog posts will double as a fun way to prepare for that.
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Information theory
The information theory is a mathematical theory in the field of probability theory and statistics , which the US-American mathematician Claude Shannon back. It deals with terms such as information
and entropy , information transfer , data compression and coding, and related topics.
In addition to mathematics, computer science and telecommunications , the theoretical consideration of communication through information theory is also used to describe communication systems in other
areas (e.g. media in journalism , the nervous system in neurology , DNA and protein sequences in molecular biology , knowledge in information science and documentation ).
Shannon's theory uses the term entropy to characterize the information content (also called information density) of messages. The more irregular a message is, the higher its entropy. In addition to
the concept of entropy, the Shannon-Hartley law according to Shannon and Ralph Hartley is fundamental for information theory . It describes the theoretical upper limit of the channel capacity , i.e.
the maximum data transmission rate that a transmission channel can achieve without transmission errors depending on the bandwidth and the signal-to-noise ratio .
Claude Shannon in particular made significant contributions to the theory of data transmission and probability theory in the 1940s to 1950s .
He wondered how one can ensure loss-free data transmission via electronic (now also optical) channels . It is particularly important to separate the data signals from the background noise.
In addition, attempts are made to identify and correct errors that have occurred during transmission. For this it is necessary to send additional redundant data (i.e. no additional information
carrying) data in order to enable the data receiver to verify or correct data.
It is doubtful, and also not claimed by Shannon, that his study A Mathematical Theory of Communication ("Information Theory ") published in 1948 is of substantial importance for questions outside of
communications engineering. The concept of entropy that he uses and is connected to thermodynamics is a formal analogy for a mathematical expression. In general, information theory can be defined as
engineering theory on a high level of abstraction. It shows the trend towards the scientification of technology, which led to the development of engineering.
The reference point of Shannon's theory is the accelerated development of electrical communications technology with its forms of telegraphy, telephony, radio and television in the first half of the
20th century. Before and next to Shannon, Harry Nyquist , Ralph Hartley and Karl Küpfmüller also made significant contributions to the theory of communications engineering. Mathematical
clarifications of relevance to information theory were provided by Norbert Wiener , who also helped it to gain considerable publicity in the context of his reflections on cybernetics.
An overarching question for communications engineers was how economically efficient and interference-free communications can be achieved. The advantages of modulation have been recognized; H.
changing the form of the message by technical means. In the technical context, two basic forms of messages - continuous and discrete - can be distinguished. These can be assigned the common forms of
presentation of information / messages: writing (discrete), language (continuous) and images (continuous).
At the end of the 1930s, there was a technical breakthrough when, with the help of pulse code modulation , it was possible to discreetly display a message that was present as a continuum in a
satisfactory approximation. With this method it became possible to telegraph speech. Shannon, who worked for Bell Telephone Laboratories, was familiar with technical developments. The great
importance of his theory for technology lies in the fact that he defined information as a “physical quantity” with a unit of measurement or counting, the bit . This made it possible to quantitatively
exactly compare the effort required for the technical transmission of information in various forms (sounds, characters, images), to determine the efficiency of codes and the capacity of information
storage and transmission channels.
The definition of the bit is a theoretical expression of the new technical possibilities to transform different forms of representation of messages (information) into a common, for technical purposes
advantageous representation of the information: A sequence of electrical impulses that can be expressed by a binary code. This is ultimately the basis for information technology on a digital basis,
as well as for multimedia. That was known in principle with information theory. In practice, however, the digital upheaval in information technology only became possible later - combined with the
rapid development of microelectronics in the second half of the 20th century.
Shannon himself describes his work as a "mathematical theory of communication". It expressly excludes semantic and pragmatic aspects of the information, ie statements about the "content" of
transmitted messages and their meaning for the recipient. This means that a "meaningful" message is just as conscientiously transmitted as a random sequence of letters. Although the Shannon theory is
usually referred to as "information theory", it does not make any direct statement about the information content of transmitted messages.
More recently, attempts have been made to determine the complexity of a message no longer just by looking at the data statistically , but rather to look at the algorithms that can generate this data.
Such approaches are in particular the Kolmogorow complexity and the algorithmic depth , as well as the algorithmic information theory of Gregory Chaitin . Classical information concepts sometimes
fail in quantum mechanical systems. This leads to the concept of quantum information .
Information theory provides mathematical methods for measuring certain properties of data. The concept of information from information theory has no direct reference to semantics , meaning and
knowledge , since these properties cannot be measured using information-theoretical methods.
See also
• Claude E. Shannon: A mathematical theory of communication . Bell System Tech. J., 27: 379-423, 623-656, 1948. (Shannon's seminal paper)
□ Claude E. Shannon, Warren Weaver: Mathematical Foundations of Information Theory , [Dt. Translated from The mathematical theory of communication by Helmut Dreßler]. - Munich, Vienna:
Oldenbourg, 1976, ISBN 3-486-39851-2
• NJA Sloane, AD Wyner: Claude Elwood Shannon: Collected Papers ; IEEE Press, Piscataway, NJ, 1993.
• Christoph Arndt: Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), 2004, ISBN 978-3-540-40855-0 ,
• Siegfried Buchhaupt: The importance of communications technology for the development of an information concept in technology in the 20th century. In: Technikgeschichte 70 (2003), pp. 277–298.
• Holger Lyre: Information Theory - A Philosophical-Scientific Introduction , UTB 2289
• Werner Heise, Pasquale Quattrocchi: Information and Coding Theory. Mathematical foundations of data compression and backup in discrete communication systems , 3rd edition, Springer,
Berlin-Heidelberg 1995, ISBN 3-540-57477-8
• John R. Pierce : An Introduction to Information Theory: Symbols, Signals and Noise ; Dover Publications, Inc., New York, second edition, 1980.
• W. Sacco, W. Copes, C. Sloyer, and R. Stark: Information Theory: Saving Bits ; Janson Publications, Inc., Dedham, MA, 1988.
• Solomon Kullback: Information Theory and Statistics (Dover Books on Mathematics), 1968;
• Alexander I. Khinchin: Mathematical Foundations of Information Theory ;
• Fazlollah M. Reza: An Introduction to Information Theory , 1961;
• Robert B. Ash: Information Theory , 1965
• Thomas M. Cover, Joy A. Thomas: Elements of Information Theory (Wiley Series in Telecommunication), 1991;
Popular science introductions
• William Poundstone: The Formula of Happiness
Web links
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Printable Multiplication Table Worksheets
Mathematics, particularly multiplication, forms the cornerstone of countless scholastic techniques and real-world applications. Yet, for numerous learners, mastering multiplication can position a
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From basics like multiplying by twos to complex concepts such as three digit multiplication our multiplication worksheets help elementary school students of all ages improve this vital skill For
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Is it possible to accommodate massive photons in the framework of a gauge-invariant electrodynamics?
The construction of an alternative electromagnetic theory that preserves Lorentz and gauge symmetries, is considered. We start off by building up Maxwell electrodynamics in (3+1)D from the assumption
that the associated Lagrangian is a gauge- invariant functional that depends on the electron and photon fields and their first derivatives only. In this scenario, as well- known, it is not possible
to set up a Lorentz invariant gauge theory containing a massive photon. We show nevertheless that there exist two radically different electrodynamics, namely, the Chern- Simons and the Podolsky
formulations, in which this problem can be overcome. The former is only valid in odd space- time dimensions, while the latter requires the presence of higher- order derivatives of the gauge field in
the Lagrangian. This theory, usually known as Podolsky electrodynamics, is simultaneously gauge and Lorentz invariant; in addition, it contains a massive photon. Therefore, a massive photon, unlike
the popular belief, can be adequately accommodated within the context of a gauge- invariant electrodynamics.
Podolsky Electrodynamic; Massive Photons; Gauge- invariant Electrodynamics
Is it possible to accommodate massive photons in the framework of a gauge-invariant electrodynamics?
M.V.S. Fonseca^* * Electronic address: marcusfo@cbpf.br ** Electronic address: alfredov@cbpf.br ; A.V. Paredes^** * Electronic address: marcusfo@cbpf.br ** Electronic address: alfredov@cbpf.br
Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ
The construction of an alternative electromagnetic theory that preserves Lorentz and gauge symmetries, is considered. We start off by building up Maxwell electrodynamics in (3+1)D from the assumption
that the associated Lagrangian is a gauge-invariant functional that depends on the electron and photon fields and their first derivatives only. In this scenario, as well-known, it is not possible to
set up a Lorentz invariant gauge theory containing a massive photon. We show nevertheless that there exist two radically different electrodynamics, namely, the Chern-Simons and the Podolsky
formulations, in which this problem can be overcome. The former is only valid in odd space-time dimensions, while the latter requires the presence of higher-order derivatives of the gauge field in
the Lagrangian. This theory, usually known as Podolsky electrodynamics, is simultaneously gauge and Lorentz invariant; in addition, it contains a massive photon. Therefore, a massive photon, unlike
the popular belief, can be adequately accommodated within the context of a gauge-invariant electrodynamics.
Keywords: Podolsky Electrodynamic; Massive Photons; Gauge-invariant Electrodynamics.
Maxwell electrodynamics, or its quantum version, i.e., QED, is widely recognized as the adequate theory for the description of the electromagnetic phenomena, because of the astonishing agreement
between theory and experiment. However, it only enjoyed this high status after some of its intrinsic problems were solved. Among them, the most remarkable one is certainly the presence of divergences
or infinities, even at the classical level [1].
This aspect of the Maxwell electromagnetic theory naturally emerges when the self-energy of an elementary (charged) particle, like the electron, for example, is considered. An object of this sort has
no internal structure, which means that it must be regarded (classically) as a geometric point. Its Coulomb energy, given by
where E is the electron electric field, like the associated self-energy, diverges.
Objects having finite extension, on the other hand, such as composite particles, must be described by internal degrees of freedom since in this case the aforementioned problem, at least in principle,
does not occur. Hadronic particles, for instance, belong to this category since their static properties, like mass, are finite and, in principle, obtained through the quark dynamics.
In spite of the mentioned success of the electromagnetic theory, it remain some intriguing questions that cannot be completely answered by a simple comparison between experiment and theory. One of
most remarkable, among others, is the question of the massless character of the photon. From a theoretical point of view the existence of massive photons is perfectly compatible with the general
principles of elementary particle physics. This possibility cannot be discarded either from an experimental viewpoint. Indeed, despite the fact that a very small value for the photon mass has not
been found experimentally up to now, this does not allows to conclude that its mass must be identically zero. In fact, the more accurate experiments currently avaliable can only set up upper bounds
on the photon mass. Incidentally, the recently recommended limit published by Particle Data Group is m[γ] < 2 × 10^-25 GeV [2]. On the other hand, using the uncertainty principle, we obtain an upper
limit on the photon rest mass equal to 10^-34 GeV, which is found by assuming that the universe is 10^10 years old [3]. Nonetheless, the relevant question, from a theoretical point of view, is that a
nonvanishing value for the photon mass is incompatible with Maxwell electrodynamics.
So, we can ask ourselves whether or not it would be possible to construct a gauge-invariant electrodynamics, such as Maxwell one, but in which a massive photon could be accommodated. At first sight,
it seems that the Proca theory [4], described by the Lagrangian
where F^µν = ∂^µA^ν ∂^νA^µ, fulfills the aforementioned requirements. Lagrangian (2) leads to massive dispersion relations for the gauge boson, implying in a Yukawa potential in the static case.
Since this potential has a finite range, the electron self-energy is finite [1]. Besides, Proca electrodynamics is Lorentz invariant. However, gauge invariance is lost, which is certainly undesirable
since, as a consequence, this model would be in disagreement with the predictions of the Standard Model SU (3) × SU (2) × U (1) [5].
Other alternative models, such as the Chern-Simons [6] and the Podolsky [7] ones, can be constructed in the same vein.
In the Chern-Simons electrodynamics a coupling between the gauge field and the field strength is introduced into the Lagrangian through the Levi-Civita tensor. This coupling yelds a massive
dispersion relation for the gauge field. As a result of this mechanism, a massive photon is generated. Nevertheless, the mentioned mechanism explicitly breaks the Lorentz invariance in four
dimensions, unless a 2-form gauge field is also introduced that mixes up with the Maxwell potential. In odd dimensions, however, this model is simultaneously Lorentz and gauge invariant.
Podolsky electrodynamics, on the other hand, seems more interesting in comparison to the above cited models since it can accomaddate a massive photon without violating the Lorentz and gauge
symmetries in (3+1)D.
There are other interesting aspects of Podolsky theory that deserves to be exploited. For instance, within its context magnetic monopoles and massive photons can coexist without conflict. That is not
the case as far as the Proca model [8] is concerned.
The aim of this paper is precisely to discuss the issue of the photon mass in the framework of some outstanding electromagnetic theories. To start off, Maxwell theory is considered in section II. In
particular, it is shown in this section that this theory can be built up via simple and general assumptions; it is also demonstrated that Lorentz and gauge invariance constrain the photon mass to be
equal to zero. In section III, we discuss the Chern-Simons theory and prove that in odd dimensions the photon can acquire mass without breaking the Lorentz and gauge symmetries. In section IV the
Podolsky electromagnetic theory is analyzed. We show that within the context of this model, massive photons are allowed while the Lorentz and gauge symmetries are preserved. It is worth mentioning
that the approach to the Podolsky model we have taken in this paper may be regarded as an alternative method to those employed by A. Accioly [9] and H. Torres-Silva [10].
We shall construct Maxwell electrodynamics based on the following three assumptions:
(i)Lorentz invariance holds.
(ii)There exists a Lagrangian
In this spirit, we consider the following gauge transformations with respect, respectively, to the bosonic field
and the matter field
In the above equations β is a local gauge parameter, β = β(x).
The requirement of the invariance of the Lagrangian with respect to these transformations, δ[gauge]
Now, since β is an arbitrary parameter, we promptly obtain
Using the Euler-Lagrange equations for the ψ field in the above expression, we then find
This result clearly shows there exists a Noetherian vector current associated to the gauge symmetry
which is conserved (∂[µ]j^µ = 0).
On the other hand, to first-order in β derivatives, we have
which can be written as
This relation tells us how the gauge field must be coupled to a conserved current in the Lagrangian.
Finally, the second-order derivative terms in the gauge parameter yield the condition
which implies that the symmetric part of the derivative term in the Lagrangian must be null, i.e,
Thus, we can write
where H[µν] is a totally antisymmetric rank-two tensor. Here
Consequently, the bosonic sector of the Lagrangian is given by
The first term in Eq. (16) is related to the vector field only, and must be bilinear in A[µ]. As a consequence, one of the Lorentz indices of H[µν] must necessarily be associated to the gauge field.
The simplest choice for the kinetic term, which is quadratic in ∂[µ]A[ν], is
i.e., the tensor H[µν] can be identified with the usual electromagnetic field strength F[µν]. Taking this into account, the corresponding Lagrangian can be written in the general form
where a and b are arbitrary constants. By analyzing the equations of motion related to (17), it is trivial to see that a convenient choice for these constants is a = -b = 1, which allows us to write
which is nothing but Maxwell Lagrangian.
The field A[µ] in (18) is massless. This raises the interesting question: Could we have chosen the tensor H[µν] such that it contained a gauge-invariant mass term related to A^µ, besides the massless
term? Since in the selection of the early H[µν] we have excluded the possibility that A^µ could be massive, this is a pertinent question. Let us then discuss this possibility.
The kinetic part of the gauge field in the Lagrangian, as commented above, must have the general form
In other words, H[µν] must be a function of A[µ] and its first derivatives only. Therefore, H^µν can be written in the alternative form
where, obviously, h^µν is an antisymmetric tensor. Accordingly,
where ε^µναβ is the Levi-Civita tensor and the quantity (?)[α] is a Lorentz vector to be determined. There are two possibilities to be considered. The first one is to assume that the mentioned
quantity is a constant vector, which implies that it would play the role of a fundamental quantity of nature. In this case, the aforementioned constant vector would single out a special direction in
space-time leading, as a consequence, to a breaking of the Lorentz symmetry. The remaining choice is (?)[α] = ∂[α], which would imply that the searched quantity should be proportional to the
electromagnetic field-strength, h^µν a F^µν. Thus, we come to the conclusion that the gauge field is massless due to the two very general assumptions considered in the construction of the Lagrangian,
in addition to the Lorentz invariance.
In the preceding section we concluded that a Lagrangian which is a functional of the electron and photon fields, as well as of their first derivatives and, besides, is invariant under local gauge
transformations and consistent with the Lorentz symmetry, confers a massless character to the vector field. Our proof, however, relied upon the fact that the space-time was endowed with (3 + 1)
dimensions. Yet, it is possible to show that in odd dimensional space-times the form of the antisymmetric tensor H[µν] need not be proportional to F[µν] only. That is the case of the so-called
Chern-Simons electrodynamics. In order to obtain the Lagrangian corresponding to this theory we suppose that the same assumptions utilized in the construction of the Maxwell theory still hold. As
long as the quantity H[µν] is concerned, we consider another alternative: the space-time has (2+1) dimensions (in particular). In such a case we have to construct an antisymmetric tensor (h^µν). This
quantity can now be expressed as follows
A Lorentz invariant term can be then constructed by contracting this term with the usual electromagnetic tensor F^µν. This means that the Lagrangian can be written in the form
where a and b are arbitrary constants. Here b has dimension of mass.
The above result may be extended to any odd dimension, because we can always construct the Chern-Simons term through the contraction of a field with n Lorentz indices with its field-strength
containing n + 1 indices. The Levi-Civita tensor, on the other hand, will have 2n + 1 indices. For instance, in a 5-dimensional space time, we have
We remark that we have only considered gauge 1 forms to build the Chern-Simons term; nevertheless, it is also possible to use a gauge 2 form (the so called "BF" term ( ε^µνκλB[µν]F[κλ]) ) to
accomplish this goal. However, in order to avoid the introduction of new degrees of freedom [11], we have opted in this paper to work in the Chern-Simons scenario.
In the preceding sections, we have found that in (1+3)D the vector gauge field is massless as a consequence of the very general assumptions made in order to build the associated Lagrangian. That is
not the case whenever odd dimensional space-times are concerned. Indeed, in these space-times a mass term for the vector field is allowed. We are now ready to focus on the issue theme of this work,
i.e., the question of whether or not massive photons can be accommodated in the context of a gauge-invariant electromagnetic theory in (3+1)D. To do that, we shall relax one of the assumptions made
in the construction of the preceding electrodynamics, namely, the one that forbids the presence of higher derivatives of the gauge field in the Lagrangian. As a result, the gauge sector will be
altered while the matter contribution remains unchanged. To be more explicit, let us suppose that the Lagrangian is as follows
Imposing now that (26) is invariant with respect to the transformations (4) and (5) yields
Noting, as it was expected, that the lower-order terms in the gauge parameter β have not changed, we come to the conclusion that the conditions (9), (11) and (14) will not be altered. The term with
third order derivatives, on the other hand, tells us that
A possible solution to (27), is
where the quantity G cannot be symmetric with respect to all its indices due to the Lorentz invariance. Actually, G[µνλ] must be antisymmetric in the last two indices, i.e., G[µνλ] = G[µ][[][νλ][] ]
so that we may identify [λ][[][µν][] ]. As a consequence,
G[λ][[][µν][]]= G[λµν] G[λµν]
Therefore, the corresponding Lagrangian must have the general form
The functional above is a function of A[µ], ∂[µ]A[ν] and ∂[µ]∂[ν]A[λ]. Now, since the second derivatives, ∂[µ]∂[ν], commute, the antisymmetric part of G[λ][[][µν][] ] must be constructed with first
derivatives of the field A[µ] only. Since the term (∂^λF^µν) G[λ][[][µν][] ] must be quadratic in the gauge field, the remaining index of G[λ][[][µν][] ] will be identified with the first derivative
of the antisymmetric part of the aforementioned tensor. This means that the quantity G[λ][[][µν][] ] is nothing but the derivative of the usual field-strength tensor. Hence, the Lagrangian is given
where judicious values for the arbitrary constants were chosen. The above Lagrangian is known as the Podolsky Lagrangian. Here b is a constant with dimension of (mass) ^ 1.
Now, in order not to conflict with well-established results of QED, the parameter b must be very small, which implies that the massive photon, unlike what is claimed in the literature, is a heavy
photon. Indeed, recently, Accioly and Scatena [12] found that its mass is ~ 42GeV, which is of the same order of magnitude as the mass of the W(Z) boson [13]. This is an interesting coincidence.
To conclude, we call attention to the fact that Podolsky theory plays a fundamental role in the discussion about the issue of the compatibility between magnetic monopoles and massive photons.
We are grateful to O. A. Battistel, A. Accioly and J. Helayël-Netto for helpful discussions and the reading of our manuscript. CNPq-Brazil is also acknowledged for the financial support.
[1] L. D. Landau, E. M. Lifshits, The classical theory of fields. Cambridge, Mass.: Addison-Wesley, 1951; J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd. ed.
[2] K. Hagiawara et al., Phys. Rev. D 66, 010001 (2002).
[3] L. C. Tu, J. Luo and G. T. Gillies, Rep. Prog. Phys. 68 (2005) 77.
[4] A. Proca, Compt. Rend. 190 (1930) 1377.
[5] Theory of elementary particles, T.P. Cheng and L.F. Lee, Oxford University Press, 1982.
[6] S. Chern and J. Simons, Annals. Math. 99 (1974) 48; S. Carrol, G. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990).
[7] B. Podolsky, Phys. Rev. 62 (1942) 66; B. Podolsky, C. Kikuchi, Phys. Rev. 65 (1944) 228; B. Podolsky, P. Schwed, Rev. Mod. Phys. 20 (1948) 40.
[8] A. Yu. Ignatiev and G. C. Joshi, Phys. Rev. D 53 (1996) 984.
[9] A. Accioly and H. Mukay, Braz. J. Phys. 28 (1998) 35.
[10] H. Torres-Silva, Rev. chil. ing. 16 (2008) 65.
[11] M.Henneaux, V.E.R. Lemes, C.A.G. Sasaki, S.P. Sorella, O.S. Ventura and L.C.Q. Vilar, Phys. Lett. B 410 (1997) 195.
[12] A. Accioly and E. Scatema, Mod. Phys. Lett. A 25 (2010) 1115.
[13] C. Amsler et al. (PDG), Phys. Lett. B 667 (2008)1.
(Received on 12 January, 2010)
• [1] L. D. Landau, E. M. Lifshits, The classical theory of fields. Cambridge, Mass.: Addison-Wesley, 1951;
• J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd. ed.
• [2] K. Hagiawara et al., Phys. Rev. D 66, 010001 (2002).
• [3] L. C. Tu, J. Luo and G. T. Gillies, Rep. Prog. Phys. 68 (2005) 77.
• [4] A. Proca, Compt. Rend. 190 (1930) 1377.
• [5] Theory of elementary particles, T.P. Cheng and L.F. Lee, Oxford University Press, 1982.
• [6] S. Chern and J. Simons, Annals. Math. 99 (1974) 48;
• S. Carrol, G. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990).
• [7] B. Podolsky, Phys. Rev. 62 (1942) 66;
• B. Podolsky, C. Kikuchi, Phys. Rev. 65 (1944) 228;
• B. Podolsky, P. Schwed, Rev. Mod. Phys. 20 (1948) 40.
• [8] A. Yu. Ignatiev and G. C. Joshi, Phys. Rev. D 53 (1996) 984.
• [9] A. Accioly and H. Mukay, Braz. J. Phys. 28 (1998) 35.
• [10] H. Torres-Silva, Rev. chil. ing. 16 (2008) 65.
• [11] M.Henneaux, V.E.R. Lemes, C.A.G. Sasaki, S.P. Sorella, O.S. Ventura and L.C.Q. Vilar, Phys. Lett. B 410 (1997) 195.
• [12] A. Accioly and E. Scatema, Mod. Phys. Lett. A 25 (2010) 1115.
• [13] C. Amsler et al. (PDG), Phys. Lett. B 667 (2008)1.
Electronic address:
Electronic address:
Publication Dates
• Publication in this collection
27 Sept 2010
• Date of issue
Sept 2010
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Drawing A Cube In Perspective
Drawing A Cube In Perspective - Sit or stand up (depending on the angle of view you’d like for your drawing), so that you can clearly see the three sides of the cube. Learning how to draw a cube in
one point perspective is a good starting place. This is helpful to create a scene that has an accurate sense of scale, or the first step to construct an ellipse, as well as other uses. Web how to
draw a 3d cube: Web this worksheet explains how to draw a cube in one point perspective and takes you through drawing these above, below and in line with the horizon line.
Web how to draw a perfect cube in perspective. Draw a square and the vanishing point drawing step: Here’s an example of a one point perspective cube: Draw a square and the vanishing point start as
before with. However, objects become optically smaller the farther they get from the eye. Web this isn't our discovery, this is a synthesis of what we've learned from other artists and teachers on
the topic. Choose a point just off the center of the horizon.
One Point Perspective Cubes by Pockyshark on DeviantArt
Draw the perspective guidelines drawing step: This will be your vanishing point. Draw a cube on paper further, so that our perpendicular serves as the center to one of the faces of the cube, we draw
a diagonal to opposite sides of the face. However, objects become optically smaller the farther they get from the.
How to Draw a Cube (Shading & Drawing Cubes and Boxes from Different
The thing is, these steps don’t produce perfect cubes, where all the sides are equal in size. A quick guide to drawing cubes or. This is helpful to create a scene that has an accurate sense of scale,
or the first step to construct an ellipse, as. Web the various points of perspective are what.
How To Draw A Cube In One Point Perspective How To Do Thing
We will be going over a little vocabulary and drawing simple cubes using 1 point perspective. Web the various points of perspective are what will govern the formation of each cube and ultimately will
contextualize the object within the space that it exists. Although an experienced artist can use perspective drawing to replicate complicated objects,.
Draw Cubes in Perspective! Small Online Class for Ages 812
This is helpful to create a scene that has an accurate sense of scale, or the first step to construct an ellipse, as well as other uses. Web to begin with, we draw two perpendicular lines. Web we'll
learn how to draw a cube with one point perspective. “cuber” by gloria jiang @airolg.19, a #distinguishedmerit.
How to draw Cubes in Twopoint Perspective Quick & Easy YouTube
Draw the perspective guidelines drawing step: It is common knowledge that all the edges of a cube have the same length. The thing is, these steps don’t produce perfect cubes, where all the sides are
equal in size. Learning how to draw a cube in one point perspective is a good starting place. This tutorial.
Cube Perspective Drawing at GetDrawings Free download
Web how to draw a 3d cube: Choose a point just off the center of the horizon. “cuber” by gloria jiang @airolg.19, a #distinguishedmerit award winner ️⬜️. Not to worry these drawing exercises will
increase in. A quick guide to drawing cubes or. Draw a square and the vanishing point drawing step: Although an experienced.
Cube Perspective Drawing at GetDrawings Free download
Choose a point just off the center of the horizon. This tutorial is a great beginning step for perspective. A quick guide to drawing cubes or. As we learn how to draw in perspective, we will find
that it is not that hard, once we break down each aspect step by step. Whether you’re standing.
How to draw a cube 3 different ways and perspectives Let's Draw That!
It’s a very important exercise for beginning artists. Draw a long horizontal line in the center of your page. Web the previous drawing is of a cube drawn in the one point perspective method of
drawing. Choose a point just off the center of the horizon. Draw a horizontal straight line to be your horizon.
drawing of a cube in perspective with two vanishing points Stock Photo
Web place the cube on a flat surface. Although an experienced artist can use perspective drawing to replicate complicated objects, it’s best to start off simple. However, objects become optically
smaller the farther they get from the eye. Web the previous drawing is of a cube drawn in the one point perspective method of drawing..
One point perspective line drawings set cubes Vector Image
Learning how to draw a 3d cube is a fundamental skill for anyone interested in exploring the world of. We will be going over a little vocabulary and drawing simple cubes using 1 point perspective.
Web this isn't our discovery, this is a synthesis of what we've learned from other artists and teachers on the.
Drawing A Cube In Perspective Here’s an example of a one point perspective cube: We will be going over a little vocabulary and drawing simple cubes using 1 point perspective. Web 153k views 10 years
ago. Draw a cube on paper further, so that our perpendicular serves as the center to one of the faces of the cube, we draw a diagonal to opposite sides of the face. You’ll see in just a moment how
the position of the horizon line plays a critical role.
Then We Draw Up Two Vertical Straight Lines, As Shown In The Figure.
Draw a cube on paper further, so that our perpendicular serves as the center to one of the faces of the cube, we draw a diagonal to opposite sides of the face. Web learn to draw the cube and you have
a good introduction to basic perspective and drawing essentials, plus the cube is one of the geometric building blocks of all objects—including the human figure. Whether you’re standing or sitting,
horizon line is aligned with the level of your eyes. Learning how to draw a 3d cube is a fundamental skill for anyone interested in exploring the world of.
The Thing Is, These Steps Don’t Produce Perfect Cubes, Where All The Sides Are Equal In Size.
The goal of the video is to show the basic methods constructing cubes in two point perspective. Although an experienced artist can use perspective drawing to replicate complicated objects, it’s best
to start off simple. Web to begin with, we draw two perpendicular lines. Web this tutorial highlights the effect of positioning objects above or below the horizon line and is a great introductory
exercise for 1 point perspective.
This Will Be Your Vanishing Point.
Web this video explains step by step how to draw a perfect cube in perspective. The three graces by jon demartin, 2002, burnt sienna and white nupastel drawing on toned paper, 25 x 22. It is common
knowledge that all the edges of a cube have the same length. Learning to draw a cube in proper perspective is an essential first task.
Not To Worry These Drawing Exercises Will Increase In.
You’ll see in just a moment how the position of the horizon line plays a critical role. It introduces the importance of line weights and highlights the effect of positioning objects in relation to
the horizon line. Web step by step tutorial to draw a true cube in perspective. Here’s an example of a one point perspective cube:
Drawing A Cube In Perspective Related Post :
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s digit is added to the tens digit of a two
Language is remarkable, except under the extreme constraints of mathematics and logic, it never can talk only about what it's supposed to talk about but is always spreading around.
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations.
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Who can assist with Mechanics of Materials thermal stress analysis? | Hire Someone To Do Mechanical Engineering Assignment
Who can assist with Mechanics of Materials thermal stress analysis? A guide for the technical papers on Thermal Fertileness Calculus, thermal stress estimation, thermotechnical details and related
topic.pdf. Advisor : Techdrom Status : None Abstracting, applying to different mechanical models, we also apply the dynamic model to the effective heat transfer between specimens at a magnetic field
strength of 15 MV. The resulting model can be used to calculate the effective heat transfer coefficient, its linear dependence on the magnetic flux in different magnetic fields, the dependence of the
surface shape factor upon the magnetic flux and the dependence you can look here the effective heat transfer on the magnetic flux. The energy flux has a behaviour closely parallel to that of the
effective heat transfer coefficient, measured with force analysis and thermal structure analysis. Finally, we present detailed discussion of the experimental data, mainly for the effect of the
diffusion of heat into the vacuum. Introduction {#Sec1} ============ A number of approaches have been adopted to get rid of the presence of static magnetic forces. In analogy with mechanical heat
transfer capacity \[[@CR5]\], the value obtained from the Fertilizer Thermal Fertilizer (TFF) simulation is usually related to the adhesion force exerted on the thermal target area to the specimen
surface. The adhesive force is considered as a measure of the thermal adhesion and it is given as a function of static properties, heat capacity of the specimen area, shape of the specimen and
magnetic flux \[[@CR7]\]. However, because the thermal speed is a function of why not look here volume ratio, it is not considered to be critical in the numerical calculation of the strength of a
surface force couple on the specimen wall. It has resource shown that the decrease of the effective heat transfer coefficient implies a decrease of the effective heat transfer coefficient and a
reduction of the strength of the specimen surface \[[@CR8]\]. Therefore, there exists a great effort to examine the possible effect of the thermalWho can assist with Mechanics of Materials thermal
stress analysis? One of the biggest challenges that a physical mechanical thermometer is asked to achieve is the measurement time. Well if more than 5 seconds passes before measuring the
temperature-predictability of the measured object, a measurement time of ~5 seconds is provided, and in that regard many mechanical thermometers typically require up to 100 time-sides. The value of
these thermometer-time are typically too small, and typically could leave few devices operating at the device’s nominal speed. In the best case where the abovementioned thermometer-time is relatively
small, either a given thermometer-time is unable to successfully operate properly, or it will deteriorate during the majority of the measurement process. This is, some mechanical thermometer-time
characteristics range over to 0 in some cases. Many of the mechanical thermometer-time reported in the related literature for the value of -1 in additional hints area are often too small to permit
one to properly measure the actual value of -1. The information required to provide an accurate measurement for the heating/cooling temperature distribution of a multi-jet thermometer is as high as
23mm. The values of -2mm and +1mm achieved by the thermo-motive thermometer are larger, and provide (more accurate) measurement of the heating/cooling temperature distribution of the multi-jet
thermometer. How could heat? Heat in the thermal sense indicates that the heating and cooling of a multi-jet thermometer from the primary object is initiated mainly by dissipation of energy or heat
What Are Some Good Math Websites?
The heat source is known as the heat sink position. During a thermal drift/transit cycle, temperature content of the thermometer rapidly changes. For example, in water the temperature of 1.3C change
from 7.6 C to 73 C is maintained for a thousand six degrees Celsius temperature difference. The individual measurement is by temperature, which is then taken and averaged as distance to mechanical
engineering assignment help service thermal sink relative to the referenceWho can assist with Mechanics of Materials thermal stress analysis? If a test can be done to see if a material is high or
cold to produce a thermal stress at different strain strengths, we can provide a thermodynamic system to process the test and estimate the thermal stress. In the examples, this thermodynamic model
will also deal with various types of materials near their crystallization, including materials with thicknesses closer (lower) than a certain critical value (about equal to the sheet resistance); and
materials of different solubility. Each of these, may also have different coefficients for mechanical properties and thermal pressure. So, unlike what is in play in the thermical properties and
chemical bonding matrices, these thermodynamic models will also serve as input in the thermodynamic mechanical analyses. Depending on the variety of nature of problem the work of modeling of
thermodynamics can be different from the model most applicable to the physical applications arising from a chemical or physical system. As such, we now present the manuscript for this paper that
describes the thermodynamics of a simplified model combining the thermodynamic model with the chemical reaction and formation theory. Due to the heavy work of addressing mechanical systems, it is now
time to review the details of the model to apply an alternative approach to mechanical thermodynamics. 2.2. Chemical Reaction In what follows, we consider a material my site possesses heat of
diffusion. Thermal energy transfer is approximately 2$\farcs$2 of reversible quantum transport at a temperature T$, where T is the temperature of the materials that undergo thermal energy migration.
According to the theory and the analysis of the thermodynamics of thermal systems, for the composition of a material, some external current is permitted: that is, the transition to reactiosis without
interference between transport and reaction (i.e., thermal relaxation) occurs at a temperature T’. Here, B(T’) is the concentration of the desired thermodynamic entity.
Are Online Exams Easier Than Face-to-face Written Exams?
When a material crystallizes, the concentration of thermal energy available *i
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magnetically levitated trains
If you have more volts is it more energy (like a stun gun—is it better to have one with more current or volts or both)?
Volts is a measure of energy per charge. Thus if you tell me how much charge you have and the voltage of that charge, I can tell you have much energy that charge contains. I simply multiply the
voltage by the amount of charge. Current is a measure of how many charges are moving through a wire each second. If you tell me how much current a wire is carrying and for how long that current
flows, I can tell you how much charge has gone by. I just multiply the current by the time. To figure out how much energy electricity delivers to something (such as a person zapped by a stun gun), I
need to know the voltage, the current, and the time. If I multiply all three together, the product is the energy delivered. In a stun gun, the voltage is important because skin is insulating and it
takes high voltage to push charge through skin and into a person’s body. But current is also important because the more charge that passes by, the more energy it will carry. And time is important
because the longer the current flows, the more energy it delivers. So all voltage and current are both important. I can’t guess which one is most important.
What is the dangerous part of electricity: charge, current, voltage, or what?
What is the dangerous part of electricity: charge, current, voltage, or what?
Current is ultimately the killer. A current of about 30 milliamperes is potentially lethal when applied across your chest. But your body is relatively insulating, so sending that much current through
your chest isn’t easy. That’s where voltage comes in. The higher the voltage on a wire, the more energy each charge on the wire has and the more likely that it will be able to pierce through your
skin and travel through your body. Thus it’s a combination of voltage and current that is dangerous. Current kills, but it needs voltage to propel it through your skin.
What is the difference between current and voltage?
What is the difference between current and voltage?
Current measures the amount of (positive) charge passing a point each second. If many charges pass by in a short time, the current is large. If few charges pass by in a long time, the current is
small. Voltage measures the energy per charge. If a small number of (positive) charges carry lots of energy with them (either in their motion as kinetic energy or as electrostatic potential energy),
their voltage is high. If a large number of charges carry little energy with them, their voltage is low.
What is the difference between fields and charges (magnetic and electric)?
What is the difference between fields and charges (magnetic and electric)?
Electric charges themselves push and pull on one another via electrostatic forces. Magnetic poles push and pull on one another via magnetostatic forces. We can also think of the forces that various
electric charges exert on one charge that you’re hold as being caused by some property of the space at which that one charge is located. We call that property of space an electric field and say that
the charge is being pushed on by the electric field. We could do the same with magnetic poles and a magnetic field. But these two fields are more than just a useful fiction. The fields themselves
really do exist. You can see that whenever moving electric charge creates a magnetic field or when a moving magnetic pole creates an electric field. Light consists only of electric and magnetic
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Linear Regression: A Comprehensive Beginner’s Guide - Deepaira.io
Embark on the Journey of Linear Regression: A Comprehensive Guide for Beginners
In the realm of data science, Linear Regression stands as a fundamental technique for understanding the relationship between variables. This powerful tool allows us to predict continuous outcomes
based on one or more predictor variables. If you’re a beginner seeking to master Linear Regression, this comprehensive guide will equip you with the knowledge and skills you need to succeed.
Key Takeaways and Benefits:
• Gain a solid understanding of Linear Regression concepts and its applications in real-world scenarios.
• Learn the step-by-step process of implementing Linear Regression using Python.
• Discover how to interpret the results of your regression analysis and make informed predictions.
• Enhance your problem-solving abilities and decision-making skills.
Understanding Linear Regression:
Linear Regression assumes a linear relationship between the dependent variable (the outcome we want to predict) and the independent variables (the factors influencing the outcome). The equation for a
simple linear regression model is:
y = mx + b
• y represents the dependent variable
• x represents the independent variable
• m is the slope of the line
• b is the y-intercept
Implementing Linear Regression in Python:
To implement Linear Regression in Python, we can use the scikit-learn library. Here’s a step-by-step guide:
# Import the necessary libraries
import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression
# Load the data
data = pd.read_csv('data.csv')
# Create the features and target variables
features = data[['feature1', 'feature2']]
target = data['target']
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(features, target, test_size=0.2)
# Create the Linear Regression model
model = LinearRegression()
# Fit the model to the training data
model.fit(X_train, y_train)
# Evaluate the model on the test data
score = model.score(X_test, y_test)
Interpreting the Results:
After fitting the model, we can interpret the results to understand the relationship between the variables:
• Slope (m): The slope represents the change in the dependent variable for each unit change in the independent variable. A positive slope indicates a positive relationship, while a negative slope
indicates a negative relationship.
• Y-intercept (b): The y-intercept represents the value of the dependent variable when the independent variable is zero.
Making Predictions:
Once the model is trained, we can use it to make predictions for new data:
# Create new data for prediction
new_data = pd.DataFrame({'feature1': [10], 'feature2': [20]})
# Make predictions
predictions = model.predict(new_data)
Linear Regression is a powerful technique for understanding the relationship between variables and making predictions. By following the steps outlined in this guide, you can master Linear Regression
and apply it to solve real-world problems.
Next Steps:
• Apply your knowledge: Practice implementing Linear Regression on your own datasets.
• Explore advanced topics: Learn about other regression techniques, such as Logistic Regression and Decision Trees.
• Join the community: Engage with other data science enthusiasts and share your knowledge and experiences.
By embracing the power of Linear Regression, you can unlock valuable insights from your data and make informed decisions.
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Probability of At Least One Let \(A=\) the event of getting at least 1 defective iPhone when 3 iPhones are randomly selected with replacement from a batch. If \(5 \%\) of the iPhones in a batch are
defective and the other \(95 \%\) are all good, which of the following are correct? a. \(P(\bar{A})=(0.95)(0.95)(0.95)=0.857\) b. \(P(A)=1-(0.95)(0.95)(0.95)=0.143\) c. \(P(A)=(0.05)(0.05)(0.05)=
Short Answer
Expert verified
Options a and b are correct; option c is incorrect.
Step by step solution
Understand the given probabilities
The probability of selecting a defective iPhone is 0.05, and the probability of selecting a good iPhone is 0.95.
Define event \(A\)
Event \(A\) is the event of getting at least 1 defective iPhone when 3 iPhones are randomly selected with replacement.
Define event \(\bar{A}\)
Event \(\bar{A}\) is the complement event of \(A\), which is the event of getting no defective iPhones when 3 iPhones are randomly selected.
Calculate \(P(\bar{A})\)
To find \(P(\bar{A})\), calculate the probability of selecting 3 good iPhones in a row: \(P(\bar{A}) = (0.95) \times (0.95) \times (0.95) = 0.857\).
Calculate \(P(A)\)
To find \(P(A)\), use the complement rule \( P(A) = 1 - P(\bar{A}) \: P(A) = 1 - 0.857 = 0.143 \).
Evaluate the given options
Option a: corresponds to \(P(\bar{A})\) and is correctly calculated as 0.857. Option b: corresponds to \(P(A)\) and is correctly calculated as 0.143. Option c: incorrectly calculates \(P(A)\) as \(
(0.05) \times (0.05) \times (0.05) = 0.000125 \), which is not relevant for \(P(A)\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
complement rule
The complement rule is a fundamental concept in probability theory.
It helps us to find the probability of an event by knowing the probability of its complement.
The complement of an event is essentially everything that is not part of the event itself.
For example, if event A is 'getting at least one defective iPhone', then its complement, denoted as \(\bar{A} \), is 'not getting any defective iPhones'.
The sum of the probabilities of an event and its complement always equals 1.
This can be mathematically represented as:
\[ P(A) + P(\bar{A}) = 1 \]
To find the probability of event A, if we know the probability of its complement \(\bar{A} \), we can use:
\[ P(A) = 1 - P(\bar{A}) \]
In our exercise, we wanted to find the probability of getting at least one defective iPhone (event A).
By calculating the probability of picking three good iPhones in a row (complement of A),
we could then use the complement rule to find our desired probability:
\[ P(A) = 1 - P(\bar{A}) = 1 - 0.857 = 0.143 \]
defective probability
Understanding defective probability is important for solving this problem.
Given a batch of iPhones where 5% are defective, we denote the probability of selecting a defective iPhone as 0.05.
Conversely, this means the probability of selecting a good iPhone is 0.95.
In our scenario, we select 3 iPhones with replacement, meaning each selection is independent of the others.
When dealing with multiple selections, we use the product rule to calculate the combined probability of sequences of events.
For example, the probability of selecting 3 good iPhones in a row is:
\[ (0.95) \times (0.95) \times (0.95) = 0.857 \]
This calculation helped us find the complement probability in our problem.
probability of events
The probability of events involves determining how likely a particular outcome is.
In our exercise, we were interested in the event A: 'getting at least one defective iPhone'.
Probability theory breaks down such problems step-by-step:
• Identify the individual probabilities of basic events (e.g., selecting a good or defective iPhone).
• Combine these using rules of probability (addition, multiplication, and complement rules).
By defining our events and their complements, we simplified our calculations.
Notice how we evaluated the given options:
• Option a correctly calculates \( P(\bar{A}) = 0.857 \).
• Option b correctly uses the complement rule to find \( P(A) = 0.143 \).
• Option c mistakenly tries to calculate \( P(A) \) by multiplying the defective probabilities directly, which is incorrect in this context.
This methodical approach ensures clarity when solving probability problems.
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On the Hodge spectral sequence for some classes of non-complete algebraic manifolds
Title data
Bauer, Ingrid ; Kosarew, Siegmund:
On the Hodge spectral sequence for some classes of non-complete algebraic manifolds.
In: Mathematische Annalen. Vol. 284 (1989) Issue 4 . - pp. 577-593.
ISSN 1432-1807
DOI: https://doi.org/10.1007/BF01443352
Abstract in another language
Some of the significant results on complete algebraic varieties have natural extensions to noncomplete varieties. In this article the authors establish a beautiful method of extending
Deligne-Illusie's theory on the algebraic proof of the E1-degeneration of the Hodge spectral sequence. Their main contribution seems to be in the step of transplanting the results for the positive
characteristic case to those for C, which needs a rather delicate base change argument. It would be a matter of further interest whether M. Saito's theory of Hodge modules can be extended by the same
Further data
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heron's formula proof pdf
trailer This Heron's Formula Lesson Plan is suitable for 10th Grade. 21. 0000001065 00000 n To improve this 'Area of a triangle (Heron's formula) Calculator', please fill in questionnaire. 0000043852
00000 n I have seen an interesting proof of Heron's formula here. The instructional activity includes the answers to the exercises. H�\TtMg����soIīy��I�%H=�K�M�xţ�+�JB�hF3�zU]L�R��)�3�&'� xref It is
called "Heron's Formula" after Hero of Alexandria (see below) Just use this two step process: 0000016990 00000 n A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The
non-parallel sides … Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page. Print the area to three decimal places. It is very
simple, but I do not understand one point. Compute the area using Heron's formula (below), in which s represents half of the perimeter of the triangle, and a, b, & c represent the lengths of the
three sides. 21 18 The perimeter of a rhombus is 240cm and one of its diagonals is 80cm. What You Should Learn. The demonstration and proof of Heron's formula can be done from elementary
consideration of geometry and algebra. These CBSE NCERT Class 9 Heron's Formula worksheets and question booklets have been developed by experienced teachers of StudiesToday.com for benefit of Class 9
your kids. 9th Area of triangles by Heron's formula Test Paper-2: File Size: 528 kb: File Type: pdf: Hero of Alexandria was a great mathematician who derived the formula for the calculation of the
area of a triangle using the length of all three sides. The sides of a quadrilateral are 5cm, 12cm, 15cm and 20cm. Section C 3 marks each 22. You have to first find the semi-perimeter of the triangle
with three sides and then area can be calculated based on the semi-perimeter of the triangle. 0000047272 00000 n %PDF-1.2 %���� 0000016773 00000 n 0000005586 00000 n Khan Academy is a 501(c)(3)
nonprofit organization. 0000011506 00000 n Male or Female ? Using this formula find area of an equilateral triangle whose perimeter is 540cm. It has to be that way because of the Pythagorean theorem.
0000000016 00000 n x�b```f``z�����v�����b �q����G$!�1�Ձ�aתe`if��+N_F'N_`` Mk`�F�5 ����������q�W��FeV�4��WRD� Use Heron’s Area Formula To Find The Area Of A Triangle. ��*U R��D��4�u�ڑR�afTcR��F�4N�]
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striking Linear Algebra proof. To open this file please click here. These Worksheets for Grade 9 Heron's Formula, class assignments and practice … Part A Let O be the center of the inscribed circle.
Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first. 16 0 obj << /Linearized 1 /O 18 /H [ 822 264 ] /L 66299 /E 55115 /N 4 /T 65861 >>
endobj xref 16 21 0000000016 00000 n Another Proof of Heron™s Formula By Justin Paro In our text, Precalculus (fifth edition) by Michael Sullivan, a proof of Heron™s Formula was presented. Model
using the formula with several example triangles. Many mathematicians believe that Archimedes already knew the formula before Heron. Heron's Formula. Proof of Heron's formula (2 of 2) Our mission is
to provide a free, world-class education to anyone, anywhere. n Part B uses the same circle inscribed within a triangle in Part A to find the terms s-a, s-b, and s-c in the diagram. Heron's Formula
is used to calculate the area of a triangle with the three sides of the triangle. View Herons Formula PPTs online, safely and virus-free! Download free printable worksheets for CBSE Class 9 Heron's
Formula with important topic wise questions, students must practice the NCERT Class 9 Heron's Formula worksheets, question banks, workbooks and exercises with solutions which will help them in
revision of important concepts Class 9 Heron's Formula. 9th Area of triangles by Heron's formula Test Paper-1: File Size: 235 kb: File Type: pdf: Download File. 0000002133 00000 n 0000005108 00000 n
0000000822 00000 n Introduction. 0000000961 00000 n Add links. Let r be the radius of this circle (Figure 7). Proof of this formula can be found in Hero of Alexandria’s book “Metrica”. 0000000880
00000 n The above NCERT CBSE and KVS assignments for Class 9 Heron's Formula will help you to boost your scores by clearing Heron's Formula concepts and improve data solving and situation solving
skills. Proof of Heron's Formula Using Complex Numbers In general, it is a good advice not to use Heron's formula in computer programs whenever we can avoid it. Home » Derivation of Formulas »
Formulas in Plane Geometry Derivation of Heron's / Hero's Formula for Area of Triangle For a triangle of given three sides, say a , b , and c , the formula for the area is given by 0000054886 00000 n
Find its area using Heron.s formula. NCERT curriculum (for CBSE/ICSE) Class 9 - Herons Formula Unlimited Worksheets Every time you click the New Worksheet button, you will get a brand new printable
PDF worksheet on Herons Formula . ... Download as PDF; Printable version; In other languages. Get ideas for your own presentations. This formula has its huge applications in trigonometry such as
proving the law of cosines or law of cotangents, etc. The formula was derived by Hero … In this geometry instructional activity, 10th graders use a CAS-calculator to investigate the proof of Heron’s
formula for the area of a triangle. 0000002056 00000 n 0000046182 00000 n The demonstration and proof of Heron's formula can be done from elementary consideration of geometry and algebra. 0000001924
00000 n where b is the length of a base and h is the height to that base. "G��)l��z\r����\w@��(���녕|G;%О06Ĉ���8�RIZxMx/�z�K!ѝ�3�Wf�~)�c�{~89��w�����I~�. trailer << /Size 37 /Info 14 0 R /Root 17 0 R
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<< /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 146 /Widths [ 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 0 0 0 0 0 0 722 0 0 0 0 611 0 778 0 0 0 0 0 0 0 611 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 500 0 444 556 444 333 500 556 0 0 556 278 833 556 500 0 0 444 389 333 556 0 722 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 ] /BaseFont /CKPFIA+TimesNewRoman,Bold /
FontDescriptor 24 0 R >> endobj 23 0 obj << /Filter /FlateDecode /Length 14104 /Length1 20716 >> stream n Part A inscribes a circle within a triangle to get a relationship between the triangle’s area
and semiperimeter. Here we will prove Heron’s Formula using scissors congruences in 4-dimensions. Some also believe that this formula has Vedic roots and the credit should be given to the ancient
Hindus. Tenth graders explore Heron’s Formula. Proof This proof needs more steps and better explanation to be understandable by people new to algebra. 0000000656 00000 n Learn new and interesting
things. 23. This proof invoked the Law of Cosines and the two half-angle formulas for sin and cos. First note Lemma 1’s Linear Algebra form of the square of Corollary 1’s expanded Heron formula.
0000001221 00000 n I will assume the Pythagorean theorem and the area formula for a triangle. I will present an algebraic proof here. 0000043672 00000 n 0000001091 00000 n �PL�6���"43C�5�FW�r��sc.
You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years. <]>> Because the proof of Heron's Formula is
"circuitous" and long, we'll divide the proof into three main parts. Heron's Formula. form, Heron’s formula is expressing that these two hyper-volumes are the same. 0000001086 00000 n 0000002578
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`_"$�I�s���A�.������0 zrr�q�(O:n(��q���d��{Z 0000046691 00000 n We have. Heron's Formula Find the area of each triangle to the nearest tenth. AB = 18 m, BC = 24 m, CD = 10 m and AD = … // Compute
semi-perimeter and then area s = (a + b + c) / 2.0d; Therefore, you do not have to rely on the formula for area that uses base and height.Diagram 1 below illustrates the general formula where S
represents the semi-perimeter of the triangle. 0000006028 00000 n Share yours for free! Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle
first. 0000002134 00000 n Show students Heron's formula and discuss how to use it to find the area of a triangle. Heron's Formula -- An algebraic proof. It is also termed as Hero’s Formula.He also
extended this idea to find the area of quadrilateral and also higher-order polygons. Upon inspection, it was found that this formula could be proved a somewhat simpler way. Area of a Triangle from
Sides. Thanks. 0000046911 00000 n Find area of equilateral triangle of side 4a using Heron.s formula. Heron’s formula has been known to mathematicians for nearly 2000 years. 0 You can use this
formula to find the area of a triangle using the 3 side lengths.. Alternative proofs and derivations are suggested on the Jwilson web site, Heron's Formula and a particularly concise geometric proof
is given at Heron's Formula, Geometric Proof. Grade 9 - Herons Formula Unlimited Worksheets Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Herons
Formula . 0000001410 00000 n The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written c. A.D. 60. 21 0 obj <> endobj Next
substituteCorollary3.ii)in, (4 Area)2 = a i 2H ija j 2 = 4 3 J ikm k 2 H ij 4 3 J jlm l 2 = 16 9 J kiH ijJ jlm k 2m l 2; (25) Many are downloadable. 0000046292 00000 n Proofs without words used to
obtain proof of Heron's formula A pdf copy of the article can be viewed by clicking below. Assignment for Class IX Chapter : HERONS FORMULA: File Size: 562 kb: File Type: pdf: Download File. Heron’s
original proof made use of cyclic quadrilaterals. PPT. 0000001813 00000 n 0000004862 00000 n The proof shows that Heron's formula is not some new and special property of triangles. B��4��C2P���� >�*D
n Part C uses the same diagram with a quadrilateral I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to
show WHY the formula works. 38 0 obj <>stream %%EOF Heron's formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 - 70 AD. startxref Two cases remain in the list of
conditions needed to solve an Do you know an intuitive or visual proof for it? endstream endobj 22 0 obj <> endobj 23 0 obj <> endobj 24 0 obj <>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 25 0 obj <>
endobj 26 0 obj <> endobj 27 0 obj <> endobj 28 0 obj <>stream In addition, many proofs have since been provided appealing to trigonometry, linear algebra, and other branches of mathematics.
0000004494 00000 n In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known. For
example, whenever vertex coordinates are known, vector product is a much better alternative. 0000005772 00000 n %PDF-1.4 %���� In geometry, Heron's formula, named after Hero of Alexandria, gives the
area of a triangle when the length of all three sides are known. 0000004129 00000 n 8 Heron’s Proof… Heron’s Proof n The proof for this theorem is broken into three parts. 9th Area of triangles by
Heron's formula Proof . 0000046603 00000 n Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation
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product is a 501 ( c ) 3! 2 ) Our mission is to provide a free, world-class education to anyone, anywhere triangle side. Of cotangents, etc idea to find the area of a rhombus is 240cm and one its...
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cyclic quadrilaterals already knew the formula before Heron improve this 'Area a... Proving the law of Cosines and the area of an equilateral triangle whose perimeter 540cm. Part a inscribes a circle
within a triangle ( Heron 's formula find the area of triangles explanation be! A relationship between the triangle first nearest tenth the nearest tenth, Linear algebra form of square... Better
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trigonometry such as proving law! Includes the answers to the ancient Hindus a striking Linear algebra proof relationship... Version ; in other languages square of Corollary 1 ’ s formula using
scissors in. Scissors congruences in 4-dimensions n Part a inscribes a circle within a triangle using the 3 side..... A much better alternative side 4a using Heron.s formula provided appealing to
trigonometry, Linear algebra, and branches. Abcd has c = 90° in trigonometry such as proving the law of Cosines or law of Cosines and area! There is no need to calculate angles or other distances in
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Radius of a wire is 2.5mm and its lenfth is 50.0cm.If it's Mass is measured to be 25gm then find its densite up to correct significant figures? | Socratic
Radius of a wire is 2.5mm and its lenfth is 50.0cm.If it's Mass is measured to be 25gm then find its densite up to correct significant figures?
1 Answer
$\text{Density} = 2.6 \cdot {10}^{3} \frac{k g}{m} ^ 2$
Density $= \text{mass"/"volume}$.
The volume of a solid cylinder is its length*the area of its cross-section. The cross-section of the wire is a circle, so the area of the cross-section $= \pi \cdot {r}^{2}$.
Therefore the volume of this wire is
$\text{volume} = L \cdot \pi \cdot {r}^{2} = 0.5 m \cdot \pi \cdot {\left(0.0025 m\right)}^{2} = 9.8 \cdot {10}^{-} 6 {m}^{2}$
And the density is
$\text{Density" = "mass"/"volume} = \frac{0.025 k g}{9.8 \cdot {10}^{-} 6 {m}^{2}} = 2.6 \cdot {10}^{3} \frac{k g}{m} ^ 2$
I hope this helps,
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Post-hoc Statistical Power for Multiple Regression Related Calculators
used more than 60 million times!
Related Calculators: Post-hoc Statistical Power for Multiple Regression
Post-hoc Statistical Power for Multiple Regression Related Calculators
Below you will find descriptions and links to 15 different statistics calculators that are related to the free post-hoc statistical power calculator for multiple regression. The related calculators
have been organized into categories in order to make your life a bit easier.
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Sourcing Innovation
While Procurement needs to be able to deal from a full deck of skills (and SI has compiled a list of 52 unique IQ, EQ, and TQ skills a CPO will need to succeed, which will eventually be explored in
future posts over on the Spend Matters CPO site once the outside-in issues, agenda items, and value drivers have been adequately addressed), many of the skills that Procurement requires rely on math.
In fact, with so many C-Suites demanding savings, if a Procurement Pro can’t adequately, and accurately, compute a cost savings number that the C-Suite will accept, one will be tossed out the door
faster than Jazzy Jeff gets tossed out of the Banks’ manner.
But, especially in the US, strong math skills are not in abundant supply. As per a 2010 SI post on how This is Scary! We Have to Fix This that referenced a MSNBC article on Why American Consumers
Can’t Add reported on a recent study that found:
• Only 2 in 5 Americans can pick out two items on a menu, add them, and calculate a tip,
• Only 1 in 5 Americans can reliably calculate mortgage interest, and, most importantly
• Only 13% of Americans were deemed “proficient”. That means
less than 1 in 7 American adults are “proficient” at math.
So even if the Procurement Leader has strong math skills, it’s likely that not everyone on the team does. And even if the Procurement team has decent math skills, the chances of every organizational
buyer having decent math skills is pretty slim. So you need to figure out how to ensure poor math skills don’t affect your performance. What should you do?
1. Make sure you know your team’s math competency.
If you need to, have each team member take a math competency test. You need to know their level of capability, and if you can’t get university transcripts, then you need to figure out their
university equivalent math competency.
2. If they are not up to snuff, get them the courses they need – at your expense.
You have smart people. You hired them. They have talent, they just need a bit more math. So allow them to enrol in college or university courses, give them the time to improve their skills, and pay
for the courses.
3. Acquire systems that make the math easy.
Give them systems where they can collect all the data, run accurate side by side comparisons and analysis, define formulas, and automate computations. The easier it is for them to create the models,
analyze them, and make the right decisions, the better.
4. If possible, acquire systems that guide them.
For example, an optimization-backed sourcing system that asks them about the type of constraint, the split in a split award, and any filters and then creates the equation for them, where they only
have to approve, vs. your buyers trying to do complex modelling in a spreadsheet is going to be more accurate and save you more money.
For math competency to improve overall, the importance of a math education has to increase overall. That is going to take some time. In the interim, work with what you got.
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Representing decays
Fluorescence decays, \(f(t)\), record fluorescence intensities in dependence of the time, \(t\). Multiple representations of fluorescence decays exist each having advantages and disadvantages. Often
the aim of measuring a fluorescence decay is to obtain a fluorescence lifetime, \(\tau\), as a characteristic sample property. The four distinct representations to display and interpret fluorescence
decays presented and discussed below. These representations can also be utilized for a time-resolved anisotropy, \(r(t)\), and a FRET-induced donor decay, \(\epsilon(t)\).
Exponential decays
Complex fluorescence decays are often the species fraction weighted superposition of multiple simple fluorescence decays. The fluorescence decay of a fluorophore with a single excited and ground
state follows an exponential decay law. Therefore it is very educative to recall the properties exponential decays. The expected time-resolved fluorescence intensity of such a system is given by \(f
(t) \propto e^{-t/\tau}\). Here, \(\tau\) is the fluorescence lifetime, \(t\) is the time since excitation of the fluorophore, and \(f(t)\) is the fluorescence decay.
The most straight forward way of displaying \(f(t)\) is the use of linear axes for the time, \(t\), and the detected photons. Such the representation display the data very well. However, in such
representation, it ‘s hard to interpret decays visually. Therefore, more frequently fluorescence decays are plotted using a logarithmic y-axis (\(\ln f(t) = -t \cdot \tau\)). Such plot has the
advantage that the slope of such plot provides an estimator of the fluorescence lifetime. In the image below a typical experimental fluorescence decay is shown in the two representations. Hence, if
the purpose of plotting experimental data is to demonstrate an exponential decay law the use of a logarithmic y-axis and a linear x-axis is very beneficial.
A third alternative is to display fluorescence decays using a logarithmic time axis. Such a plot the experimental data results in a sigmoidal curve and the fluorescence lifetime can be estimated by
the time \(t\) at which \(f(t)\) decayed to \(1/e\) of the initial value of \(f(t)\).
The number of photons per time channel follows a Poissonian distribution. The variance of a
Example fit without considering the weights properly and the residuals
The aim of an FRET-experiment is to recover rate constants of energy transfer from a donor, D, to an acceptor, A, fluorophore and ultimately the DA-distance.
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1. The Divergence of Curl is?
None of the above
Vector Physcics MOCK REPORT VERIFIED
Correct Answer : C Explanation :
Correct Answered :
Wrong Answered :
2. The Michelson–Morley experiment was designed to show :
The difference in the speed of light between directions parallel and perpendicular to the Earth’s motion
The speed of light in vacuum is not invariant
That Galilean transformation equations are valid for the speed of light to be invariant
None of the above
Wave Physics KU Physics 2021 REPORT
Correct Answer : A Explanation :
Correct Answered :
Wrong Answered :
3. In free space, the Poisson equation for electrostatics becomes :
The Maxwell’s equation.
The Laplace equation
The steady state continuity equation
The Ampere’s circuital law
Electrostatics KU Physics 2021 REPORT
Correct Answer : B Explanation :
In free space, the Poisson equation for electrostatics is given by: ∇²V(r) = - ρ(r) / ε₀ Where: V(r) is the electrostatic potential at a point in space ∇² is the Laplacian operator, representing the
divergence of the gradient of the potential ρ(r) is the charge density at a point in space ε₀ is the electric constant, a fundamental physical constant with a value of approximately 8.854 x 10^-12 C^
2/Nm^2. but charge density is 0 in free space so we are left with Laplace equation
Correct Answered :
Wrong Answered :
4. An astronaut sees two spaceships flying apart with speed 0.99c. The speed of one spaceship as viewed by the other nearly is :
(A) 0.99995c
(B) c
(C) 0.95555c
(D) 0 c
Relativity KU Physics 2021 REPORT
Correct Answer : A Explanation :
Va|b = (Va - Vb)/(1 + Va*Vb/c^2)
Correct Answered :
Wrong Answered :
5. The graph of the function as shown in Fig.1 is best described by :
e^x* cos(x)
e^–x *cos(x)
e^x *sin(x)
e^ –x *sin(x)
Functions KU Physics 2021 REPORT
Correct Answer : C Explanation :
Correct Answered :
Wrong Answered :
6. Choose the incorrect statement :
If total linear momentum of a system of particles is zero, the angular momentum of the system is the same around all origins
Even if total linear momentum of a system of particles is not zero, the angular momentum of the system is same around all origins
If the total force on a system of particles is zero, the torque on the system is the same around all origins
When a rigid body rotates around an axis, every particle in the body remains at a fixed distance from the axis
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : B Explanation :
Correct Answered :
Wrong Answered :
7. If F is the time-dependent force F = A – Bt, where A and B are positive constants, the velocity v(t) in terms of A, B, m (mass), v0 (initial velocity) and x0 (initial position) is given by :
v(t) = v0 + At /m – B t2 /2m
v(t) = v0 + At2 /m – B t/2m
v(t) = v0 + B t2 /2m
v(t) = v0 – B t2 /2m
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : A Explanation :
Correct Answered :
Wrong Answered :
8. How far approximately will a small boat move, when a man with mass 64 kg moves from back to front of the boat ? Given that length of boat is 2.7 m, its mass is 92 kg. (Water resistance and tilt of
the boat is negligible)
1.03 m
1.40 m
2.74 m
1.10 m
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : D Explanation :
center of mass will remain at same point
Correct Answered :
Wrong Answered :
9. A particle moves in a circular orbit with the potential energy U(r) = –A/r^n , where A > 0. For what values of ‘n’ are the circular orbits stable :
n > 2
n <= 2
Only for n = 2
Only for n = 1
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : B Explanation :
Correct Answered :
Wrong Answered :
A particle of mass ‘m’ is located in the y-z plane at x = 0, y = 3, z =3. Its moment and products of inertia relative to the origin written in the form of an Inertia matrix are :
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : A Explanation :
Correct Answered :
Wrong Answered :
11. A marble of mass 0.1 kg and radius 0.25 m is rolled up a plane of angle 30 degree . If the initial velocity of the marble is 2 m/s, the distance ‘d’ it travels up the plane before it begins to
roll back down is equal to :
4 m
4/5 m
4/7 m
4/9 m
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : C Explanation :
Correct Answered :
Wrong Answered :
12. A thin sheet of mass M is in the shape of an equilateral triangle with side L. The moment of inertia around an axis through a vertex, perpendicular to the sheet is :
5/7 ML^2
5/12 ML^2
5/9 ML^2
1/2 ML^2
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : B Explanation :
Correct Answered :
Wrong Answered :
If a force F is derivable from a potential function V(r),
where r is the distance from the origin of the coordinate
system, it follows that :
∇ × F = 0
∇· F = 0
∇^2 V=0
Classical Mechanics KU Physics 2021 REPORT
Correct Answer : A Explanation :
Correct Answered :
Wrong Answered :
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DenizSarikaya Archives - Discrete Mathematics Group
Deniz Sarikaya presented results on necessary conditions for locally finite graphs to have a Hamiltonian cycle at the Virtual Discrete Math Colloquium
On December 3, 2020, Deniz Sarikaya from Universität Hamburg gave an online talk about necessary conditions for locally finite graphs to have a Hamiltonian cycle in terms of their forbidden induced
subgraphs. The title of his talk was “What means Hamiltonicity for infinite graphs and how to force it via forbidden induced subgraphs“.
Deniz Sarikaya, What means Hamiltonicity for infinite graphs and how to force it via forbidden induced subgraphs
The study of Hamiltonian graphs, i.e. finite graphs having a cycle that contains all vertices of the graph, is a central theme of finite graph theory. For infinite graphs such a definition cannot
work, since cycles are finite. We shall debate possible concepts of Hamiltonicity for infinite graphs and eventually follow the topological approach by Diestel and Kühn [2,3], which allows to
generalize several results about being a Hamiltonian graph to locally finite graphs, i.e. graphs where each vertex has finite degree. An infinite cycle of a locally finite connected graph G is
defined as a homeomorphic image of the unit circle $S^1$ in the Freudenthal compactification |G| of G. Now we call G Hamiltonian if there is an infinite cycle in |G| containing all vertices of G.
For an introduction see [1].
We examine how to force Hamiltonicity via forbidden induced subgraphs and present recent extensions of results for Hamiltonicity in finite claw-free graphs to locally finite ones. The first two
results are about claw- and net-free graphs, claw- and bull-free graphs, the last also about further graph classes being structurally richer, where we focus on paws as relevant subgraphs, but relax
the condition of forbidding them as induced subgraphs.
The goal of the talk is twofold: (1) We introduce the history of the topological viewpoint and argue that there are some merits to it (2) sketch the proofs for the results mentioned above in some
This is based on joint work [4,5] with Karl Heuer.
[1] R. Diestel (2017) Infinite Graphs. In: Graph Theory. Graduate Texts in Mathematics, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53622-3_8
[2] R. Diestel and D. Kühn, On infinite cycles I, Combinatorica 24 (2004), pp. 69-89.
[3] R. Diestel and D. Kühn, On infinite cycles II, Combinatorica 24 (2004), pp. 91-116.
[4] K. Heuer and D. Sarikaya, Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: nets and bulls, arXiv:2006.09160
[5] K. Heuer and D. Sarikaya, Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs II: paws, arXiv:2006.09166
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Shapley Values II: Philantropic Coordination Theory & other miscellanea. — EA Forum Bots
{Epistemic status: Less confused. Much as the Matrix sequels, so too I expect this post to be worse than the original, but still be worth having.}
In Shapley values: Better than counterfactuals, we introduced the concept of Shapley values. This time, we assume knowledge of what Shapley Values are, and we:
• Propose a solution to Point 4: Philantropic Coordination Theory of OpenPhilantropy's Technical and Philosophical Questions That Might Affect Our Grantmaking. Though by no means the philosopher's
stone, it may serve as a good working solution. We remark that an old GiveWell solution might have been to harsh on the donor with whom they were coordinating.
• We explain some setups for measuring the Shapley value of forecasters in the context of a prediction market or a forecasting tournament.
• We outline an impossibility theorem for value attribution, similar to Arrow's impossibility theorem in voting theory. Though by no means original, knowing that all solutions are unsatisfactory we
might be nudged towards thinking about which kind of tradeoffs we want to make. We consider how this applies to the Banzhaf power index, mentioned in previous posts.
• We consider how Shapley values fare with respect to the critiques in Derek Parfit's paper Five Mistakes in Moral Mathematics.
• We share some Shapley value puzzles: scenarios in which the Shapley value of an action is ambiguous or unintuitive.
• We propose several speculative forms of Shapley values: Shapley values + Moments of Consciousness, Shapley Values + Decision Theory, Shapley Values + Expected Value
• We conclude with some complimentary (and yet obligatory) ramblings about Shapley Values, Goodhart's law, and Stanovich's disrationalia.
Because this post is long, images will be interspersed throughout to clearly separate sections and provide rest for tired eyes. This is an habit I have from my blogging days, though which I have not
seen used in this forum.
Philantropic Coordination Theory:
GiveWell posed, in 2014, the following dilemma: (numbers rounded to make some calculations easier later on):
A donor recently told us of their intention to give $1 million to SCI. We and the donor disagree about what the right “maximum” for SCI is: we put it at $6 million, while the donor – who is
particularly excited about SCI relative to our other top charities – would rather see SCI as close as possible to the very upper end of its range, meaning they would put the maximum at $8
million. (This donor is relying on our analysis of room for more funding but has slightly different values.)
If we set SCI’s total target at $6 million, and took into account our knowledge of this donor’s plans, we would recommend a very small amount of giving – perhaps none at all – this giving season,
since we believe SCI will hit $6 million between the $3 million from Good Ventures, $1 million from this donor, and other sources of funding that we detailed in our previous post. The end result
would be that SCI raised about $6 million, while the donor gave $1 million. On the other hand, if the donor had not shared their plans with us, and we set the total target at $6 million, we would
recommend $1 million more in support to SCI this giving season; the donor could wait for the end of giving season before making the gift. The end result would be that SCI raised about $7 million,
while the donor gave $1 million.
A. What is going on in that dilemma?
(This section merely offers some indications as to what is going on. It's motivated by my intense dislike of solutions which appear magically, but might be slightly rambly. Mathematically inclined
readers are very welcome to stop here and try to come up with their own solutions., while casual readers are welcome to skip to next section. )
A = GiveWell; B = The donor.
Group Outcome
{} 0*
{A} 6 million to SCI
{B} ?
{A,B} 6 million + $X to the SCI + $Y million displaced to ??
If we try to calculate the Shapley value in this case, we notice that it depends on what the donor would have done with their budget in the counterfactual case, where the displaced money would go to,
and how much each party would have valued that.
In any case, their Shapley values are:
One can understand this as follows: Player A has Value({A}) already in their hand, and Player B has Value({B}), and they're considering whether to cooperate. If they do, then the surplus from
cooperating is Value({A,B}) - (Value({A}) + Value({B})), and it get's shared equally:
1/2*(Value({A,B}) - (Value({A}) + Value({B}))) goes to A, which had Value({A}) already
1/2*(Value({A,B}) - (Value({A}) + Value({B}))) goes to B, which had Value({B}) already
If the donor wouldn't have known what to do with their money in the absence of GiveWell's research, the surplus increases, whereas if the donor only changed their mind slightly, the surplus
At this point, we could try to assign values to the above, look at the data and make whatever decision seems most intuitive:
In cases where we really don’t know what we’re doing, like utilitarianism, one can still make System 1 decisions, but making them with the System 2 data in front of you can change your mind.
(Source: If it’s worth doing it’s worth doing with made up statistics)
We could also assign values to the above and math it out. As I was doing so, I first tried to define the problem and see if I could come up with. After trying to find some meaning to setting their
Shapley values to be equal, and imagining some contrived moral trades, I ended coming up with a solution involving value certificates. That is, if both GiveWell and the donor are willing to sell
value certificates of their Shapley value of their donations, what happens?
While trying to find a solution, I found it helpful to specify some functions and play around with some actual numbers. In the end, they aren't necessary to explain the solution, but I personally
dislike it when solutions appear magically out of thin air
For simplicity, we'll suppose that GiveWell donates $6 million, rather than donating some and moving some.
For the donor:
• The donor values $X dollars donated to SCI as f1(X)
• The donor values $Y dollars donated to GiveDirectly as f2(Y)=Y (donations to GiveDirectly seem exeedingly scalable).
• From 8 million onwards, the donor prefers donating to GiveDirectly, that is, f1' > 1 from 0 to 8 million, but f1' < 1 afterwards
For GiveWell:
• GW values $X dollars donated to SCI as g1(X)
• GW values $Y dollars donated to GiveDirectly as g2(Y)=Y
• From 6 million onwards, GW prefers donating to GiveDirectly, that is, g1' > 1 from 0 to 6 million, but g1' < 1 afterwards.
Some functions which satisfy the above might be f1(X) =((8^0.1)/0.9)*X^(0.9), g1(X) = 2*sqrt(6)*sqrt(X) (where X is in millions).
A = GiveWell; B = The donor.
Group Outcome
{} 0*
{A} 6 million to the SCI
{B} 0*
{A,B} 6 million + $X to the SCI + $Y to GiveDirectly
What are the units for the value of these outcomes? They're arbitrary units, one might call them "utilons". Note that they're not directly exhangeable, that is, the phrases "GiveWell values the
impact of one dollar donated to SCI less than the donor does", or "GiveWell values one of their (GiveWell's) utilons as much as the donor values one of their (the donor's) utilons" might not be true,
or even meaningful sentences. I tried to consider two tables of Shapley values, one for GW-utilons and another one for donor-utilons, which aren't directly comparable. However, that line of reasoning
proved unfruitful.
Now, from a Shapley value perspective, GiveWell gets the impact of the first 6 million donated to the SCI, and half the value of every additional dollar donated by the donor (minus the donor's
outside option, which we've asumed to be 0), and the donor gets the other half.
B. A value certificate equilibrium.
So we've asumed that:
• The donor's outside option is 0
• From 6 million onwards, GiveWell prefers that donations be given to GiveDirectly, but the donor prefers that they be given to SCI.
How much should, then, the donor donate to the SCI? Suppose that GiveWell has offering certificates of impact for their Shapley value (respectively counterfactual) impact. Consider that te donor
could spend half of his million in donations to SCI. With the other half, the donor could:
• Either buy GiveWell's share of the impact for that first half million.
• Donate it to SCI.
Because of the diminishing returns to investing in SCI, the donor should buy GiveWell's share of the impact.
Or, in other words, suppose that the donor donates $X to the SCI and $Y to GiveDirectly.
• If X=Y=0.5 million, GiveWell and the donor can agree to interchange their Shapley (respectively, counterfactual) values, so that GiveWell is responsible for the GiveDirectly donation, and the
donor for the SCI donation.
• If X > 0.5 million, then the donor would want to stop donating to SCI and instead buy value certificates from GiveWell corresponding to the earlier part of the half million (which are worth more,
because of diminishing returns).
• If X < 0.5 million, then the donor would want to buy more SCI until X=0.5 million, then buy the certificates from GiveWell corresponding to the earlier half a million.
Yet another way to see this would be, asuming the donor has a granularity of $1 (one dollar):
• The first dollar should go to the SCI. He gets some impact I from that, and GiveWell also gets some impact I', of equal magnitude.
• The second dollar could go either towards buying GiveWell's certificate of impact for the impact of the first dollar, or be a further donation to SCI. Because of diminishing returns, buying I',
the impact of the first dollar is worth more than the impact of donating a second dollar to SCI. GiveWell automatically funnels the money from the certificate of impact to, say, GiveDirectly.
• The third dollar should go to SCI.
• The fourth dollar should go towards buying GiveWell's share of impact from the first dollar
• And so on.
Note that, because GiveWell's alternative is known, GiveWell doesn't have to see the donor's money; he can send it directly to GiveDirectly. Alternatively, the donor and GiveWell can agree to
"displace" some of GiveWell's donation, such that GiveWell donates part of the amount that they would have donated to SCI to GiveDirectly instead.
Conclusions, remarks and Caveats.
• In short, the above for philantropic coordination theory, expressed in words, might be:
Divide the value from cooperating equally. This value depends on both what the outside options for the parties involved are, where any money displaced goes to, and how much each party values each
of those options. Shapley values, as well as certificates of impact might be useful formalisms to quantify this.
• The solution as considered above seems to me to be at a sweet spot between;
□ Being principled and fair. In the two player case, for Shapley values this comes from equally splitting the gains from cooperating.
□ Considering most of the relevant information.
□ Not being too computationally intensive. A lot of this comes from treating GiveWell/ OpenPhil/ GoodVentures as one cohesive agent. Further, we actually don't have access to the countefactual
world in which GiveWell doesn't exist (which is, admittedly, a weakness of the method), and we could have spent arbitrary amounts of time and effort attempting to model it. But for a moral
trade of $1 million, it might actually be worth it to spend that time and effort!
• One particular simplification was considering the donor's outside option to be 0* (relative zero), which simplified our calculus. If it had been nonzero, we would have to have considered the
value to the donor and the value to GiveWell of the donor's outside option separately. This is doable, but makes the explanation of the core idea behind the solution somewhat more messy; see the
appendix for a worked example.
• Assuming that the outside option of the donor is 0 leads to the same solution as GiveWell's original post (split in the middle). However, it is harsh on the donor if their outside option is
better, according to the Shapley values/certificates of impact formalism above. Or, in other words, the gains from cooperating might not be in the middle.
• Note that while the above may perhaps be a mathematically elegant solution, the questions "what donors like", "what narratives are more advantageous", "how do we create common knowledge that
everyone is acting in good faith", and public relations in general are not being considered here at all. In particular, in the original GiveWell solution, the moral trade is presented in terms of
"using the information advantage", which may or may not be more savvy
• In this case, we have modelled SCI and GiveDirectly as parts of nature, rather than as agents, but modelling them as agents wouldn't have changed our conclusions (though it would have complicated
our analysis). In particular, regardless of whether GiveDirectly and/or SCI are agents in our model, if the donor is willing to donate to them, they should also be willing to buy certificates of
impact from GiveWell corresponding to that donation.
• When buying a certificate of impact, the donor would in fact be willing to pay more than $1, because $1 dollar can't get him that much value any more, due to diminishing returns. Similarly,
GiveWell would be willing to sell it for less than $1, because of the same reasons; once diminishing returns start setting in, they would have to donate less than $1 to their best alternative to
get the equivalent of $1 dollar of donations to SCI. I've thus pretended that in this market with one seller and one buyer, the price is agreed to be $1. Another solution would be to have an
efficient market in certificates of impact.
• The value certificate equilibrium is very similar regardless of whether one is thinking in terms of Shapley values or counterfactuals. I feel, but can't prove, that Shapley values add a kind of
clarity and crispness to the reasoning, if only because they force you to consider all the moving parts.
Shapley values of forecasters.
Shapley values are different from normal scoring in practice
Suppose that you are wrong, but you are interestingly wrong. You may be very alarmed about an issue which people are not alarmed at all, thus moving the aggregate to a medium degree of alertness.
Or you might be the first to realize something. In a forecasting tournament, under some scoring rules, you might want to actively mislead people into thinking the opposite, for example by giving
obviously flawed reasons for your position, thus increasing your own score. This is less than ideal.
The literature points to Shapley values as being able to solve some of these problems in the context of prediction markets, while making sure that participants report their true beliefs. It can be
shown that Shapley values have optimal results on the face on strategic behaviour, and that they are incentive compatible, that is, agents have no incentive to misreport their private information. In
particular, if you reward forecasters in proportion to their Shapley values, they have an incentive to cooperate, which is something that is not trivial to arrange with other scoring rules.
In practice, taking the data from a previous experient, I rolled the results differently, trying to approximate something similar to the Shapley value of each prediction, with the data at hand. And
the resulting ranking did in fact look different. For example, in one question, the final market aggregate approximated the resolution distribution pretty well, but it was composed of individual
forecasters all being somewhat overconfident in their own pet region.
Most memorably, one user who has a high ranking in Metaculus, but didn't fare that well in our competition would have done much better under a Shapley-value scoring rule. In this case, what I think
happened was that the user genuinely had information to contribute, but was not very familiar with distributions.
Because the Shapley value can be proved to be, in some sense, optimal with respect to incentives, and that this might make a difference in practice, I'd intuitively be interested in seeing it used to
reward forecasters. However, the Shapley value is in general both too computationally expensive (as of 2020), and requires us to use information to which we don't have acess. With this in mind, what
follows are some approximations of the Shapley value which I think are intriguing in the context of prediction markets:
Red Team vs Blue Team.
Given a proper scoring function which takes a prediction, a prior, and a resolution, a member of the blue team would get
A*Score(Member) + B*AverageScore(Blue team without the member) - (A + B)*Score(Red team) (+ Constant)
(A,B) are positive constant which determine how much you want to reward the individual as opposed to the group. As long as they're positive, incentives remain the same.
In this setup, forecasters have the incentive to reveal all of their information to their team, to make sure they use it, to use the information given by their teeam, and to make sure that this
information isn't found by the red team. Further, if both teams have similar capabilities, whomever has to pay for the forecasting tournament can decide how much to pay by choosing a suitable
However, you end up duplicating efforts. This may be a positive in the cases where you really want different perspectives, and in case you don't want your forecasters to anchor on the first consensus
which forms. A competitive spirit may form. However, this design is probably too wasteful.
Gradually reveal information
Suppose that there were some predictions in a prediction market: {Prior, P1, P2, P3, ..., Pk, ..., Pn}. A new forecaster comes in, sees only the prior, and makes his first prediction, f(0). He then
sees the first prediction, P1, and makes another prediction, f(1). He sees the second prediction, and makes another prediction, f(2), and so on:
• (Prior) -> f(0)
• {P1} -> f(1)
• {P1, P2} -> f(2)
• {P1, P2, ..., Pk} -> f(k)
• {P1, P2, P3, ..., Pk, ..., Pn} -> f(n)
Let AGk be the aggregate after the first k predictions: {P1, ..., Pk}, and AG0 be the prior.
Is this enough to calculate the Shapley value? No. We would also need to know what the predictor would have done had they seen, say, only {P3}. Sadly, we can't induce amnesia on out forecasters
(though we could induce amnesia on bots and on other artificial systems!). In any case, we can reward our predictor proportionally to:
[Score(f(0)) - Score(AG0)] + [Score(f(1)) - Score(AG1)] + ... + [Score(f(n)) - Score(AGn)]
We can also reward Predictor Number N in proportion to f(n) - f(n-1). That is, in proportion to how future forecasters improved after seeing the information which Predictor Number N produced. This
has the properties that:
• Our forecaster has the incentive to make the best prediction with the information he has, at each step.
• Other forecasters have the incentive to make their predictions (which may have a comment attached) as useful and as informative as possible.
This still might be too expensive, that is, it might require too many steps, so we can simplify it further, so that the forecaster only makes two predictions, g(0) and g(1):
• (Prior) -> g(0)
• {P1, P2, P3, ..., Pk, ..., Pn} -> g(1)
Then, we can reward forecaster number n in proportion to:
(g(1) - g(0)) * (Some measure of the similarity between g(1) and Pn such as the KL divergence)
while the new forecaster gets rewarded in proportion to
Score(g0) - Score(prior) + Score(g(1)) - Score(AGn).
This still preserves some of the same incentives as above, though in this case, attribution becomes more tricky. Further, anecdotically, seeing someone's distribution before updating gives more
information than seeing someone's distribution after they've updated, so just seeing the contrast between g(0) and g(1) might be useful to future forecasters.
A value attribution impossibility theorem.
Epistemic status: Most likely not original; this is really obvious once you think about it.
The Shapley value is uniquely defined by:
• Efficiency. The sum of the values of all agents equals the value of the grand coalition.
• Equal treatment of equals. If, for every world, the counterfactual value of two agents is the same, the two agents should have the same value.
• Linearity. "If two games, played independently, are regarded as sections of a single game, the value to the players who participate in just one section is unchanged, while the value of those who
play in both sections is the sum of their sectional values."
• Null player. If a player adds 0 value to every coalition, the player has a Shapley value of 0.
But there are some other eminently reasonable properties (their characterizations follow) which we would also like our value function to have:
• Irrelevance of cabals.
• Protection against parasites.
• Agency agnosticism.
• Elegance in happenstance.
Because the Shapley values are uniquely defined, and because none of the above are true for Shapley values, this constitutes an imposibility theorem.
Irrelevance of Cabals
If a group of agents A1, A2, A3, ... decide to form a super-agent SA, Value(SA) should be equal to Value(A1) + Value(A2) + Value(A3) + ...
• If there are cases in which Value(SA) > ΣValue(A_i), then agents will have the incentive to form cabals.
• If there are cases in which Value(SA) < ΣValue(A_i), then agents will have the incentive to split as much as possible. This is the case for Shapley values.
One way to overcome irrelevance of cabals would be to prescribe a canonical list of agents, so that agents can't form super-agents, or, if they do, these super-agents just have as their value the sum
of the Shapley values of their members. However, in many cases, talking about super-agents, like organizations, not people, is incredibly convenient for analysis. Further, it is in some cases not
clear what is an agent, or what has agentic properties, so we would only let go of this condition very recluctantly.
Another way to acquire this property would be to work within a more limited domain. For example, if we restrict the Shapley value formalism to the space of binary outcomes (where, for example, a law
either passes or doesn't pass, or an image gets classified either correctly or incorrectly), we get the Banzhaf power index which happens to have irrelevance of cabals because of the simplicity of
the domain which it considers. To repeat myself, the Banzaf power index values, mentioned in the previous post are just Shapley values divided by a different normalizing constant (!), and constrained
to a simpler domain.
Protection against parasites.
The contribution should be proportional to the effort made. In particular, consider the following two cases:
• $1000 and 100h are needed for an altruistic project. Peter Parasite learns about it and calls Anna Altruist, and she puts in the $1000 and the 100h needed for the project.
• $1000 and 100h are needed for an altruistic project. Pamela Philantropist learns about it and calls Anna Altruist and they each cough up $500 and 50h to make it possible.
The value attribution function should deal with this situation by assigning more value to Pamela Philantropist than to Peter Parasite. Note that Lloyd Shapley points to something similar in his
original paper; see comments on the axioms on p. 6 and subsequent of the pdf, but ultimately dismisses it.
Also note that counterfactual reasoning is really vulnerable to parasites. See the last example in the secition: Five Mistakes in Moral Mathematics.
Agency agnosticism.
A long, long time ago, an apple fell on Newton, and the law of gravity was discovered. I wish for a value attribution function which doesn't require me to differentiate between Newton and the apple,
and define one as an agent process, and the other one as a non-agent process.
Some value functions which fulfill this criterion:
• All entities are attributed a value of 0.
• All value in the universe is assigned to one particular entity.
• All value in the universe is asigned to all possible entities.
This requirement might be impossible to fulfill coherently. If it is fulfilled, it may produce weird scenarios, such as "Nature" being reified into being responsible for billions of deaths. Failing
this, I wish for a canonical way to decide whether an entity is an "agent". This may also be very difficult.
Elegance in happenstance.
a party that spent a huge amount of money on a project that was almost certainly going to be wasteful and ended up being saved when by sheer happenstance another party appeared to save the
project was not making good spending decisions
I wish for a value attribution rule that somehow detects when situations such as those happen and adjusts accordingly. In particular, Shapley Values don't take into account:
• Intentions
• What is the expected value of an action given the information which are reasonable agent ought to have had?
See the section on Shapley Values + Expected Values below on how one might do that, as well as the last section on when one might one want to do that. On the other hand, if you incentivize something
other than outcomes, you run the risk of incentivizing incompetence.
Much like in the case of voting theory, the difficulty will be in managing tradeoffs, rather than in choosing the one true voting system to rule them all (pun intended).
With that in mind, one of the most interesting facts about Arrow's impossibility theory is that there are voting methods which aren't bound by it, like Score Voting. Quoting from Wikipedia:
As it satisfies the criteria of a deterministic voting method, with non-imposition, non-dictatorship, monotonicity, and independence of irrelevant alternatives, it may appear that it violates
Arrow's impossibility theorem. The reason that score voting is not a counter-example to Arrow's theorem is that it is a cardinal voting method, while the "universality" criterion of Arrow's
theorem effectively restricts that result to ordinal voting methods
As such, I'm hedging my bets: impossibility theorems must be taken with a grain of salt; they can be stepped over if their assumptions do not hold.
The Indian Mathematician Brahmagupta describes the solution to the quadratic equation as follows:
18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the
square. [The result is] the middle [number].
to describe the solution to the quadratic equation ax^2 + bx = c.
I read Parfit's piece with the same admiration, sadness and sorrow with which I read the above paragraph. On the one hand, he is oftent clearly right. On the other hand, he's just working with very
rudimentary tools: mere words.
With that in mind, how do Shapley values stand up to critique? Parfit proceeds by presenting several problems, and Toby Ord suggested that Shapley values might perform inadequately on some of them;
I'll sample the Shapley solution to those I thought might be the trickiest ones:
The First Rescue Mission
I know all of the following. A hundred miners are trapped in a shaft with flood-waters rising. These men can be brought to the surface in a lift raised by weights on long levers. If I and three
other people go to stand on some platform, this will provide just enough weight to raise the lift, and will save the lives of these hundred men. If I do not join this rescue mission, I can go
elsewhere and save, single-handedly, the lives of ten other people. There is a fifth potential rescuer. If I go elsewhere, this person will join the other three, and these four will save the
hundred miners.
Do Shapley Values solve this satisfactorily? They do. If you go to save the other ten, your Shapley value is in fact higher. You can check this on http://shapleyvalue.com/; input 100 for every group
with four participants, 10 for every group of less than four people which contains you. For the total, try with either 110, or 100, corresponding to whether you remain or leave.
The Second Rescue Mission
As before, the lives of a hundred people are in danger. These people can be saved if I and three other people join in a rescue mission. We four are the only people who could join this mission. If
any of us fails to join, all of the hundred people will die. If I fail to join, I could go elsewhere and save, single-handedly, fifty other lives.
Do Shapley Values solve this satisfactorily? They do. By having a stronger outside option, your Shapley value is in fact increased, even if you end up not taking it. Again, you can check on the link
above, inputing 50 whenever you're in the group, and 100 (or 50 again) when the whole group is involved.
Simultaneous headshots
X and Y simultaneously shoot and kill me. Either shot, by itself, would have killed.
Do Shapley Values solve this satisfactorily? Maybe? They each get responsiblity for half a death; whether this is satisfactory is up to discussion. I agree that the solution is counter-intuive, but
I'm not sure it's it's wrong. In particular, consider the question with the signed reversed:
I die of an unexpected heart attack. X and Y simultaneously revive me (they both make a clone of me with the memory backup I had just made the day before, but the law sadly only allows one
instance per being, so one has to go).
In this case, I find that my intuition is reversed; X and Y "should focus on cases which are less likely to be resurrected", and I see it as acceptable that they each get half the impact. I take this
as a sign that my intuition here isn't that reliable in this case; resurrecting someone should probably be as good as kiling them is bad.
Consider also:
You have put a bounty of 1 million dollars on the death of one of your hated enemies. As before, X and Y killed them with a simultaneous headshot. How do you divide the bounty?
On the other hand, we can add a fix on Shapley values to consider intentions (that is, expected values), which maybe fixes this problem. We can also use different variations of Shapley values
depending on whether we want to coordinate with, award, punish, incentivize someone, or to attribute value. For this, see the last and the second to last sections. In conclusion, this example is
deeply weird to me, perhaps because the "coordinate" and "punish" intuitions go in different directions.
German Gift
Statement of the problem: X tricks me into drinking poison, of a kind that causes a painful death within a few minutes. Before this poison has any effect, Y kills me painlessly.
Do Shapley Values solve this satisfactorily? They do. They can differentiate between the case where Y kills me painlessly because I've been poisoned (in which case she does me a favor), and the case
where Y intended to kill me painlessly anyways. Depending on how painful the death is (for example, a thousand and one times worse than just dying), Y might even end up with a positive Shapley value
even in that second case.
Third Rescue Mission
Statement of the problem: As before, if four people stand on a platform, this will save the lives of a hundred miners. Five people stand on this platform.
Do Shapley Values solve this satisfactorily? They do. They further differentiate cleanly between the cases where:
• All five people see the opportunity at the same time. In this case, each person gets 1/5th of the total.
• Four people detect the opportunity at the same time. Seeing them, a fifth person joins in. In this case, the initial four people each get 1/4th, and the fifth person gets 0.
An additional problem:
Here is an additional problem which I also find counterintuitive (though I'm unsure on how much to be confused about it):
X kills me. Y resurrects me. I value my life at 100 utilons.
Here, the Shapley value of X is -50, and the Shapley value of Y is 50. Note, however, that the moment in which Y has an outside option to save someone else, their impact jumps to 100.
Ozzie Gooen comments:
Note that in this case case, the counterfactuals would be weird too. The counterfactual value of X is 0, because Y would save them. The counterfactual value of Y would be +100, for saving them.
So if you were to try to assign value, Y would get lots, and X wouldn’t lose anything. X and Y could then scheme with each other to do this 100 times and ask for value each time
Overall, I think that Shapley values do pretty well on the problems posed by Parfit on Five Mistakes on Moral Mathematics. It saddens me to see that Parfit had to resort to using words, which are
unwieldy, for considering hypothesis of like the following:
(C7) Even if an act harms no one, this act may be wrong because it is one of a set of acts that together harm other people. Similarly, even if some act benefits no one, it can be what someone
ought to do, because it is one of a set of acts that together benefit other people.
Shapley value puzzles
[The first four puzzles of this section, and the commentary in between, were written by Ozzie Gooen.]
Your name is Emma. You see 50 puppies drowning in a pond. You think you only have enough time to save 30 puppies yourself, but you look over and see a person in the distance. You yell out, they come
over (their name is Phil), and together you save all the puppies from drowning.
Calculate the Shapley values for:
The correct answer, of course, for both, should have been “an infinitesimal fraction” of the puppies. In your case, your parents were necessary for you to exist, so they should get some impact. Their
parents too. Also, there were many people responsible for actions that led to your being there through some chaotic happenstance. Also, in many worlds where you would have not been there, someone
else possibly would have; they deserve some Shapley value as well.
In moral credit assignment, it seems sensible that all humans should be included. That includes all those who came before, many of whom were significant in forming the exact world we have today.
However, maybe we want a more intuitive answer for a very specific version of the Shapley value; we’ll only include value from the moment when we started the story above.
Now the answer is Emma: 40 puppies, Phil: 10 puppies. In total, you share 50 saved puppies. You can tell by trying it out in this calculator.
Now that we’ve solved all concerns with Shapley values, let’s move on to some simpler examples.
You (Emma again) are enjoying a nice lonely stroll in the park when you hear a person talking loudly on their cell phone. Their name is Mark. You stare to identify the voice, and you spot some
adorable puppies drowning right next to him. You yell at Mark to help you save the puppies, but he shrugs and walks away, continuing his phone conversation. You save 30 puppies. However, you realize
that if it weren’t for Mark, you wouldn’t have noticed them at all.
Calculate the Shapley values for:
You (Emma again) are enjoying a nice lonely stroll in the park when you hear a rock splash in a pond. You look and notice some 30 adorable puppies drowning right to it. You save all of the puppies.
You realize that if it weren’t for the rock, you wouldn’t have noticed them at all.
Calculate the Shapley values for:
You (Emma again), are enjoying a nice stroll in the park. Alarmedly, 29 paperclip satisficers* inform you that a paperclip is going to be lost forever, and 30 adorable puppies will drown unless you
do something about it. You, together with the paperclip satisficers, spend three grueling hours saving the 30 puppies and the paperclip.
Calculate the Shapley values for:
• You (Emma)
• Each paperclip satisficer
You (Emma again), decide that this drowning puppies business must stop, and create the Puppies Liberation Front. You cooperate with the Front for the Liberation of Puppies, such that the PLF gets the
puppies out of the water, and the FLP dries them, and both activities are necessary to rescue a puppy. Together, you rescue 30 puppies.
Calculate the Shapley values for:
• The Puppies Liberation Front:
• The Front for the Liberation of Puppies:
The Front for the Liberation of Puppies splits off a subgroup in charge of getting the towels: The Front for Puppie Liberation. Now;
• The Puppies Liberation Front gets the puppies out of the water
• The Front for the Liberation of Puppies dries them
• The Front for Puppie Liberation makes sure there are enough clean & warm towels for every puppie.
All steps are necessary. Together, you save 30 puppies. Calculate the Shapley value of:
• The Puppies Liberation Front:
• The Front for the Liberation of Puppies:
• The Front for Puppie Liberation:
Your name is Emma. Phil sees 30 puppies drowning in a pond, and he yells at you to come and save them. To your frustration, Phil just watches while you do the hard work. But you realize that without
Phil’s initial shouting, you would never have saved the 30 puppies.
Calculate the Shapley values for:
You are Emma, again. You finally find the person who has been trying to drown so many puppies, Lucy. You ask how many puppies she threw into the water: 100. Relieved, you realize you (and you alone)
have managed to save all of them.
Calculate the Shapley values for:
Speculative Shapley extensions.
Shapley values + Decision theory.
Epistemic status: Very biased. I am quite convinced that some sort of timeless decision theory is probably better. I also think that it is more adequate than competing theories in domains where other
agents are simulating you, like philantropic coordination questions.
In the previous post, Toby Ord writes:
In particular, the best things you have listed in favour of the Shapley value applied to making a moral decision correctly apply when you and others are all making the decision 'together'. If the
others have already committed to their part in a decision, the counterfactual value approach looks better.
e.g. on your first example, if the other party has already paid their $1000 to P, you face a choice between creating 15 units of value by funding P or 10 units by funding the alternative. Simple
application of Shapley value says you should do the action that creates 10 units, predictably making the world worse.
One might be able to get the best of both methods here if you treat cases like this where another agent has already committed to a known choice as part of the environment when calculating Shapley
values. But you need to be clear about this. I consider this kind of approach to be a hybrid of the Shapley and counterfactual value approaches, with Shapley only being applied when the other
agents' decisions are still 'live'. As another example, consider your first example and add the assumption that the other party hasn't yet decided, but that you know they love charity P and will
donate to it for family reasons. In that case, the other party's decision, while not yet made, is not 'live' in the relevant sense and you should support P as well.
Note that the argument, though superficially about Shapley values, is actually about which decision theory one is using; Toby Ord seems to be using CDT (or, perhaps, EDT), whereas I'm solidly in the
camp of updateless/functional/timeless decision theories. From my perspective, proceeding as the comment above suggests would leave you wide open to blackmail, would incentivize commitment races and
other nasty things (i.e., if I'm the other party in the example above, by commiting to donate to my pet cause, I can manipulate Toby Ord to donate money to the charity I love from family reasons,
(and to take it away from more effective charities (unless, of course, Toby Ord has previously commited to not negotiating with such blackmailers, and I know that))). I'm not being totally fair here,
and timeless decision theories also have other bullets to bite (perhaps transparent Newcomb and counterfactual mugging are the more antiintuitive examples (though I'd bite the bullet for both (as a
pointer, see In logical time, all games are iterated games)))
But, as fascinating as they might be, we don't actually have to have discussions about decision theories so full of parentheses that they look like Lisp programs. We can just earmark the point of
disagreement. We can specify that if you're running a causal decision theory, you will want to consider only agents whom you can causally affect, and will only include such agents on your Shapley
value calculations, whereas if you're running a different decision theory you might consider a broader class of agents to be "live", including some of those who have already made their decisions. In
both cases, you're going to have to bite the bullets of your pet decision theory.
Personally, and only half-jokingly, there is a part of me which is very surprised to see decision theory in general and timeless decision theories in particular be used for a legit real life problem,
namely philantric coordination theory, as the examples I'm most familiar with all involve Omegas which can predict you almost perfectly, Paul Elkmans who can read your facial microexpressions, and
other such contrived scenarios.
Shapley values + Moments of consciousness.
One way to force irrelevance of cabals is to define a canonical definition of agent, and have the value of the group just be the sum of the values of the individuals. One such canonical definition
could be over moments of consciousness, that is, you consider each moment of consciousness to be an agent, and you compute Shapley values over that. The value attributed to a person would be the sum
of the Shapley values of each of their moments of consciousness. Similarly, if you need to compute the value of a group, you compute the Shapley value of the moments of consciousness of the
integrants the the group.
On the one hand, the exact result of this procedure are particularly uncomputable as of 2020. And yet, using your intuition, you can see that the Shapley value of a person would be proportional to
the number of necesary moments of consciousness which the person contributes, so not all is lost. On the other hand, it buys you irrelevance of cabals, and somewhat solves the parasite problem:
• $1000 and 100h are needed for an altruistic project. Peter Parasite learns about it and calls Anna Altruist, and she puts in the $1000 and the 100h needed for the project.
• $1000 and 100h are needed for an altruistic project. Pamela Philantropist learns about it and calls Anna Altruist and they each cough up $500 and 50h to make it possible.
So taking into account moments-of-consciousness would assign more value to those who put in more (necessary) effort. Your milleage may vary with regards to whether you consider this to be a positive.
Shapley Values + Sensitivity Analysis + Expected Values
Epistemic Status: Not original, but I don't know where I got the idea from.
Given some variable of interest, average the Shapley values over different values of that variable, proportional to their probabilities. In effect, don't report the Shapley values directly, but do a
Sensitivity Analysis on them.
In the cases in which you want to punish or reward an outcome, and in particular in cases where you want to point to a positive or negative example for other people, you might want to act not on the
Shapley value from the world we ended up living in, but to take the expected value of the Shapley Value by the agent under consideration. Expected values either with the information you had, or from
the information they had.
If you the agent's information for those expected values, this allows you to punish evil but incompetent people, celebrate positive but misguided acts (which you might want to do for e.g., small
kids). If you use your own information for those expected values, this also allows you to celebrate competent but unlucky entrepeneurs, and punish evil and competent people even when, by chance, they
don't succeed.
However, Shapley values are probably be too unsophisticated to be used in situations which are primarily about social incentives.
Some complimentary (and yet obligatory) ramblings about Shapley Values, Goodhart's law, and Stanovich's disrationalia.
So one type of Goodhart's law is Regressional Goodhart.
When selecting for a proxy measure, you select not only for the true goal, but also for the difference between the proxy and the goal. Example: height is correlated with basketball ability, and
does actually directly help, but the best player is only 6'3", and a random 7' person in their 20s would probably not be as good
Other examples relevant to the discussion at hand:
• If you're optimizing your counterfactual value, you're also optimizing for the difference between the counterfactual values and the more nuanced thing which you care about.
• If you're optimizing your Shapley value, you're also optimizing for the difference between the Shapley values and the more nuanced thing which you care about.
• If you're optimizing for cost-effectiveness, you're also optimizing for the difference between being cost-effective and the more nuanced thing which you care about.
• If you're optimizing for being legible, you're also optimizing for the difference between being legible and the more nuanced thing which you care about.
• More generally, if you're optimizing for a measure of impact, you're also optimizing for the difference between what you care about and the measure of impact.
Of course, one could take this as a fully general counterargument against optimization and decide to become chaotic good. Instead, one might recognize that, even though you might cut yourself with a
lightsaber, lightsabers are powerful, and you would want to have one. Or, in other words, to be really stupid, you need a theory (for an academic treatment on the matter, consult work by Keith
Stanovich). Don't let Shapley values be that theory.
One specific way one can be stupid with Shapley values is by not disambiguating between different use cases, and not notice that Shapley values are imperfect at the social ones:
• Coordinate. Choose what to do together with a group of people.
• Award. Pick a positive exemplar and throw status at them, so that people know what is your or your group's idea of "good".
• Punish. Pick a negative exemplar and act so as to make clear to your group that this exemplar corresponds to your group's idea of "bad".
• Incentivize. You want to see more of X. Create mechanisms so that people do more of X.
• Attribute. Many people have contributed to X. How do you divide the spoils?
To conclude, commenters in the last post emphasized that one should not consider Shapley Values as the philosopher's stone, the summum bonum, and I've smoothed my initial enthusiasm somewhat since
then. With that in mind, we considered Shapley values in a variety of cases, from the perhaps useful in practice (philantropic coordination theory, forecasting incentives), to the exceedingly
speculative, esoteric and theoretical.
Assorted references
Appendix: A more complicated worked example.
Suppose that you have two players, player a and Player B, and three charities: GiveDirectly, SCI, and the Not Quite Optimal Charity (NQOC).
Charity Value for A Value for B
GD y=x y=x
SCI y=2x if x<10, y=x/2 + 15 if x>=10 y=2x
NQOC 1/10 1/5
Or, in graph form:
• I don't actually think that the diminishing returns are likely to be that brutal and that sudden, but assuming they are simplified calculations.
• I don't actually know whether 1/5th to 1/10th of the value of a donation to GiveDirectly is a reasonable estimate of what an informed donor would have made in the absence of GiveWell's analysis.
Then suppose that:
• Player A has 10 million, and information
• Player B has 1 million, and not much information.
Then the outcomes might look like
Group Value for A Value for B
{} 0* 0*
{A} 20 20
{B} 1/10 1/5
{A,B} 20 + X/2 + Y 20 + 2X + Y
where X and Y are the quantities which player B donates to SCI and GiveDirectly, respectively, and they sum up to 1. So the value of cooperating is
• X/2 + Y -1/10 according to A's utility function
• 2X + Y - 1/5 according to B's utility function
In particular, if B donates only to SCI, then the value of cooperating is:
• X/2 -1/10 according to A's utility function
• 2X - 1/5 according to B's utility function
Now, we have the interesting scenario in which A is willing to sell their share of the impact for X/4 - 1/20, but B is willing to buy it for more, that is, for X - 1/10. In this case, suppose that
they come to a gentleman's agreement and decide that the fair price is (X/4 - 1/20)/2 + (X - 1/10)/2, which simplified, is equal to (5 X)/8 - 3/40.
Now, player B then donates X to SCI and buys player A's certificates of impact for that donation (which are cheaper than continuing to donate). If they spend the million, then X + (5 X)/8 - 3/40 = 1,
so X≈0.66, that is, 0.66 million are donated to SCI, whereas 0.34 million are spent buying certificates of impact from player A, which then donates that to GiveDirectly.
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DAX Visual Calculations: Bringing Excel Simplicity to Power BI
If you’ve worked with DAX (Data Analysis Expressions) measures in Power BI, there’s a good chance you have a love-hate relationship with them. On one hand, DAX is easy to start learning and
incredibly powerful. On the other hand, there’s no way to directly refer to data inside of a visual, making calculations like running sums and moving averages difficult to build. Thankfully, that’s
about to change as Microsoft rolls out visual calculations for Power BI which will allow you to build DAX calculations that are defined inside of a specific visual, similar to how calculations are
done in Excel.
Benefits of Visual Calculations
Visual calculations are built inside of a report’s visuals, so their code is executed directly on the visual they’re built in. Since this code is specific to a single visual and directly references
the data in that visual, it’s easier to write the DAX and easier to maintain. On top of this, visual calculations operate on the aggregated data already found inside the visual, which usually results
in better performance than a standard DAX measure that has to operate on the detail level of the source data.
Building Visual Calculations
Many existing DAX functions can be used in visual calculations and will work similarly to calculated columns. Templates are also available to make it easier to write common visual calculations such
as running sum, moving average, and percent of grand total. Additional details can be found in the example below.
As of today, visual calculations are still in preview mode, so you’ll need to first enable them by going to File > Options & Settings > Preview features and checking the box next to Visual
Calculations. After you’ve done this, creating a new visual calculation is as easy as selecting a visual and pressing the New Calculation button on the “Home” tab.
In this example, we’re going to quickly build a running sum calculation that resets at the beginning of each year. We’ll be performing this calculation on the Sum of Units Sold data in Microsoft’s
financials test dataset, found by going to File > New > Report > Use sample data > Load sample data. Below are the steps and the simple DAX needed to accomplish this:
1. Select your visual. The visual in this example is a Matrix showing the Sum of Units Sold with a Year/Quarter/Month date hierarchy on the row axis.
2. Press the New Calculation button on the “Home” tab.
3. The visual calculation window will open. This window contains a visual preview that shows your visual, a formula bar where you can build your calculations, and a visual matrix that displays the
results of the visual calculations as you create them.
4. In the formula bar, give your calculation a name and create the DAX code. Here you can see that the calculation is as simple as calling the “RUNNINGSUM” function and referring to the Sum of Units
Sold as the column to execute the running sum on. The “HIGHESTPARENT” property at the end of the calculation resets the calculation at the beginning of each year because, for this visualization,
the highest parent on the row axis is the year. Note that you could also press the fx button next to the formula bar and choose “Running sum” from the list of common visual calculation templates
rather than manually typing the calculation.
In this example, I’ve named my calculation “Running Total by Year.” In a normal DAX measure, this name would not be recommended because it’s ambiguous (running total of what?). However, because
visual calculations are used only in the visual they’ve been defined on and the Sum of Units Sold column is the only other calculation in this visual, this name is fine.
5. Once you’ve created your calculation, click Back to Report at the top of the page and you’ll see your updated visual with the new visual calculation. Notice how the calculation resets in January
When to Utilize Visual Calculations
Visual calculations are a fantastic option when you need to build an Excel-like calculation that specifically references a column, measure, or another visual calculation in a visual. They’re usually
easier to create than standard DAX calculations, and allow you to focus on the results rather than spending time figuring out complex filter contexts and all the ways your new calculation might need
to interact with your data model.
Visual calculations aren’t perfect though. As of this writing, there are a handful of limitations, such as:
• Underlying data that can’t be exported from visuals using visual calculations.
• Filters that cannot be applied to the results of visual calculations.
• Some visuals, like tree maps, geographic maps, and small multiples, are unsupported.
There are also times when using a visual calculation will work, but you might be causing yourself more effort in the future by using them. For example, if you have 10 different visuals that all need
a running sum of units sold by year, it will likely take you longer to build your running sum visual calculations for all 10 visuals than it would be to build a single standard DAX measure and apply
that to each visual. Also, if you ever need to change the calculation in the future, it would probably take you longer to change the 10 visual calculations than it would to change the single standard
DAX measure. Generally, if visual calculations are supported for your use case, and the calculation you’re building isn’t going to be repeated many times throughout your report, it’s worth your time
to explore the visual calculation option.
What Will You Use Visual Calculations For?
Unless you’re already a DAX wizard, we think you’ll find visual calculations to be a powerful and time-saving tool when building Power BI reports. But don’t take our word for it…give them a try in
your next report and see for yourself! For additional information on visual calculations, see Microsoft’s visual calculations article.
Looking for more? Visit our blog to find more best practices, insights, and updates from our experts.
Snow Fox Data
Snow Fox Data is a premier data strategy, data science and analytics solutions provider. Our team of data architects, data scientists, data engineers, and data analysts are passionate about helping
businesses make a difference with data.
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Season 8 Episode 0
At the start of the previous season, we talked about how 91 is not a prime number. But how can you tell? In this episode, we'll factorise some numbers!
Further Reading
Quadratic Sieve
The quadratic sieve algorithm is described in (much) more detail here.
There’s a step that’s a little misleading in the Wikipedia page; the page says at one point "we can then factor 1649=gcd(194,1649)gcd(34,1649)". This step doesn’t necessarily follow from what we have
on the previous line. In general, we're hoping that those greatest common divisors are the factorisation, but it’s possible that they’re not. I first learnt about it in the Oxford Mathematics Part A
short option Number Theory lecture notes.
The very large number at the start of the stream is called RSA-129. You can read about it here.
I misspoke slightly on the stream – RSA-129 was not part of the RSA Factoring Challenge issued by RSA Laboratories. It was set as a separate RSA Challenge by Martin Gardner back in 1977. There's a
description of the effort to factorise the number here, with an enigmatic title; The Magic Words are Squeamish Ossifrage.
Make a perfect square
Find a non-empty subset of the following four numbers which multiply together to give a square number.
$$30, 150, 12, 10.$$
(Harder) Prove that if four numbers have no prime factors other than 2 or 3 or 5, then there is a non-empty subset of those numbers which multiply together to give a square number.
(Harder) Prove that if \(n+1\) numbers have just \(n\) distinct prime factors between them, then there is a non-empty subset of those numbers which multiply together to give a square number.
Highest Common Factor
I said that I would give you a link for Euclid’s Algorithm.
Perhaps it’s surprising that “find any non-trivial factor of this composite number” is usually rather hard, while “find the largest number that’s a factor of both of these two numbers” is usually
rather easy. You might think that you need to be able to do the first thing in order to do the second thing, but you don’t.
Follow-up question! This suggests a factorisation technique where we multiply together all the primes less than \(n\) and solve the “easy” problem of finding the largest number that’s a factor of
both that product and \(n\). Why isn’t this a good idea?
While I was trying to give an overview of the algorithm, I mentioned Al-Khwarizmi. This was a well-meaning attempt to get some history of mathematics into the livestream, but I was way off the mark
with my dates; I think I said that Al-Khwarizmi knew this algorithm before Euclid, but that’s impossible! Al-Khwarizmi is still worth reading about though. We get the word algorithm from his name.
That's maybe why I was thinking of him.
A Tale of Two Sieves
The Quadratic Sieve was developed by Carl Pomerance, and they've written a great overview article on the method. It’s 13 pages, and even just the first four pages give you a good bit of context. Get
it here; A Tale of Two Sieves.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.
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ETS Quant Practice Test 2 Q.6
I’m came across this question from ETS Quant Practice test.
The question states that the averages of lists X and Y are equal. In the solution explanation, they solve for s in terms of t and say s could be negative, zero, or positive, leading to answer D
My question: Since X is a list (therefore should be ordered) and s is listed after 5 in X, shouldn’t s be always positive? Or it’s only when they clearly state that a list is ordered that we can
consider it to be in order ?
A “list” isn’t usually a thing in math unless sufficient context is given. You can’t inherit definitions from other areas of math i guess and treat a list as something like an indexed set.
Anyway, for your specific question:
\frac{2 + 5 + s + t}{4} = \frac{2 + 5 + t}{3} \implies \boxed{3s - t = 7}
I think you it’s pretty evident now what the answer is.
2 Likes
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Plans for Mixture model package - help in the process
Replied on Mon, 11/08/2021 - 10:49
First off, let me say that I'm pleased to hear of your interest in developing a package around OpenMx. OpenMx has a lot of capabilities that rarely see use, since only a power user would know how to
write a script to use them.
You probably know this already, but OpenMx's Newton-Raphson optimizer requires analytic first and second derivatives of the fitfunction being optimized. Presently, only two kinds of models are able
to automatically provide such analytic derivatives to the optimizer: IRT models (as you saw), and models that use the GREML expectation and fitfunction. However, `mxFitFunctionAlgebra()` has support
for user-provided analytic fitfunction derivatives. For perspective, let's take a look at a modification of the example syntax for `mxComputeEM()` :
mm <- mxModel(
"Mixture", data4mx, class1, class2,
mxAlgebra((1-Posteriors) * Class1.fitfunction, name="PL1"),
mxAlgebra(Posteriors * Class2.fitfunction, name="PL2"),
mxAlgebra(PL1 + PL2, name="PL"),
mxAlgebra(PL2 / PL, recompute='onDemand',
initial=matrix(runif(N,.4,.6), nrow=N, ncol = 1), name="Posteriors"),
mxAlgebra(-2*sum(log(PL)), name="FF"),
# write some expression here that evaluates to a row vector containing the first partial derivatives of 'FF' w/r/t all free parameters
# write some expression here that evaluates to a matrix of second partial derivatives of 'FF' w/r/t all free parameters
The above should work, once the comments have been replaced with appropriate MxAlgebra expressions. However, I have never tried anything quite like this before.
As to whether or not it makes a difference to use Newton-Raphson versus quasi-Newton ("gradient descent")...? Well, since Newton-Raphson uses analytic first and second fitfunction derivatives, it
will typically be faster (fewer major iterations, fewer fitfunction evaluations) and more numerically accurate. However, OpenMx's Newton-Raphson optimizer does not handle lower and upper bounds ("box
constraints") on parameters very well, and is incompatible with explicit MxConstraints. In contrast, OpenMx's three quasi-Newton optimizers have full support for constrained optimization. Also, note
that the quasi-Newton optimizers can all optionally use analytic first derivatives of the fitfunction.
Replied on Wed, 11/10/2021 - 17:50
In reply to Newton-Raphson by AdminRobK
Newton-Raphson and HMM
I think the same, we want to make some of these complex models more accesible.
Thank you, I see how to make some examples from these models run with Newton-Raphson, but it would take me longer than I want for the first edition of the package. So that it works for all the models
I want to make available. Will work on that addition for a future version.
On another issue that is higher priority in our models, is the addition of Hiddent Markov Models (HMM). I am working on applying HMM with multiple subjects in time series. The example from the
**mxExpectationHiddenMarkov()** function is applied with a single time series
start_prob <- c(.2,.4,.4)
transition_prob <- matrix(c(.8, .1, .1,
.3, .6, .1,
.1, .3, .6), 3, 3)
noise <- .05
# simulate a trajectory
state <- sample.int(3, 1, prob=transition_prob %*% start_prob)
trail <- c(state)
for (rep in 1:500) {
state <- sample.int(3, 1, prob=transition_prob[,state])
trail <- c(trail, state)
# add noise
trailN <- sapply(trail, function(v) rnorm(1, mean=v, sd=sqrt(noise)))
classes <- list()
for (cl in 1:3) {
classes[[cl]] <- mxModel(paste0("cl", cl), type="RAM",
mxPath("one", "ob", value=cl, free=FALSE),
mxPath("ob", arrows=2, value=noise, free=FALSE),
m1 <-
mxModel("hmm", classes,
mxData(data.frame(ob=trailN), "raw"),
mxMatrix(nrow=3, ncol=1, free=c(F,T,T),
lbound=0.001, ubound=.99,
labels=paste0('i',1:3), name="initial"),
mxMatrix(nrow=length(classes), ncol=length(classes),
labels=paste0('t', 1:(length(classes) * length(classes))),
components=sapply(classes, function(m) m$name),
transition="transition", scale="softmax"),
m1$transition$free[1:(length(classes)-1), 1:length(classes)] <- TRUE
m1 <- mxRun(m1)
But for most scenarios we have multiple subjects time series. My idea was to define the classes as random intercept models as multilevel models based on the time variables. Or should the HMM be
define in another way with multiple subjects?
But when I define the model like this I get an error
Running hmm with 9 parameters
Error in runHelper(model, frontendStart, intervals, silent, suppressWarnings, :
hmm.fitfunction: component class1.fitfunction must be in probability units
Here is my model so far
dat <- rio::import("schizophrenia_markov.sav")
dat$time <- as.factor(dat$time)
nclass <- 2
class <- list()
for(cl in 1:nclass){
class[[cl]] <- mxModel(paste0("class",cl),
mxModel(paste0('time',cl), type="RAM",
latentVars = c('time'),
mxData(data.frame(time=unique(dat$time)), 'raw', primaryKey='time'),
mxPath('time', arrows=2, values=1)),
manifestVars = c('DEP'),
mxData(dat[,c("DEP","time")], 'raw'),
mxPath('one', 'DEP'),
mxPath('DEP', arrows=2, values=1),
mxPath(paste0('time',cl,'.time'), 'DEP', values=1,
free=FALSE, joinKey='time'),
hmm <- mxModel("hmm", class,
mxData(dat[,c("DEP","time")], 'raw'),
mxMatrix(nrow=length(class), ncol=1, free=c(F,T),
lbound=0.001, ubound=.999, values=1,
mxMatrix(nrow=length(class), ncol=length(class),
labels=paste0('t', 1:(length(class) * length(class))),
components=sapply(class, function(m) m$name),
transition="transition", scale="softmax"),
hmm$transition$free[1:(length(class)-1), 1:length(class)] <- TRUE
hmmFit <- mxRun(hmm)
Thanks for all the help!
File attachments
Replied on Wed, 11/10/2021 - 19:06
jpritikin Joined: 05/23/2012
One way to get this model working is to group all the observations that happen at the same time into a single row. This would remove the multilevel structure from the data. Each row would be
independent. The way you coded it, the rows are not independent because there are multiple rows that happened at the same time.
Replied on Wed, 11/17/2021 - 20:14
In reply to multilevel by jpritikin
HMM with multiple subjects and multiple time points
I was able to make the random intefcept mode work with the wide data format
dat <- rio::import("schizophrenia_markov.sav")
###### random intercept SEM structure
dat$time <- as.factor(dat$time)
dat2 <- spread(dat[,c("ID","DEP","time")], key="time", value="DEP")
colnames(dat2) <- c("ID", paste0("t",0:6))
dat3 <- data.frame(dat2[,2:8])
dataRaw <- mxData( observed=dat3, type="raw" )
vars <- colnames(dat3)
nclass <- 2
class <- list()
for(cl in 1:nclass){
class[[cl]] <- mxModel(paste0("class",cl),
latentVars = c("int"),
mxPath( from="int", arrows=2,
free=T, values=1),
mxPath( from="one", to="int", arrows=1,
free=TRUE, values=1 ),
mxPath( from="int", to=vars, arrows=1,
free=F, values=1 ),
mxPath( from=vars, arrows=2,
free=T, values=rep(0, length(vars)),
mxPath( from="one", to=vars,
free=F, values=rep(0, length(vars)),
labels=c(paste0("mean",cl,1:length(vars))) ),
hmm <- mxModel("hmm", class,
mxMatrix(nrow=length(class), ncol=1, free=c(F,T),
lbound=0.0001, ubound=.9999,
mxMatrix(nrow=length(class), ncol=length(class),
labels=paste0('t', 1:(length(class) * length(class))),
components=sapply(class, function(m) m$name),
transition="transition", scale="softmax"),
hmm$transition$free[1:(length(class)-1), 1:length(class)] <- TRUE
hmmFit <- mxRun(hmm)
hmmFit <- mxTryHard(hmm, extraTries=50, maxMajorIter=5000, exhaustive=T)
But this is still not the model I am looking to replicate from HMM. As the model I am looking to replicate is to have multiple subjects with multiple time points. From here the "initial" would be the
proportions in each latent class at the first time point, and the "transition" would be the transition between latent classes over time.
So, as another option I attempted to use the long format data, and add the time variable as second level predictor to the initial and transition. But this didnt work as I could only add the time
predictor to the initial states and from the data, instead of the second level structure model (similar to this [post](https://github.com/OpenMx/OpenMx/blob/3cb6593af0754be134e107cf4f42315d8c18db2b/
inst/models/passing/xxm-3.R) )
dat$time <- as.factor(dat$time)
nclass <- 2
class <- list()
for(cl in 1:nclass){
class[[cl]] <- mxModel(paste0("class",cl),
manifestVars = c('DEP'),
mxData(dat[,c("DEP","time")], 'raw'),
mxPath('one', 'DEP'),
mxPath('DEP', arrows=2, values=1),
#mxPath(paste0('time',cl,'.time'), 'DEP', values=1,free=FALSE,joinKey='time'),
hmm <- mxModel("hmm", class,
## MLM model structure
mxModel(paste0('time'), type="RAM",
latentVars = c('time'),
mxData(data.frame(time=unique(as.factor(dat$time))), 'raw',
mxPath('time', arrows=2, values=1)),
mxData(dat[,c("DEP","time")], 'raw'),
mxMatrix( type = "Full", nrow = 2, ncol = 2,
free=c(TRUE, FALSE,TRUE,TRUE), values=1,
labels = c("p11","p21", "p12", "p22"),
name = "initialM" ),
mxMatrix(nrow=2, ncol=1, labels=c(NA, "data.time"), values=1,
mxAlgebra(initialM %*% initialV, name="initial"),
mxMatrix(nrow=length(class), ncol=length(class),
labels=paste0('t', 1:(length(class) * length(class))),
mxMatrix(nrow=2, ncol=2, labels=c("data.time", "data.time","data.time", "data.time"), values=1,
mxAlgebra(transitionM %*% transitionV, name="transition"),
#mxPath('time.time', 'initial', free=FALSE, values=1, joinKey="time"),
components=sapply(class, function(m) m$name),
transition="transition", scale="softmax"),
hmm$transitionM$free[1:(length(class)-1), 1:length(class)] <- TRUE
hmmFit <- mxTryHard(hmm, extraTries=50, maxMajorIter=5000, exhaustive=T)
From the **schizophrenia_markov.sav**, the results I am trying to rplicate are
Initial states
0.9919 0.0081
Class[-1] 1 2
1 0.8351 0.1649
2 0.0013 0.9987
Not sure how to set up the **mxExpectationHiddenMarkov()** correctly here
Appreciate the help
Replied on Thu, 11/18/2021 - 09:24
AdminNeale Joined: 03/01/2013
Similar to this post?
The “this post” you refer to is just a test script. Can you please edit the link?
Replied on Thu, 11/18/2021 - 15:39
In reply to Similar to this post? by AdminNeale
Right link, wrong wording
Sorry for the lack of clarity. Did menat to include that link, as the example code for multilevel regression. But use the wrong wording calling it a post
Trying to apply the multilevel structure in the HMM for estimation with multiple subjects and multiple time points
Replied on Fri, 05/13/2022 - 06:03
R package in course
After talking with colleagues, I found that Caspar van Lissa has added some user friendly mixture modeling with OpenMx in his package tidySEM.
I am working with him on improving and adding more features to this package, including tutorial vignettes.
But I havent been able to work out a Hidden Markov Model when I have multiple items over time, and estimate the transitions between latent states.
Have try:
- random intercept model
- transpose the data
- CFA over time
- growth curve
But none of this have match the expected basic results for a HMM. Would anyone here have a new idea on how to implement this?
Once I have a working example we can translate this into the tidySEM package
Thank you
Replied on Fri, 05/13/2022 - 16:44
mhunter Joined: 07/31/2009
See example script
If I understand correctly what you're after, then there's an example script for that. It's [https://github.com/OpenMx/OpenMx/blob/master/inst/models/passing/HMM-multigroup.R](https://github.com/
OpenMx/OpenMx/blob/master/inst/models/passing/HMM-multigroup.R) which is a multi-subject HMM test script. Essentially, each person has a tall format data set with different rows being different
times, and different columns being different variables. Each person has a hidden Markov model that happens to be the same model. Then the people are combined into a multigroup model where each
"group" is a person.
One thing to be careful about: free parameters with the same labels are constrained to be equal. If you leave the free parameters unlabeled, they will be different. If you leave your parameters
unlabeled by accident, then you could end up estimating separate free parameters for everyone, and it could be quite a large number of parameters.
Good luck!
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8 Best AP Calculus BC Prep Book (2022) » Exam Cave
8 Best AP Calculus BC Prep Book (2022)
The AP BC Calculus exam has two sections. The first section takes 105 minutes and has 45 questions. It accounts for 50% of the overall exam score. You can’t use calculators in the first 30 questions,
but you can do so with the second 14 questions
The second section lasts 90 minutes and consists of six questions. You get two problems to solve for 30 minutes with the use of a calculator, but you won’t use the calculator for the last four
questions. So, the question is: what’s the best way to prepare?
In this detailed review, find out more about some of the best AP BC Calculus prep books, where to get them, and some common questions asked about BC Calculus prep books.
Best AP BC Calculus Exam Prep Books
Some of the Best AP Calculus BC prep books include:
1. Cracking the AP Calculus BC Exam
A comprehensive AB Calculus BC exam prep book
Cracking the AP Calculus BC Exam by Princeton Review 2020 Edition is a must-have when preparing for your Calculus exam. It features three full-length practice tests and has access to online
You should get this guide as a supplement to your course textbooks when you’re planning to revise for your Calculus exam. The guide also comes with a step by step walkthrough of the must-know
formulas and sample questions.
Students get questions from previous years to the current exam year. These questions have a detailed answer explanation to help you improve on similar questions in the future and give you an insight
into where you might have one wrong. The end of the chapter has tricks and tips to help with preparation. You also get valuable tactics to help you focus when you have little to no time to study
• Has challenging practice tests and detailed content review
• Provides a balanced and complete approach
• Online student tool pages
• Up to date information on course changes
• A cheat sheet for all crucial formulas
• Engaging activities to assess your progress
• Not ideal for the free-response section
Publisher: The Princeton Review
Year: 2019
Number of Pages: 768 pages
Grade: A+
2. Barron’s AP Calculus
A detailed study guide with a focus on different topics
Barron’s AP Calculus has excellent online reviews thanks to its high-quality study aids. It’s ideal for both AB calculus and BC calculus exam preparation.
The AP BC Calculus section gives you a targeted focus on different topics. One different aspect is the section that teaches you how to use a calculator properly. It’s essential to learn how to use
the required tool as it comes in handy during the exam.
Students also get three practice exams for the Calculus BC test. You’ll find this book excellent as it gives you content to study for two exams at once. That means if you are taking the AP Calculus
AB exam, you don’t have to buy another book.
On the downside, the book is suitable if you’re looking to polish up on the things you already know about BC calculus. Average students may find it challenging to solve the practice questions. Also,
the book hasn’t been updated since 2019, which means you may lose out on some information.
• A quality books that covers two topics at once without sacrificing on quality
• Has a dedicated section and practice exams
• Comes with examples to practice
• Comes with the new exam format
• Lets you learn how to use the graphing calculator properly
• Previous buyers found that it contains similar content with other Barron’s test prep books
• It’s still in its 15th edition and hasn’t been updated
Publisher: Barron’s Educational Series
Year: 2019
Number of Pages: 672 pages
Grade: B+
3. AP Calculus BC Lecture Notes-Bita Korsunsky
A perfect book with each lesson targeting a specific formula or skill
Featuring simple to follow review notes, the AP Calculus BC Lecture Notes comes with explanations and examples. In this book are full slides and review of all Calculus BC curriculum topics. You also
get a list of theorems and formulas that you need for the test.
The accurate and concise explanations of the example problems help you prepare ahead of the exam. Students who find it difficult to understand Calculus will find this book resourceful. It also serves
as a reinforcement for those experienced in the subject.
However, unlike other study guides and prep books, the AP Calculus BC Lecture Notes doesn’t come with practice or quiz tests. You also don’t have access to online resources. The book is best for
supplementing with other study guides and books.
• An organized and simple to follow approach
• Comes with theorems and formulas you’ll need for the test
• Accurate and concise explanations
• Doesn’t have practice tests
• Lacks online resources
Publisher: CreateSpace Independent Publishing Platform
Year: 2018
Number of Pages: 187 pages
Grade: B
4. Multiple Choice Questions to Prepare for the AP Calculus BC Exam
A preparation workbook with multiple choice questions for the AP Calculus BC exam
The Multiple Choice Questions to Prepare for the AP Calculus BC Exam is written by an award-winning Calculus teacher, Rita Korsunsky. She has an excellent history and 95% of her students get a score
of 5.
The guide comes with multiple-choice questions like those you’ll find in the actual Calculus BC exam. It also meets the College Board requirements, which is a plus.
Students get six multiple-choice exams, tips for taking the AP test, thermos and reference for the formula, and answer keys for simplicity.
What’s more, you can download step-by-step solutions for this book in PowerPoint format. You’ll get everything you need to score a 5 in your exam. Find the well-organized formulas and theorems that
are easy to view.
One aspect that makes this guide an excellent option is the section that provides some helpful tips to avoid common mistakes made during the BC exam. The book has fantastic reviews online as previous
buyers find it helpful in reinforcing their understanding of Calculus.
• Provides useful test-taking strategies
• Helpful explanation of the practice tests
• Up to date book
• Downloadable PowerPoint material for further studying
• Practice questions
• Reasonably priced and an easy to understand format
• It comes with an additional solution CD for a detailed explanation
Publisher: CreateSpace Independent Publishing Platform
Year: 2013
Number of Pages: 158
Grade: B
5. Calculus 12th Edition
A detailed book with examples, exercises, and applications
The Calculus 12th Edition by Howard Anton is one of the popular Calculus BC textbooks in the market. This book makes it simple for you to understand the critical concepts of Calculus as it covers all
the core functions of trigonometry, algebra, geometry, and elementary functions.
All this is to ensure that you’re prepared for the exam. Unlike other guides, this textbook uses visual, verbal, and algebraic approaches to teach the fundamental Calculus concepts.
• Has fantastic exercises, examples, and applications
• Provides a balance between clarity of explanations and rigor
• Presents concepts in a visual, verbal, numerical, and algebraic point of view
Publisher: Willey, 12th Edition
Year: 2021
Number of
Grade: B-
6. AP Calculus BC Lecture Notes Volume 1 and 2
A prep book with summarized Calculus BC exam
The AP Calculus BC Lecture Notes Volume 1 and 2 is another fantastic way to prepare for your BC exam within a short period.
Students get printouts of PowerPoint slides of the Calculus exam. These notes are fantastic for review and learning when you need to prepare efficiently and quickly. You’ll love how this book targets
specific content and skills. You get every Calculus concept, thanks to the detailed explanations of theorems.
The guide also illustrates step-by-step methods to solve different problems. Once you finish studying the notes, you’ll find all the theorems and formulas you need for AP Calculus BC test
Another impressive aspect is the multiple-choice type questions that resemble those you’ll find in the actual AP test. These problems have a similar difficulty level like those on the Calculus BC
• Detailed topics and notes
• An easy and clear book
• Explains challenging concepts easily
• User-friendly book
• Ideal resources for students and teachers
• May need you to go through weeks before the exam
Publisher: CreateSpace Independent Publishing Platform
Year: 2014
Number of Pages: 187
Grade: B-
7. 5 Steps to a 5: AP Calculus BC, 2022
A perfect book for the perfect score
The 5 Steps to a 5 is one book that offers skilled and practical knowledge to better understand what AP tests entail. As a student, you get helpful test-taking strategies. Being a revised syllabus
means that you get a score that matches the AP Calculus BC exam format.
Inside the book are three complete AP practice tests for the BC exam, three separate study plans, and an application feature that has extra practice questions. You also get assignment notification to
check your test readiness.
At the start, you get to create your customized study plan that is followed up by a diagnostic test to determine your knowledge level. Afterward, you can create your strategy using this book as a way
to prepare for the exam.
It comes with a content review section with topics outlined in the College Board’s outline. The review section is comprehensive as it connects one section to the other to follow up on what you’ve
You also get to review all the different theorems and formulas in the appendix. It’s an excellent and affordable guide that promises to help in your AP Calculus BC exam.
• Has an online app with features to help prep for the exam
• Reasonably priced
• Comes with practice tests
• Ideal for both teachers and students
• Uses a systematic steeply-step approach
Publisher: Mc-Graw Hill Education
Year: 2021
Number of Pages: 464
Grade: B
8. AP Calculus AB and BC Crash Course
A prep book to get you a high score in less time
Termed as the complete AP Calculus BC book, the AP Calculus AB and BC Crash Course provide a narrow focus for accelerated review. It comes with extensive practice tests with tricks and tips on how to
handle the questions.
Students get two full-length practice tests online. The best part is that you get the necessary information for the test without any filler content. You also get expert test-taking strategies that
cover free-response and multiple-choice sections.
Unlike other books, the Crash Course has its course and content layout focused on the exam entirely. It also acts as a good resource to complement your course books.
• Ideal for last-minute revision
• Focused content without the fluff
• Has helpful tips to handle the free-response and multiple-choice sections
• Some questions lack accuracy
• Only ideal as a supplement to other books
Publisher: Research and Education Association
Year: 2021
Number of Pages: 256
Grade: B-
AP BC Calculus Prep Book FAQs
Some of the frequently asked questions include:
What is the best AP BC Calculus Book?
That depends on what you’re looking for in a prep book. Some books are ideal for last-minute revision, while others are detailed and provide comprehensive information. You need to choose a book
depending on what you are looking for, your budget, and study needs.
What Should I Study for in the BC Exam?
You should study how to answer the free-response and multiple-choice questions. It’s also essential to learn more about how to use a graphing calculator.
Is AP Calculus BC Difficult?
In my experience, Calculus BC is not very challenging with lots of practice and studying. Go for it!
Final Thoughts
Although AP BC Calculus is a tough course, with the right mix of practice and knowledge, you can easily score a 5 in your exam. The above Calculus prep books will challenge you with free-response and
multiple-choice questions.
Remember to start reading early to avoid a last-minute rush as you’ll find it challenging to understand all the concepts and formulas.
Was this article helpful? Share it!
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Balancing Graphs Using Geometric Invariant Theory
Clayton Shonkwiler
Colorado State University
SIAM Minisymposium on Interactions Among Analysis, Optimization, and Network Science
October 5, 2024
\(A \in \mathbb{C}^{d \times d}\) is normal if \(AA^\ast = A^\ast A\).
\(0 = AA^\ast - A^\ast A = [A,A^\ast]\).
Define the non-normal energy \(\operatorname{E}:\mathbb{C}^{d \times d} \to \mathbb{R}\) by
\(\operatorname{E}(A) := \|[A,A^\ast]\|^2.\)
Obvious Fact.
The normal matrices are the global minima of \(\operatorname{E}\).
Theorem [with Needham]
The only critical points of \(\operatorname{E}\) are the global minima; i.e., the normal matrices.
\(\operatorname{E}\) is not quasiconvex!
Theorem [with Needham]
The only critical points of \(\operatorname{E}\) are the global minima; i.e., the normal matrices.
Let \(\mathcal{F}: \mathbb{C}^{d \times d} \times \mathbb{R} \to \mathbb{C}^{d \times d}\) be negative gradient descent of \(\operatorname{E}\); i.e.,
\(\mathcal{F}(A_0,0) = A_0 \qquad \frac{d}{dt}\mathcal{F}(A_0,t) = -\nabla \operatorname{E}(\mathcal{F}(A_0,t))\).
Theorem [with Needham]
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathcal{F}(A_0,t)\) exists, is normal, has the same eigenvalues as \(A_0\), and is real if \(A_0\) is.
\(\mathbb{C}^{d \times d}\) is symplectic, with symplectic form \(\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)\).
A symplectic manifold is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \(\omega \in \Omega^2(M)\).
Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)
dx \wedge dy \left( \textcolor{12a4b6}{a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}}, \textcolor{d9782d}{c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y}} \right) = ad
- bc
(a,b) = a \vec{e}_1 + b \vec{e}_2 = a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}
(c,d) = c \vec{e}_1 + d \vec{e}_2 = c \frac{\partial}{\partial x} + d \frac{\partial}{\partial y}
\(\mathbb{C}^{d \times d}\) is symplectic, with symplectic form \(\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)\).
Consider the conjugation action of \(\operatorname{SU}(d)\) on \(\mathbb{C}^{d \times d}\): \(U \cdot A = U A U^\ast\).
This action is Hamiltonian with associated momentum map \(\mu: \mathbb{C}^{d \times d} \to \mathscr{H}_0(d)\) given by
\(\mu(A) := [A,A^\ast]\).
So \(\operatorname{E}(A) = \|\mu(A)\|^2\).
Geometric Invariant Theory (GIT)
The GIT quotient consists of group orbits which can be distinguished by \(G\)-invariant (homogeneous) polynomials.
\(\mathbb{C}^* \curvearrowright \mathbb{CP}^2\)
\(t \cdot [z_0:z_1:z_2] = [z_0: tz_1:\frac{1}{t}z_2]\)
Roughly: identify orbits whose closures intersect, throw away orbits on which all \(G\)-invariant polynomials vanish.
\( \mathbb{CP}^2/\!/\,\mathbb{C}^* \cong\mathbb{CP}^1\)
Let \(T \simeq \operatorname{U}(1)^{d-1}\) be the diagonal subgroup of \(\operatorname{SU}(d)\). The conjugation action of \(T\) on \(\mathbb{C}^{d \times d}\) is also Hamiltonian, with momentum map
\(A \mapsto \mathrm{diag}([A,A^\ast])\).
\([A,A^\ast]_{ii} = \|A_i\|^2 - \|A^i\|^2\), where \(A_i\) is the \(i\)th row of \(A\) and \(A^i\) is the \(i\)th column.
If \(A = \left(a_{ij}\right)_{i,j} \in \mathbb{R}^{d \times d}\) such that \(\mathrm{diag}([A,A^\ast]) = 0\), then \(\widehat{A} = \left(a_{ij}^2\right)_{i,j}\) is the adjacency matrix of a balanced
Define the unbalanced energy \(\operatorname{B}(A) := \|\mathrm{diag}([A,A^\ast])\|^2 = \sum \left(\|A_i\|^2 - \|A^i\|^2\right)^2\).
Let \(\mathscr{F}(A_0,0) = A_0, \frac{d}{dt}\mathscr{F}(A_0,t) = - \nabla \operatorname{B}(\mathscr{F}(A_0,t))\) be negative gradient flow of \(\operatorname{B}\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
This is “local”: \(a_{ij}\) is updated by a multiple of \((\|A_j\|^2-\|A^j\|^2)-(\|A_i\|^2-\|A^i\|^2)\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
Theorem (with Needham)
For any \(A_0 \in \mathbb{C}^{d \times d}\), the matrix \(A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)\) exists, is balanced, has the same eigenvalues and principal minors as \(A_0\), and has
zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:
Theorem (with Needham)
For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1,
and has zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:
Theorem (with Needham)
For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1,
and has zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:
Theorem (with Needham)
For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1,
and has zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
By doing gradient flow \(\overline{\mathscr{F}}\) on the unit sphere, we can preserve weights:
Theorem (with Needham)
For any non-nilpotent \(A_0 \in \mathbb{C}^{d \times d}\) with \(\|A\|^2=1\), the matrix \(A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)\) exists, is balanced, has Frobenius norm 1,
and has zero entries whenever \(A_0\) does.
If \(A_0\) is real, so is \(A_\infty\), and if \(A_0\) has all non-negative entries, then so does \(A_\infty\).
Three proofs of the Benedetto–Fickus theorem
Dustin Mixon, Tom Needham, Clayton Shonkwiler, and Soledad Villar
Sampling, Approximation, and Signal Analysis (Harmonic Analysis in the Spirit of J. Rowland Higgins), Stephen D. Casey, M. Maurice Dodson, Paulo J. S. G. Ferreira and Ahmed Zayed, eds., Birkhäuser,
Cham, 2023, 371–391
Balancing Graphs Using Geometric Invariant Theory
By Clayton Shonkwiler
Balancing Graphs Using Geometric Invariant Theory
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Automated Sublinear Amortised Resource Analysis of Data Structures (AUTOSARD)
FWF Project Number: P 36623
Wider Research Context. Amortised analysis is a method for the worst-case cost analysis of (probabilistic) data structures, where a single data structure operation is considered as part of a larger
sequence of operations. The cost analysis of sophisticated data structures, such as self-adjusting binary search trees, has been a main objective already in the initial proposal of amortised
analysis. Analysing these data structures requires sophisticated potential functions, which typically contain sublinear expressions (such as the logarithm). Apart from our pilot project, the analysis
of these data structures has so far eluded automation.
Objectives. We target an automated analysis of the most common data structures with good, ie. sublinear, complexity, such as balanced trees, Fibonacci heaps, randomised search trees, skip lists, skew
heaps, Union-find, etc. Our goals are the verification of textbook data structures, the confirmation and improvement (on coefficients) of previously reported complexity bounds, as well as the
automated analysis of realistic data structure implementations. Initially, we will only consider strict (first-order) functional programming languages. Later on, we will extend our research towards
lazy evaluation in order to allow for the analysis of persistent data structures. Moreover, we will also consider probabilistic data structures.
Methods. Based on the success of our pilot study, we envision the following steps for automatising amortised analysis: (i) Fix a parametrised potential function; (ii) derive a (linear) constraint
system over the function parameters from the AST of the program; (iii) capture the required non-linear reasoning trough explicit background knowledge integrable into the constraint solving mechanism;
and finally (iv) find values for the parameters by an (optimising) constraint solver.
Pilot Study
AUTOSARD is an outgrow from our pilot study on (automated) amortised cost analysis of (randomised) splay trees, pairing heaps and randomised meldable heaps.
Automated Expected Amortised Cost Analysis of Probabilistic Data Structures
Lorenz Leutgeb, Georg Moser, Florian Zuleger
Proceedings of 34rd Computer Aided Verification, pages 70--91, 2022
ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures
Lorenz Leutgeb, Georg Moser, Florian Zuleger
Proceedings of 33rd Computer Aided Verification, pp 99-122, 202
Type-Based Analysis of Logarithmic Amortised Complexity
Martin Hofmann, Lorenz Leutgeb, Georg Moser, David Obwaller, Florian Zuleger
To appear in 'Mathematical Structures in Computer Science'.
Duration: 4 Years
The project will start on April 1, 2023.
To be announced
The project is coordinated by
• Georg Moser (Theoretical Computer Science; University of Innsbruck)
• Florian Zuleger (Formal Methods in Systems Engineering; Vienna University of Technology)
Further (prospective) project members will be announced here, timely.
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Probability Integral Transformation (sample-based version) — pit_sample
Probability Integral Transformation (sample-based version)
Uses a Probability Integral Transformation (PIT) (or a randomised PIT for integer forecasts) to assess the calibration of predictive Monte Carlo samples. Returns a p-values resulting from an
Anderson-Darling test for uniformity of the (randomised) PIT as well as a PIT histogram if specified.
A vector with the true observed values of size n
nxN matrix of predictive samples, n (number of rows) being the number of data points and N (number of columns) the number of Monte Carlo samples. Alternatively, predictions can just be a vector
of size n.
the number of draws for the randomised PIT for integer predictions.
A vector with PIT-values. For continuous forecasts, the vector will correspond to the length of true_values. For integer forecasts, a randomised PIT will be returned of length length(true_values) *
Calibration or reliability of forecasts is the ability of a model to correctly identify its own uncertainty in making predictions. In a model with perfect calibration, the observed data at each time
point look as if they came from the predictive probability distribution at that time.
Equivalently, one can inspect the probability integral transform of the predictive distribution at time t,
$$ u_t = F_t (x_t) $$
where \(x_t\) is the observed data point at time \(t \textrm{ in } t_1, …, t_n\), n being the number of forecasts, and \(F_t\) is the (continuous) predictive cumulative probability distribution at
time t. If the true probability distribution of outcomes at time t is \(G_t\) then the forecasts \(F_t\) are said to be ideal if \(F_t = G_t\) at all times t. In that case, the probabilities \(u_t\)
are distributed uniformly.
In the case of discrete outcomes such as incidence counts, the PIT is no longer uniform even when forecasts are ideal. In that case a randomised PIT can be used instead: $$ u_t = P_t(k_t) + v * (P_t
(k_t) - P_t(k_t - 1) ) $$
where \(k_t\) is the observed count, \(P_t(x)\) is the predictive cumulative probability of observing incidence k at time t, \(P_t (-1) = 0\) by definition and v is standard uniform and independent
of k. If \(P_t\) is the true cumulative probability distribution, then \(u_t\) is standard uniform.
The function checks whether integer or continuous forecasts were provided. It then applies the (randomised) probability integral and tests the values \(u_t\) for uniformity using the Anderson-Darling
As a rule of thumb, there is no evidence to suggest a forecasting model is miscalibrated if the p-value found was greater than a threshold of p >= 0.1, some evidence that it was miscalibrated if 0.01
< p < 0.1, and good evidence that it was miscalibrated if p <= 0.01. However, the AD-p-values may be overly strict and there actual usefulness may be questionable. In this context it should be noted,
though, that uniformity of the PIT is a necessary but not sufficient condition of calibration.
Sebastian Funk, Anton Camacho, Adam J. Kucharski, Rachel Lowe, Rosalind M. Eggo, W. John Edmunds (2019) Assessing the performance of real-time epidemic forecasts: A case study of Ebola in the Western
Area region of Sierra Leone, 2014-15, doi:10.1371/journal.pcbi.1006785
# \dontshow{
data.table::setDTthreads(2) # restricts number of cores used on CRAN
# }
## continuous predictions
true_values <- rnorm(20, mean = 1:20)
predictions <- replicate(100, rnorm(n = 20, mean = 1:20))
pit <- pit_sample(true_values, predictions)
## integer predictions
true_values <- rpois(50, lambda = 1:50)
predictions <- replicate(2000, rpois(n = 50, lambda = 1:50))
pit <- pit_sample(true_values, predictions, n_replicates = 30)
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Lesson 10
Drawing Triangles (Part 2)
10.1: Using a Compass to Estimate Length (5 minutes)
The purpose of this warm-up is to remind students that a compass is useful for transferring a length in general, and not just for drawing circles. As students discuss answers with their partners,
monitor for students who can clearly explain how they can use the compass to compare the length of the third side.
Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by time to discuss their answers with their partner. Follow with a whole-class discussion. Provide access to
geometry toolkits and compasses.
Student Facing
1. Draw a \(40^\circ\) angle.
2. Use a compass to make sure both sides of your angle have a length of 5 centimeters.
3. If you connect the ends of the sides you drew to make a triangle, is the third side longer or shorter than 5 centimeters? How can you use a compass to explain your answer?
Activity Synthesis
Ask previously identified students to share their responses to the final question. Display their drawing of the angle for all to see. If not mentioned in students’ explanations, demonstrate for all
to see how to use the compass to estimate the length of the third side of the triangle.
10.2: Revisiting How Many Can You Draw? (15 minutes)
Students continue to practice drawing triangles from given conditions and categorizing their results. This activity focuses on the inclusion of a single angle and two sides. Again, they do not need
to memorize which conditions result in unique triangles, but should begin to notice how some conditions (such as the equal side lengths) result in certain requirements for the completed triangle.
There is an optional blackline master that can help students organize their work at trying different configurations of the first set of measurements. If you provide students with a copy of the
blackline master, ask them to determine whether any of the configurations result in the same triangle, as well as whether any one configuration results in two possible triangles.
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find
all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions. Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible
mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class.
Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Use the applet to draw triangles.
1. Draw as many different triangles as you can with each of these sets of measurements:
1. One angle measures \(40^\circ\), one side measures 4 cm and one side measures 5 cm.
2. Two sides measure 6 cm and one angle measures \(100^\circ\).
2. Did either of these sets of measurements determine one unique triangle? How do you know?
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find
all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions. Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible
mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class.
Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
1. Draw as many different triangles as you can with each of these sets of measurements:
1. One angle measures \(40^\circ\), one side measures 4 cm, and one side measures 5 cm.
2. Two sides measure 6 cm, and one angle measures \(100^\circ\).
2. Did either of these sets of measurements determine one unique triangle? How do you know?
Anticipated Misconceptions
Some students may draw two different orientations of the same triangle for the first set of conditions, with the \(40^\circ\) angle in between the 4 cm and 5 cm sides. Prompt them to use tracing
paper to check whether their two triangles are really different (not identical copies).
If students struggle to create more than one triangle from the first set of conditions, prompt them to write down the order they already used for their measurements and then brainstorm other possible
orders they could use.
Activity Synthesis
Ask one or more students to share how many different triangles they were able to draw with each set of conditions. Select students to share their solutions.
If not brought up in student explanations, point out that for the first problem, one possible order for the measurements (\(40^\circ\), 5 cm, 4 cm) can result in two different triangles (the bottom
two in the solution image). One way to show this is to draw a 5 cm segment and then use a compass to draw a circle with a 4 cm radius centered on the segment’s left endpoint. Next, draw a ray at a \
(40^\circ\) angle centered on the segment’s right endpoint. Notice that this ray intersects the circle twice. Each one of these points could be the third vertex of the triangle. While it is helpful
for students to notice this interesting aspect of their drawing, it is not important for students to learn rules about the number of possible triangles given different sets of conditions.
If the optional blackline master was used, ask students:
• “Which configurations made identical triangles?” (the top left and bottom left)
• “Which configurations made more than one triangle?” (the bottom right)
If not mentioned by students, explain to students that the top left and bottom left configurations result in the same triangle, because in both cases the \(40^\circ\) angle is in between the 4 cm and
5 cm sides and that the bottom right configuration results in two different triangles, because the arc intersects the ray in two different places.
MLR 1 (Stronger and Clearer Each Time): Before discussing the second set of conditions as a whole class, have student pairs share their reasoning for why there were no more triangles that could be
drawn with the given measures, with two different partners in a row. Have students practice using mathematical language to be as clear as possible when sharing with the class, when and if they are
called upon.
Engagement: Develop Effort and Persistence. Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills
10.3: Three Angles (15 minutes)
This activity focuses on including three angle conditions. The goal is for students to notice that some angle conditions result in a large number of possible triangles (all scaled copies of one
another) or are impossible to create. Students are not expected to learn that the angles must sum to 180 degrees in a triangle, but are not barred from noticing this fact.
Arrange students in groups of 2. Tell students that they should attempt to create a triangle with the given specifications. If they can create one, they should attempt to either create at least one
more or justify to themselves why there is only one. If they cannot create any, they should show some valid attempts to include as many pieces as they can and be ready to explain why they cannot
include the remaining conditions.
Give students 5 minutes of quiet work time followed by time to discuss the triangles they could make with a partner. Follow with a whole-class discussion. Provide access to geometry toolkits and
If using the digital lesson, students should still try to create a triangle with the given specifications. If they can create one, they should attempt to either create at least one more or justify to
themselves why there is only one. If they cannot create any, they should be ready to explain some of their attempts and why they cannot include the remaining conditions.
Student Facing
Use the applet to draw triangles. Sides can overlap.
1. Draw as many different triangles as you can with each of these sets of measurements:
1. One angle measures \(50^\circ\), one measures \(60^\circ\), and one measures \(70^\circ\).
2. One angle measures \(50^\circ\), one measures \(60^\circ\), and one measures \(100^\circ\).
2. Did either of these sets of measurements determine one unique triangle? How do you know?
Student Facing
Are you ready for more?
Using only the point, segment, and compass tools provided, create an equilateral triangle. You are only successful if the triangle remains equilateral while dragging its vertices around.
Arrange students in groups of 2. Tell students that they should attempt to create a triangle with the given specifications. If they can create one, they should attempt to either create at least one
more or justify to themselves why there is only one. If they cannot create any, they should show some valid attempts to include as many pieces as they can and be ready to explain why they cannot
include the remaining conditions.
Give students 5 minutes of quiet work time followed by time to discuss the triangles they could make with a partner. Follow with a whole-class discussion. Provide access to geometry toolkits and
If using the digital lesson, students should still try to create a triangle with the given specifications. If they can create one, they should attempt to either create at least one more or justify to
themselves why there is only one. If they cannot create any, they should be ready to explain some of their attempts and why they cannot include the remaining conditions.
Student Facing
1. Draw as many different triangles as you can with each of these sets of measurements:
1. One angle measures \(50^\circ\), one measures \(60^\circ\), and one measures \(70^\circ\).
2. One angle measures \(50^\circ\), one measures \(60^\circ\), and one measures \(100^\circ\).
2. Did either of these sets of measurements determine one unique triangle? How do you know?
Student Facing
Are you ready for more?
Using only a compass and the edge of a blank index card, draw a perfectly equilateral triangle. (Note! The tools are part of the challenge! You may not use a protractor! You may not use a ruler!)
Anticipated Misconceptions
If students struggle to get started, remind them of Lin’s technique of using the protractor and a ruler to make an angle that can move along a line.
Activity Synthesis
Select students to share their drawings and display them for all to see. Ask students:
• “Were there any sets of measurements that produced a unique triangle?” (no)
• “Which combinations of angles could not be drawn?” (the angles in the second problem, \(50^\circ, 60^\circ, 100^\circ\))
• “Why is there more than one triangle that can be made with the measurements in the first problem?” (because there are no side lengths mentioned, so we can create scaled copies of the triangles
with the same angles but with shorter or longer side lengths)
MLR 1 (Stronger and Clearer Each Time): Before discussing the second set of conditions as a whole class, have student pairs share their reasoning for why there were no triangles that could be drawn
with the given measures, with two different partners in a row. Have students practice using mathematical language to be as clear as possible when sharing with the class, when and if they are called
Lesson Synthesis
• How was a compass useful in drawing triangles today? (It helps find all the points a certain distance away.)
• What strategies did you use to include two given side lengths and a given angle? (Draw one of the side lengths and use a protractor to draw the angle at one end, then use a compass to finish the
• What strategies did you use to include three given angles? (Draw one angle then use a protractor and ruler to slide along one side of the first angle.)
10.4: Cool-down - Finishing Noah’s Triangle (5 minutes)
Student Facing
A triangle has six measures: three side lengths and three angle measures.
If we are given three measures, then sometimes, there is no triangle that can be made. For example, there is no triangle with side lengths 1, 2, 5, and there is no triangle with all three angles
measuring \(150^\circ\).
Sometimes, only one triangle can be made. By this we mean that any triangle we make will be the same, having the same six measures. For example, if a triangle can be made with three given side
lengths, then the corresponding angles will have the same measures. Another example is shown here: an angle measuring \(45^\circ\) between two side lengths of 6 and 8 units. With this information,
one unique triangle can be made.
Sometimes, two or more different triangles can be made with three given measures. For example, here are two different triangles that can be made with an angle measuring \(45^\circ\) and side lengths
6 and 8. Notice the angle is not between the given sides.
Three pieces of information about a triangle’s side lengths and angle measures may determine no triangles, one unique triangle, or more than one triangle. It depends on the information.
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Skilful Skyful Part 3: Flying Geese : Carolyn Gibbs Quilts
Skilful Skyful Part 3: Flying Geese
I’m going to share all the tips and tricks for making and using Flying Geese units in this, the third instalment of my Skilful Skyful series.
These little units crop up in lots of blocks, and also in borders.
They have a large triangle flanked by two smaller triangles, and always finish as a rectangle which is twice as wide as it is high e.g. 6″ x 3″ or 7″ x 3 ½”
Finished Flying Geese unit
Flying Geese unit including seam allowances
Remember that the finished size will need a quarter inch seam allowance adding all round.
Don’t worry about that extra part at the top; it is correct, and you need it – the extra will be hidden in the seam allowance to leave the point of the triangle on the seamline.
So to finish 3″ x 6″ the completed unit needs to be 3 ½” x 6 ½”.
To finish 1 ½” x 3″ the completed unit needs to be 2″ x 3½”.
Cutting sizes
The size and shape of the pieces that you need to cut to make the Flying Geese partly depends on which method you are going to use. You could cut out the triangles separately, but as they have bias
edges, these will easily stretch and distort as they are sewn. It is much better to “quick piece” them, by stitching squares, which are then cut apart to make multiple units.
I will tell you in the free block patterns which sizes to cut, but you might be interested to know how to work it out yourself for other blocks, so here goes;
(Note that one other popular method of making Flying Geese is the “flip corners” method. I don’t use this, as you throw away a lot of triangles that are trimmed off, which I think is wasteful)
Now, in order to work out the size of pieces to cut to make a Flying Geese unit, it helps to be able to identify the nature of these three triangles.
Do this by comparing them to the two types you learnt to make in parts 1 & 2.
A half square triangle unit contains two triangles which each have two shorter edges on the straight grain of the fabric, and one longer edge on the bias.
Can you identify any single triangles like these on this photo?
The two smaller triangles on the outside of a Flying Geese unit are also like this, so they are half-square triangles.
A quarter square triangle unit contains four triangles which are still right angled triangles, but are oriented differently.
These each have one long edge on the straight grain, and two shorter edges on the bias.
This is like the larger triangle in the middle of the Flying Geese unit, so that is also a quarter-square triangle.
In Skilful Skyful instalment 1, you learnt that to cut a square of the correct size to make a half square triangle unit, you need to add ⅞” to the finished size.
In Skilful Skyful instalment 2, you learnt that to cut a square of the correct size to make a quarter square triangle unit, you need to add 1 ¼” to the finished size.
So, by looking at this diagram, you can work out what size of square to cut to make your Flying Geese unit:
• To finish with a Flying Geese which is 3″ x 6″, you need to cut small squares which are 3 ⅞” and large squares which are 7 ¼”
• To finish with a Flying Geese which is 1 ½ x 3″, you need to cut small squares which are 2 ⅜” and large squares which are 4 ¼”
How many do you need to cut?
Remember that the smaller squares will end up being cut along one diagonal, so will make two small triangles.
The larger squares will end up being cut along both diagonals, so will make four larger triangles.
Sizes for trimming
Now I know that some of you don’t like cutting these strange sizes, and prefer to cut larger and then trim down afterwards. That’s OK – but it doesn’t usually work with the quick-piecing methods. If
you just round up to the nearest inch or half inch, you wouldn’t necessarily end up with a good Flying Geese point in a place that can be trimmed to a quarter inch seam. You would need to add twice
as much to the small units as to the bigger ones. I will give you a couple of examples here of overlarge sizes that can be trimmed down; any other sizes, please ask!
• To finish with a Flying Geese which is 3″ x 6″, you could cut small squares which are 4″ and large squares which are 7 ½” and then trim down the units afterwards to 3 ½” x 6 ½”.
• To finish with a Flying Geese which is 1 ½ x 3″, you need to cut small squares which are 2 ½” and large squares which are 4 ½” and then trim down the units afterwards to 2″ x 3 ½”
Making Four at a time
I wish I knew who had developed this method – its very clever!
If you start with one large square and four small squares, it makes four Flying Geese at a time.
Step by step instructions are below – or you can watch me demonstrate the method in this video:
Quick-piecing Flying Geese units
Draw a diagonal line across the wrong side of all four small squares.
Place two of the small squares right sides down onto opposite corners of the large square, lined up with the edges – they will overlap just a little at the centre. Make sure that the drawn lines run
from corner to corner of the large square.
Pin like this, a little away from the lines, so that you can keep the pins in place while you stitch.
Stitch ¼” away from both sides of the lines.
Then cut along the line.
Open out the flaps, and press with the seam in the direction indicated by the pattern.
Here, it is pressed towards the small triangle, even though this is behind the lighter fabric.
You may be worried that this breaks a rule about always pressing behind the darker fabric, but it means that when this unit is combined into the block, it will meet other diagonal seams pressed in
the opposite direction. This will reduce bulk, and give perfect points.
There is more about this in another page of the website; Pressing for Perfect Points if you are interested.
Now take the other two small squares, and place each of them right sides down onto the one of the strange shaped pieces you have.
Make sure that the diagonal line comes from the corner of the large triangle to the middle, as shown here.
Pin and stitch ¼” away from each side of the line.
Cut along the line, open out the flaps, and press the seam towards the small triangle. This pressing direction ensures that the position of the triangle point will be visible when the units are
assembled later.
You have now made four Flying Geese units!
Once you have made these four Flying Geese units, you have done most of the work to make the easier block provided for this instalment: Evening Star. All it needs are some small squares for the
corners, and a large one for the middle.
Evening Star
The centre block can be made in a contrasting colour as here, or in the same colour as the star points – in which case the block name becomes Single Star.
Instructions for both are provided in the free download pattern.
If you would like to make a project using this simple block, I have designed “Baby Stars” which would be a perfect gift for a new arrival.
Baby Stars
Pressing Direction for Flying Geese
The Evening Star pattern includes guidance telling you to press the first seams towards the large triangle, and the second seams towards the small triangles.
Why have I suggested that the seams are pressed like this? It is to avoid chopping off the points of the Flying Geese when they are stitched into a block.
To stitch the horizontal block seam correctly, it should pass through the intersection of the two diagonal seams – as shown by the black stitching in the left-hand Flying Geese units. If it is too
low (like the red line), the point of the Flying Goose will be hidden in the seam allowance.
Provided the second Flying Geese seam is pressed away from the large triangle, as shown in the first diagrams, the two seams can easily be seen, and your new seam can be positioned to go exactly
through the crossover (even if this is slightly more or less than the ideal ¼” seam allowance – the point is more important).
If however, you have pressed both towards the large triangle, as shown on the right hand diagram, this crossover point can’t be seen – so it is much more difficult to stitch the new seam in exactly
the right place.
This video explains this in more detail:
Pressing for perfect points: Flying Geese
As an aside, although you need to press “in/out” for this particular block, it actually doesn’t normally matter for the first one you stitch, but the second one must be pressed out, if the crossover
is to be visible. (Don’t think of this as right and left, by the way, as you may not always stitch the same side first, particularly if you are quick-piecing)
As explained in the video, which you choose is determined by several factors; reducing bulk at the point, and/or ensuring that the diagonal seams meet the seams on the next units pressed in the
opposite direction. I have worked out the best pressing directions for you in all the patterns I provide, so just follow the instructions, and you should achieve perfect points.
The first block, Evening Star, is easy, as it had quite a lot of squares, and you only had to worry about getting the point at the middle of the Flying Geese correct.
Once you have mastered this, you can move onto a more challenging block: Dutchman’s Puzzle. Can you see that it is entirely made from Flying Geese, arranged in pairs?
For this one, you need to be careful about the points at the sides of the Flying Geese as well, in addition to the pinwheel at the centre.
As with Evening Star, the key to keeping the points is to pin and stitch with the point of the Flying Geese units on top so that you can see the triangle point and aim to stitch exactly through it
(even if this means a slight deviation from your ¼” seam – the point is more important!)
The free pattern includes full directions advising which way the seam allowances should be pressed to achieve the best results (and why!).
The same units can be rearranged to make another block: Mosaic.
Although the centre now does not have the tricky pinwheel found in Flying Dutchman, more care needs to be given to ensuring that the points meet properly at the sides.
Consideration of the pressing direction will help ensure that the diagonal seams meet pressed in opposite directions – but as the quick piecing method is used, you need to be more careful than usual.
Instructions for this are included on the same pattern as Dutchman’s Puzzle.
Many other blocks use Flying Geese units too; why not try Rising Star or Festival Star? Each of these patterns is only £2.
This was a quilt I made many years ago, which included an enormous number of Flying Geese units. (pattern not available)
Notice the Hanging Diamonds quilting pattern – an effective use of an overall grid.
If you are a slightly more experienced quilter, and would like to make a larger project which includes Flying Geese units, consider making this lap quilt: Jack-in-a-Box. This uses a different method
to construct the Flying Geese units, which produces half-square triangle units at the same time – so they are all ready when you need to make the border.
This is the third instalment of the Skilful Skyful series. By working through these tutorials, you can build up your skills in machine-stitched block patchwork, making a set of 12″ blocks which could
be combined if you wished into a Sampler Quilt.
Other pages about machine-stitched patchwork can be found in the Techniques section
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Polynomial order
Let $P$ be an irreducible polynomial of degree $d\ge 1$ over a prime finite field ${\mathrm{𝔽}}_{p}$. The order of $P$ is the smallest positive integer $n$ such that $P\left(x\right)$ divides $
{x}^{n}-1$. $n$ is also equal to the multiplicative order of any root of $P$. It is a divisor of ${p}_{d}-1$. The polynomial $P$ is a primitive polynomial if $n={p}^{d}-1$.
This tool allows you to enter a polynomial and compute its order. If you enter a reducible polynomial, the orders of all its non-linear factors will be computed and presented.
The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: computes the order of an irreducible polynomial over a finite field F[p]. interactive exercises, online calculators and plotters, mathematical recreation and games
• Keywords: interactive mathematics, interactive math, server side interactivity, algebra, coding, polynomials, finite_field, factorization, roots, order, cyclic_code
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, Sommersemester 2017
Update: The course will be taught in English.
Ankündigungen / Announcements
• 11.10.2017: Klausureinsicht for last Monday's exam will be Thursday from 13:00 to 14:00 in my office (1.301 in RUD25). If you would like to see your graded exam but cannot make it during that
time, feel free to e-mail me for an appointment.
• 26.07.2017: Klausureinsicht for yesterday's exam will be Thursday from 11:00 to 12:00 in my office (1.301 in RUD25). If you would like to see your graded exam but cannot make it during that time,
feel free to e-mail me for an appointment.
• 29.06.2017: Two versions of the solution to problem 2(a) on the Take-home midterm are now posted below
• 30.05.2017: The final exam will take place on 25.07.2017 (registration deadline 11.07.2017). The date for the second attempt is 9.10.2017 (registration deadline 25.09.2017). See the math
department's Prüfungsangelegenheiten page for administrative details.
• 4.05.2017: There are now some lecture notes about nets and convergence posted below. They cover a subset of the material from the lectures on 27.4, 3.5 and 4.5.
previous announcements (no longer relevant)
topics covered in the lectures (updated regularly)
Lecture notes on nets and convergence --- this is a topic that is not particularly well covered in most of the standard introductory books on topology, but it fits in naturally and leads to very
elegant proofs of fundamental results such as the Bolzano-Weierstrass theorem and Tychonoff's theorem.
Übungsblätter / Problem sets
The grader for this course is Daniel Platt; you can contact him at plattd at math dot hu dash berlin dot de if you have questions about your graded homework.
• Problem Set 1 (distributed 19.04.2017, due 26.04.2017)
• Problem Set 2 (distributed 27.04.2017, due 3.05.2017)
• Discussion of #5 on Problem Set 2 (convergence in the box topology)
• Problem Set 3 (distributed 3.05.2017, due 10.05.2017) --- the version handed out in class had small errors in Problems 2(b) and 3(b), which are corrected in this version
• Problem Set 4 (distributed 10.05.2017, due 17.05.2017)
• Problem Set 5 (distributed 17.05.2017, due 24.05.2017)
• Problem Set 6 (distributed 24.05.2017, due 31.05.2017) --- this is a revised version with a minor correction to Problem 2(c)
• Problem Set 7 (distributed 31.05.2017, due 7.06.2017)
• Problem Set 8 (distributed 8.06.2017, due 14.06.2017) --- this is a revised version with a minor correction to Problem 1(h)
• Take-home midterm
• Take-home midterm solutions to problem 2(a) --- this problem turned out to be a bit subtler than we'd realized ahead of time, so we sat down to write up a solution and ended up with two versions:
Chris's version ; Felix's version
• Problem Set 9 (distributed 28.06.2017, due 5.07.2017)
• Problem Set 10 (distributed 5.07.2017, due 12.07.2017)
• Problem Set 11 (distributed 12.07.2017, due date extended to 20.07.2017)
• Practice exam / Probeklausur: This is meant to be representative of the format and type of questions you can expect on the actual exam, though it is somewhat longer and in some areas perhaps more
difficult. You may consider it an upper bound for the possible difficulty of the actual exam. Problem 5(d) also concerns topics to be discussed in the final week of lectures (which have therefore
not appeared on any problem sets before).
Other useful links
Somebody miscalculated the fundamental group. (Hermannplatz, 8.06.2017)
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ent Notes
• Bonds: is a debt instrument issued by governments or corporations to raise money
□ The successful investor must be able to:
☆ Understand bond structure
☆ Calculate bond rates of return
☆ Understand interest rate risk
☆ Differentiate between real and nominal returns
• Components:
□ Bond – Security that obligates the issuer to make specified payments to the bondholder.
□ Face Value – Payment at the maturity of the bond. Also called “principal ” or “par value ”
□ Coupon – The interest payments paid to the bondholder.
□ Coupon Rate – Annual interest payment as a percentage of face value.
□ Asked Price – The price that investors need to pay to buy the bond.
□ Bid Price – The price asked by an investor who owns the bond and wishes to sell it.
□ Spread – The difference between the bid price and the asked price.
□ The spread is how a seller of a bond makes a profit.
☆ Note: While Treasury bonds are quoted in 32nds, corporate bonds are quoted in decimals.
• Calculating yields
□ Bond Pricing: The value of a bond is the present value of all cash flows generated by the bond (coupons and repayment of face value), discounted at the required rate of return.
□ Current Yield:
□ Yield to Maturity:
• Relationship of interest rates and bond prices
□ When the interest rate rises, the present value of the payments to be received by the bondholder falls and bond prices fall.
□ When the interest rate decreases, the present value of the payments to be received by the bondholder increases and bond prices rise.
□ Interest rate risk – The risk in bond prices due to fluctuations in interest rates.
• Risks of bonds, Credit Rating, Default Risk and how they affect yield
□ Default Risk – The risk that a bond issuer may default on his bonds.
☆ Companies compensate investors for bearing this added risk in the form of higher interest rates on their bonds.
□ Default Premium – The additional yield on a bond that investors require for bearing default risk.
☆ Usually the difference between the promised yield on a corporate bond and the yield on a U.S. Treasury bond with the same coupon and maturity.
□ Credit agency – An agency that rates the safety of most bonds.
□ Investment-grade bonds – Bonds rated Baa or above by Moody’s or BBB or above by Standard & Poor’s.
□ Junk bond – Bond with a rating below Baa or BBB
• Types of Bonds :
□ Zero-Coupon Bonds – Bonds that are issued well below face value with no coupon payment. At maturity investors receive $1,000 face value for the bond.
☆ Are corporate bonds the only bonds which can be offered as zero-coupon bonds?
□ Floating-Rate Bonds – Bonds with coupon payments that are tied to some measure of current market rates. A common example would be a bond with coupon rate tied to the short-term Treasury rate
plus 2%.
□ Convertible Bonds – Bonds that allow the holder to exchange the bond at a later date for a specified number of shares of common stock.
Yield to Maturity – Interest rate for which the present value of the bond’s payments equals the price.
□ Current Yield – Annual coupon payments divided by bond price.
□ Stocks:
☆ Primary market – Market for the sale of new securities by corporations.
☆ Secondary Market – Market in which previously issued securities are traded among investors.
☆ Initial Public Offering (IPO) – First offering of stock to the general public.
☆ Primary Offering – Occurs when a corporation sells shares in the firm.
☆ Market cap (market capitalization) – The total value of a company’s outstanding shares.
☆ P/E Ratio – Ratio of stock price to earnings per share.
☆ Dividend Yield – The ratio of dividends paid and share price. Tells the investor how much dividend income they can expect for every $1 invested in the stock.
☆ Dividend discount model (continuous & uneven growth)
○ Dividend Discount Model – Discounted cash-flow model which states that today’s stock price equals the present value of all expected future dividends.
○ Dividend Yield
○ Sustainable Growth Rate
■ Payout Ratio – Fraction of earnings paid out as dividends.
■ Plowback Ratio – Fraction of earnings retained by the firm
★ Note: Plowback Ratio = 1 – Payout Ratio
■ g (sustainable growth rate) – The firm’s growth rate if it plows back a constant fraction of earnings, maintains a constant return on equity, and keeps its debt ratio constant.
☆ Technical Analysis – Investors who attempt to identify undervalued stocks by searching for patterns in past stock prices.
☆ Fundamental Analysis – Investors who attempt to find mispriced securities by analyzing fundamental information, such as accounting performance and earnings prospects.
○ Note: In a market with many talented and competitive analysts, any bargains will be quickly eliminated.
☆ Random walk – Security prices change randomly, with no predictable trends or patterns.
○ They are equally likely to offer a high or low return on any particular day, regardless of what has occurred on previous days.
Net Present Value:
☆ NPV (calc and rule)
○ Opportunity Cost of Capital – Expected rate of return given up by investing in a project
○ Net Present Value – Present value of cash flows minus initial investments.
○ Net Present Value Rule – Managers increase shareholders’ wealth by accepting all projects that are worth more than they cost. Therefore, they should accept all projects with a
positive net present value.
☆ Payback Period – Time until cash flows recover the initial investment of the project.
○ RULE: Says a project should be accepted if its payback period is less than a specified cutoff period.
○ Discounted Payback Rule – This is the number of periods before the present value of prospective cash flows equals or exceeds the initial investment.
☆ IRR (definition & rule/how to use)
○ Definition: The discount rate that gives the project a zero NPV is known as the project’s internal rate of return, or IRR . It is also termed the discounted cashflow (DCF) rate of
○ RULE: IRR Rule – Managers increase shareholders’ wealth by accepting all projects which offer a rate of return that is higher than the opportunity cost of capital.
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Fast Fourier Transforms
2019 May 12
See all posts
Trigger warning: specialized mathematical topic
Special thanks to Karl Floersch for feedback
One of the more interesting algorithms in number theory is the Fast Fourier transform (FFT). FFTs are a key building block in many algorithms, including extremely fast multiplication of large numbers
, multiplication of polynomials, and extremely fast generation and recovery of erasure codes. Erasure codes in particular are highly versatile; in addition to their basic use cases in fault-tolerant
data storage and recovery, erasure codes also have more advanced use cases such as securing data availability in scalable blockchains and STARKs. This article will go into what fast Fourier
transforms are, and how some of the simpler algorithms for computing them work.
The original Fourier transform is a mathematical operation that is often described as converting data between the "frequency domain" and the "time domain". What this means more precisely is that if
you have a piece of data, then running the algorithm would come up with a collection of sine waves with different frequencies and amplitudes that, if you added them together, would approximate the
original data. Fourier transforms can be used for such wonderful things as expressing square orbits through epicycles and deriving a set of equations that can draw an elephant:
Ok fine, Fourier transforms also have really important applications in signal processing, quantum mechanics, and other areas, and help make significant parts of the global economy happen. But come
on, elephants are cooler.
Running the Fourier transform algorithm in the "inverse" direction would simply take the sine waves and add them together and compute the resulting values at as many points as you wanted to sample.
The kind of Fourier transform we'll be talking about in this post is a similar algorithm, except instead of being a continuous Fourier transform over real or complex numbers, it's a discrete Fourier
transform over finite fields (see the "A Modular Math Interlude" section here for a refresher on what finite fields are). Instead of talking about converting between "frequency domain" and "time
domain", here we'll talk about two different operations: multi-point polynomial evaluation (evaluating a degree \(< N\) polynomial at \(N\) different points) and its inverse, polynomial interpolation
(given the evaluations of a degree \(< N\) polynomial at \(N\) different points, recovering the polynomial). For example, if we are operating in the prime field with modulus 5, then the polynomial \
(y = x² + 3\) (for convenience we can write the coefficients in increasing order: \([3,0,1]\)) evaluated at the points \([0,1,2]\) gives the values \([3,4,2]\) (not \([3, 4, 7]\) because we're
operating in a finite field where the numbers wrap around at 5), and we can actually take the evaluations \([3,4,2]\) and the coordinates they were evaluated at (\([0,1,2]\)) to recover the original
polynomial \([3,0,1]\).
There are algorithms for both multi-point evaluation and interpolation that can do either operation in \(O(N^2)\) time. Multi-point evaluation is simple: just separately evaluate the polynomial at
each point. Here's python code for doing that:
def eval_poly_at(self, poly, x, modulus):
y = 0
power_of_x = 1
for coefficient in poly:
y += power_of_x * coefficient
power_of_x *= x
return y % modulus
The algorithm runs a loop going through every coefficient and does one thing for each coefficient, so it runs in \(O(N)\) time. Multi-point evaluation involves doing this evaluation at \(N\)
different points, so the total run time is \(O(N^2)\).
Lagrange interpolation is more complicated (search for "Lagrange interpolation" here for a more detailed explanation). The key building block of the basic strategy is that for any domain \(D\) and
point \(x\), we can construct a polynomial that returns \(1\) for \(x\) and \(0\) for any value in \(D\) other than \(x\). For example, if \(D = [1,2,3,4]\) and \(x = 1\), the polynomial is:
\[ y = \frac{(x-2)(x-3)(x-4)}{(1-2)(1-3)(1-4)} \]
You can mentally plug in \(1\), \(2\), \(3\) and \(4\) to the above expression and verify that it returns \(1\) for \(x= 1\) and \(0\) in the other three cases.
We can recover the polynomial that gives any desired set of outputs on the given domain by multiplying and adding these polynomials. If we call the above polynomial \(P_1\), and the equivalent ones
for \(x=2\), \(x=3\), \(x=4\), \(P_2\), \(P_3\) and \(P_4\), then the polynomial that returns \([3,1,4,1]\) on the domain \([1,2,3,4]\) is simply \(3 \cdot P_1 + P_2 + 4 \cdot P_3 + P_4\). Computing
the \(P_i\) polynomials takes \(O(N^2)\) time (you first construct the polynomial that returns to 0 on the entire domain, which takes \(O(N^2)\) time, then separately divide it by \((x - x_i)\) for
each \(x_i\)), and computing the linear combination takes another \(O(N^2)\) time, so it's \(O(N^2)\) runtime total.
What Fast Fourier transforms let us do, is make both multi-point evaluation and interpolation much faster.
Fast Fourier Transforms
There is a price you have to pay for using this much faster algorithm, which is that you cannot choose any arbitrary field and any arbitrary domain. Whereas with Lagrange interpolation, you could
choose whatever x coordinates and y coordinates you wanted, and whatever field you wanted (you could even do it over plain old real numbers), and you could get a polynomial that passes through them.,
with an FFT, you have to use a finite field, and the domain must be a multiplicative subgroup of the field (that is, a list of powers of some "generator" value). For example, you could use the finite
field of integers modulo \(337\), and for the domain use \([1, 85, 148, 111, 336, 252, 189, 226]\) (that's the powers of \(85\) in the field, eg. \(85^3\) % \(337 = 111\); it stops at \(226\) because
the next power of \(85\) cycles back to \(1\)). Futhermore, the multiplicative subgroup must have size \(2^n\) (there's ways to make it work for numbers of the form \(2^{m} \cdot 3^n\) and possibly
slightly higher prime powers but then it gets much more complicated and inefficient). The finite field of intergers modulo \(59\), for example, would not work, because there are only multiplicative
subgroups of order \(2\), \(29\) and \(58\); \(2\) is too small to be interesting, and the factor \(29\) is far too large to be FFT-friendly. The symmetry that comes from multiplicative groups of
size \(2^n\) lets us create a recursive algorithm that quite cleverly calculate the results we need from a much smaller amount of work.
To understand the algorithm and why it has a low runtime, it's important to understand the general concept of recursion. A recursive algorithm is an algorithm that has two cases: a "base case" where
the input to the algorithm is small enough that you can give the output directly, and the "recursive case" where the required computation consists of some "glue computation" plus one or more uses of
the same algorithm to smaller inputs. For example, you might have seen recursive algorithms being used for sorting lists. If you have a list (eg. \([1,8,7,4,5,6,3,2,9]\)), then you can sort it using
the following procedure:
• If the input has one element, then it's already "sorted", so you can just return the input.
• If the input has more than one element, then separately sort the first half of the list and the second half of the list, and then merge the two sorted sub-lists (call them \(A\) and \(B\)) as
follows. Maintain two counters, \(apos\) and \(bpos\), both starting at zero, and maintain an output list, which starts empty. Until either \(apos\) or \(bpos\) is at the end of the corresponding
list, check if \(A[apos]\) or \(B[bpos]\) is smaller. Whichever is smaller, add that value to the end of the output list, and increase that counter by \(1\). Once this is done, add the rest of
whatever list has not been fully processed to the end of the output list, and return the output list.
Note that the "glue" in the second procedure has runtime \(O(N)\): if each of the two sub-lists has \(N\) elements, then you need to run through every item in each list once, so it's \(O(N)\)
computation total. So the algorithm as a whole works by taking a problem of size \(N\), and breaking it up into two problems of size \(\frac{N}{2}\), plus \(O(N)\) of "glue" execution. There is a
theorem called the Master Theorem that lets us compute the total runtime of algorithms like this. It has many sub-cases, but in the case where you break up an execution of size \(N\) into \(k\)
sub-cases of size \(\frac{N}{k}\) with \(O(N)\) glue (as is the case here), the result is that the execution takes time \(O(N \cdot log(N))\).
An FFT works in the same way. We take a problem of size \(N\), break it up into two problems of size \(\frac{N}{2}\), and do \(O(N)\) glue work to combine the smaller solutions into a bigger
solution, so we get \(O(N \cdot log(N))\) runtime total - much faster than \(O(N^2)\). Here is how we do it. I'll describe first how to use an FFT for multi-point evaluation (ie. for some domain \(D
\) and polynomial \(P\), calculate \(P(x)\) for every \(x\) in \(D\)), and it turns out that you can use the same algorithm for interpolation with a minor tweak.
Suppose that we have an FFT where the given domain is the powers of \(x\) in some field, where \(x^{2^{k}} = 1\) (eg. in the case we introduced above, the domain is the powers of \(85\) modulo \(337
\), and \(85^{2^{3}} = 1\)). We have some polynomial, eg. \(y = 6x^7 + 2x^6 + 9x^5 + 5x^4 + x^3 + 4x^2 + x + 3\) (we'll write it as \(p = [3, 1, 4, 1, 5, 9, 2, 6]\)). We want to evaluate this
polynomial at each point in the domain, ie. at each of the eight powers of \(85\). Here is what we do. First, we break up the polynomial into two parts, which we'll call \(evens\) and \(odds\): \
(evens = [3, 4, 5, 2]\) and \(odds = [1, 1, 9, 6]\) (or \(evens = 2x^3 + 5x^2 + 4x + 3\) and \(odds = 6x^3 + 9x^2 + x + 1\); yes, this is just taking the even-degree coefficients and the odd-degree
coefficients). Now, we note a mathematical observation: \(p(x) = evens(x^2) + x \cdot odds(x^2)\) and \(p(-x) = evens(x^2) - x \cdot odds(x^2)\) (think about this for yourself and make sure you
understand it before going further).
Here, we have a nice property: \(evens\) and \(odds\) are both polynomials half the size of \(p\), and furthermore, the set of possible values of \(x^2\) is only half the size of the original domain,
because there is a two-to-one correspondence: \(x\) and \(-x\) are both part of \(D\) (eg. in our current domain \([1, 85, 148, 111, 336, 252, 189, 226]\), 1 and 336 are negatives of each other, as \
(336 = -1\) % \(337\), as are \((85, 252)\), \((148, 189)\) and \((111, 226)\). And \(x\) and \(-x\) always both have the same square. Hence, we can use an FFT to compute the result of \(evens(x)\)
for every \(x\) in the smaller domain consisting of squares of numbers in the original domain (\([1, 148, 336, 189]\)), and we can do the same for odds. And voila, we've reduced a size-\(N\) problem
into half-size problems.
The "glue" is relatively easy (and \(O(N)\) in runtime): we receive the evaluations of \(evens\) and \(odds\) as size-\(\frac{N}{2}\) lists, so we simply do \(p[i] = evens\_result[i] + domain[i]\cdot
odds\_result[i]\) and \(p[\frac{N}{2} + i] = evens\_result[i] - domain[i]\cdot odds\_result[i]\) for each index \(i\).
Here's the full code:
def fft(vals, modulus, domain):
if len(vals) == 1:
return vals
L = fft(vals[::2], modulus, domain[::2])
R = fft(vals[1::2], modulus, domain[::2])
o = [0 for i in vals]
for i, (x, y) in enumerate(zip(L, R)):
y_times_root = y*domain[i]
o[i] = (x+y_times_root) % modulus
o[i+len(L)] = (x-y_times_root) % modulus
return o
We can try running it:
>>> fft([3,1,4,1,5,9,2,6], 337, [1, 85, 148, 111, 336, 252, 189, 226])
[31, 70, 109, 74, 334, 181, 232, 4]
And we can check the result; evaluating the polynomial at the position \(85\), for example, actually does give the result \(70\). Note that this only works if the domain is "correct"; it needs to be
of the form \([x^i\) % \(modulus\) for \(i\) in \(range(n)]\) where \(x^n = 1\).
An inverse FFT is surprisingly simple:
def inverse_fft(vals, modulus, domain):
vals = fft(vals, modulus, domain)
return [x * modular_inverse(len(vals), modulus) % modulus for x in [vals[0]] + vals[1:][::-1]]
Basically, run the FFT again, but reverse the result (except the first item stays in place) and divide every value by the length of the list.
>>> domain = [1, 85, 148, 111, 336, 252, 189, 226]
>>> def modular_inverse(x, n): return pow(x, n - 2, n)
>>> values = fft([3,1,4,1,5,9,2,6], 337, domain)
>>> values
[31, 70, 109, 74, 334, 181, 232, 4]
>>> inverse_fft(values, 337, domain)
[3, 1, 4, 1, 5, 9, 2, 6]
Now, what can we use this for? Here's one fun use case: we can use FFTs to multiply numbers very quickly. Suppose we wanted to multiply \(1253\) by \(1895\). Here is what we would do. First, we would
convert the problem into one that turns out to be slightly easier: multiply the polynomials \([3, 5, 2, 1]\) by \([5, 9, 8, 1]\) (that's just the digits of the two numbers in increasing order), and
then convert the answer back into a number by doing a single pass to carry over tens digits. We can multiply polynomials with FFTs quickly, because it turns out that if you convert a polynomial into
evaluation form (ie. \(f(x)\) for every \(x\) in some domain \(D\)), then you can multiply two polynomials simply by multiplying their evaluations. So what we'll do is take the polynomials
representing our two numbers in coefficient form, use FFTs to convert them to evaluation form, multiply them pointwise, and convert back:
>>> p1 = [3,5,2,1,0,0,0,0]
>>> p2 = [5,9,8,1,0,0,0,0]
>>> x1 = fft(p1, 337, domain)
>>> x1
[11, 161, 256, 10, 336, 100, 83, 78]
>>> x2 = fft(p2, 337, domain)
>>> x2
[23, 43, 170, 242, 3, 313, 161, 96]
>>> x3 = [(v1 * v2) % 337 for v1, v2 in zip(x1, x2)]
>>> x3
[253, 183, 47, 61, 334, 296, 220, 74]
>>> inverse_fft(x3, 337, domain)
[15, 52, 79, 66, 30, 10, 1, 0]
This requires three FFTs (each \(O(N \cdot log(N))\) time) and one pointwise multiplication (\(O(N)\) time), so it takes \(O(N \cdot log(N))\) time altogether (technically a little bit more than \(O
(N \cdot log(N))\), because for very big numbers you would need replace \(337\) with a bigger modulus and that would make multiplication harder, but close enough). This is much faster than schoolbook
multiplication, which takes \(O(N^2)\) time:
5 | 15 25 10 5
9 | 27 45 18 9
8 | 24 40 16 8
1 | 3 5 2 1
So now we just take the result, and carry the tens digits over (this is a "walk through the list once and do one thing at each point" algorithm so it takes \(O(N)\) time):
[15, 52, 79, 66, 30, 10, 1, 0]
[ 5, 53, 79, 66, 30, 10, 1, 0]
[ 5, 3, 84, 66, 30, 10, 1, 0]
[ 5, 3, 4, 74, 30, 10, 1, 0]
[ 5, 3, 4, 4, 37, 10, 1, 0]
[ 5, 3, 4, 4, 7, 13, 1, 0]
[ 5, 3, 4, 4, 7, 3, 2, 0]
And if we read the digits from top to bottom, we get \(2374435\). Let's check the answer....
Yay! It worked. In practice, on such small inputs, the difference between \(O(N \cdot log(N))\) and \(O(N^2)\) isn't that large, so schoolbook multiplication is faster than this FFT-based
multiplication process just because the algorithm is simpler, but on large inputs it makes a really big difference.
But FFTs are useful not just for multiplying numbers; as mentioned above, polynomial multiplication and multi-point evaluation are crucially important operations in implementing erasure coding, which
is a very important technique for building many kinds of redundant fault-tolerant systems. If you like fault tolerance and you like efficiency, FFTs are your friend.
FFTs and binary fields
Prime fields are not the only kind of finite field out there. Another kind of finite field (really a special case of the more general concept of an extension field, which are kind of like the
finite-field equivalent of complex numbers) are binary fields. In an binary field, each element is expressed as a polynomial where all of the entries are \(0\) or \(1\), eg. \(x^3 + x + 1\). Adding
polynomials is done modulo \(2\), and subtraction is the same as addition (as \(-1 = 1 \bmod 2\)). We select some irreducible polynomial as a modulus (eg. \(x^4 + x + 1\); \(x^4 + 1\) would not work
because \(x^4 + 1\) can be factored into \((x^2 + 1)\cdot(x^2 + 1)\) so it's not "irreducible"); multiplication is done modulo that modulus. For example, in the binary field mod \(x^4 + x + 1\),
multiplying \(x^2 + 1\) by \(x^3 + 1\) would give \(x^5 + x^3 + x^2 + 1\) if you just do the multiplication, but \(x^5 + x^3 + x^2 + 1 = (x^4 + x + 1)\cdot x + (x^3 + x + 1)\), so the result is the
remainder \(x^3 + x + 1\).
We can express this example as a multiplication table. First multiply \([1, 0, 0, 1]\) (ie. \(x^3 + 1\)) by \([1, 0, 1]\) (ie. \(x^2 + 1\)):
1 | 1 0 0 1
0 | 0 0 0 0
1 | 1 0 0 1
The multiplication result contains an \(x^5\) term so we can subtract \((x^4 + x + 1)\cdot x\):
- 1 1 0 0 1 [(x⁴ + x + 1) shifted right by one to reflect being multipled by x]
And we get the result, \([1, 1, 0, 1]\) (or \(x^3 + x + 1\)).
Addition and multiplication tables for the binary field mod \(x^4 + x + 1\). Field elements are expressed as integers converted from binary (eg. \(x^3 + x^2 \rightarrow 1100 \rightarrow 12\))
Binary fields are interesting for two reasons. First of all, if you want to erasure-code binary data, then binary fields are really convenient because \(N\) bytes of data can be directly encoded as a
binary field element, and any binary field elements that you generate by performing computations on it will also be \(N\) bytes long. You cannot do this with prime fields because prime fields' size
is not exactly a power of two; for example, you could encode every \(2\) bytes as a number from \(0...65536\) in the prime field modulo \(65537\) (which is prime), but if you do an FFT on these
values, then the output could contain \(65536\), which cannot be expressed in two bytes. Second, the fact that addition and subtraction become the same operation, and \(1 + 1 = 0\), create some
"structure" which leads to some very interesting consequences. One particularly interesting, and useful, oddity of binary fields is the "freshman's dream" theorem: \((x+y)^2 = x^2 + y^2\) (and the
same for exponents \(4, 8, 16...\) basically any power of two).
But if you want to use binary fields for erasure coding, and do so efficiently, then you need to be able to do Fast Fourier transforms over binary fields. But then there is a problem: in a binary
field, there are no (nontrivial) multiplicative groups of order \(2^n\). This is because the multiplicative groups are all order \(2^n\)-1. For example, in the binary field with modulus \(x^4 + x + 1
\), if you start calculating successive powers of \(x+1\), you cycle back to \(1\) after \(\it 15\) steps - not \(16\). The reason is that the total number of elements in the field is \(16\), but one
of them is zero, and you're never going to reach zero by multiplying any nonzero value by itself in a field, so the powers of \(x+1\) cycle through every element but zero, so the cycle length is \(15
\), not \(16\). So what do we do?
The reason we needed the domain to have the "structure" of a multiplicative group with \(2^n\) elements before is that we needed to reduce the size of the domain by a factor of two by squaring each
number in it: the domain \([1, 85, 148, 111, 336, 252, 189, 226]\) gets reduced to \([1, 148, 336, 189]\) because \(1\) is the square of both \(1\) and \(336\), \(148\) is the square of both \(85\)
and \(252\), and so forth. But what if in a binary field there's a different way to halve the size of a domain? It turns out that there is: given a domain containing \(2^k\) values, including zero
(technically the domain must be a subspace), we can construct a half-sized new domain \(D'\) by taking \(x \cdot (x+k)\) for \(x\) in \(D\) using some specific \(k\) in \(D\). Because the original
domain is a subspace, since \(k\) is in the domain, any \(x\) in the domain has a corresponding \(x+k\) also in the domain, and the function \(f(x) = x \cdot (x+k)\) returns the same value for \(x\)
and \(x+k\) so we get the same kind of two-to-one correspondence that squaring gives us.
\(x\) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
\(x \cdot (x+1)\) 0 0 6 6 7 7 1 1 4 4 2 2 3 3 5 5
So now, how do we do an FFT on top of this? We'll use the same trick, converting a problem with an \(N\)-sized polynomial and \(N\)-sized domain into two problems each with an \(\frac{N}{2}\)-sized
polynomial and \(\frac{N}{2}\)-sized domain, but this time using different equations. We'll convert a polynomial \(p\) into two polynomials \(evens\) and \(odds\) such that \(p(x) = evens(x \cdot
(k-x)) + x \cdot odds(x \cdot (k-x))\). Note that for the \(evens\) and \(odds\) that we find, it will also be true that \(p(x+k) = evens(x \cdot (k-x)) + (x+k) \cdot odds(x \cdot (k-x))\). So we can
then recursively do an FFT to \(evens\) and \(odds\) on the reduced domain \([x \cdot (k-x)\) for \(x\) in \(D]\), and then we use these two formulas to get the answers for two "halves" of the
domain, one offset by \(k\) from the other.
Converting \(p\) into \(evens\) and \(odds\) as described above turns out to itself be nontrivial. The "naive" algorithm for doing this is itself \(O(N^2)\), but it turns out that in a binary field,
we can use the fact that \((x^2-kx)^2 = x^4 - k^2 \cdot x^2\), and more generally \((x^2-kx)^{2^{i}} = x^{2^{i+1}} - k^{2^{i}} \cdot x^{2^{i}}\) , to create yet another recursive algorithm to do this
in \(O(N \cdot log(N))\) time.
And if you want to do an inverse FFT, to do interpolation, then you need to run the steps in the algorithm in reverse order. You can find the complete code for doing this here: https://github.com/
ethereum/research/tree/master/binary_fft, and a paper with details on more optimal algorithms here: http://www.math.clemson.edu/~sgao/papers/GM10.pdf
So what do we get from all of this complexity? Well, we can try running the implementation, which features both a "naive" \(O(N^2)\) multi-point evaluation and the optimized FFT-based one, and time
both. Here are my results:
>>> import binary_fft as b
>>> import time, random
>>> f = b.BinaryField(1033)
>>> poly = [random.randrange(1024) for i in range(1024)]
>>> a = time.time(); x1 = b._simple_ft(f, poly); time.time() - a
>>> a = time.time(); x2 = b.fft(f, poly, list(range(1024))); time.time() - a
And as the size of the polynomial gets larger, the naive implementation (_simple_ft) gets slower much more quickly than the FFT:
>>> f = b.BinaryField(2053)
>>> poly = [random.randrange(2048) for i in range(2048)]
>>> a = time.time(); x1 = b._simple_ft(f, poly); time.time() - a
>>> a = time.time(); x2 = b.fft(f, poly, list(range(2048))); time.time() - a
And voila, we have an efficient, scalable way to multi-point evaluate and interpolate polynomials. If we want to use FFTs to recover erasure-coded data where we are missing some pieces, then
algorithms for this also exist, though they are somewhat less efficient than just doing a single FFT. Enjoy!
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program called "call_info.cpp" solution
In this assignment you will implement a program called “call_info.cpp” that uses
three functions, input, output, and process. You must use input and output
parameters when implementing this program. The function input will get the input
from the user, the function process will perform the necessary calculations required
by your algorithm, and the function output will print the results and any output that
needs to be printed.
The program “call_info.cpp” will calculate the net cost of a call (net_cost), the tax on
a call (call_tax) and the total cost of the call (total_cost). The program should accept
a cell phone number (cell_num), the number of relay stations(relays), and the
length in minutes of the cal (call_length). Please consider the following
1) The tax rate (in percent) on a call (call_rate) is simply based on the number
of relay stations (relays) used to make the call (1<= relays <=5 then tax_rate = 1%; 6<= relays <=11 then tax_rate = 3%; 12<= relays <=20 then tax_rate = 5%; 21<= relays <=50 then tax_rate = 8%;
relays 50 then tax_rate =12%) . 2) The net cost of a call is calculated by the following formula: net_cost = ( relays / 50.0 * 0.40 * call_length). 3) The tax on a call is calculated by the following
formula: call_tax = net_cost * tax_rate / 100. 4). The total cost of a call (rounded to the nearest hundredth) is calculated by the following formula: total_cost = net_cost + call_tax . All tax and
cost calculations should be rounded to the nearest hundredths. Use the following format information to print the variables: Field Format ====================================== Cell Phone XXXXXXXXX
Number of Relay Stations �� XXXXXX Minutes Used ����� XXXXXX Net Cost ���� XXXXXXX.XX Call Tax XXXXX.XX Total Cost of Call �� XXXXXXX.XX Handing in your program Electronically submit "call_info.cpp"
in the Assignments area of Blackboard before the due date and time. Remember, no late assignments will be accepted. Input Example: (Your program should prompt the user for input) Enter your Cell
Phone Number: 9548267184 Enter the number of relay stations: 40 Enter the length of the call in minutes: 56 Output Example: (Your output should look lit this)
***************************************************** Cell Phone Number: 9548267184 ***************************************************** Number of Relay Stations: 40 Length of Call in Minutes: 56
Net Cost of Call: 17.92 Tax of Call: 1.43 Total Cost of Call: 19.35 Ask the user if more calculations are necessary with the following prompt: Would you like to do another calculation for another
employee (enter y or n)?
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- Algebra
Early Edge: Proportions
In this Early Edge video lesson, you'll learn more about Proportions, so you can be successful when you take on high-school Math & Arithmetic.
This site has a tutorial and then some examples of how to write proportions to solve for missing lengths of similar polygons
Ton Conversion
This site helps to explain the different types of tons and how to convert them.
In this Early Edge video lesson, you'll learn more about Introduction to Algebra, so you can be successful when you take on high-school Math & Algebra.
In this Early Edge video lesson, you'll learn more about Two-Step Equations, so you can be successful when you take on high-school Math & Algebra.
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Periodic nonlinear Schrödinger equation and invariant measures EQUATION DE SCHRODINGER MESURES INVARIANTES EQUATIONS DE KORTEWEG-DE VRIES FONCTIONS PERIODIQUES BOURGAIN M/93/50 IHES 09/1993 A4 18 f.
EN TEXTE PREPUBLICATION M_93_50.pdf 1993 On the Cauchy and invariant measure problem for the periodic Zakharov system PROBLEME DE CAUCHY EQUATION DE SCHRODINGER MECANIQUE STATISTIQUE BOURGAIN M/93/63
IHES 11/1993 A4 14 f. EN TEXTE PREPUBLICATION M_93_63.pdf 1993 Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations SYSTEMES HAMILTONIENS EQUATIONS D’ONDES NON
LINEAIRES EQUATION DE SCHRODINGER BOURGAIN M/94/18 IHES 03/1994 A4 22 f. EN TEXTE PREPUBLICATION M_94_18.pdf 1994 Invariant measures for the 2D-defocusing nonlinear Schrödinger equation EQUATION DE
SCHRODINGER MESURES INVARIANTES MECANIQUE STATISTIQUE PROBLEME DE CAUCHY BOURGAIN M/94/28 IHES 04/1994 A4 13 f. EN TEXTE PREPUBLICATION M_94_28.pdf 1994 Gibbs measures and quasi-periodic solutions
for nonlinear Hamiltonian partial differential equations SYSTEMES HAMILTONIENS MESURES DE GIBBS EQUATIONS DIFFERENTIELLES NON LINEAIRES EQUATION DE SCHRODINGER PERTURBATION BOURGAIN M/95/13 IHES 02/
1995 A4 11 f. EN TEXTE PREPUBLICATION M_95_13.pdf 1995 Tendances nouvelles en mécanique : Quatre conférences sur la mécanique céleste et les instabilités MECANIQUE CELESTE RELATIVITE GENERALE
EXPERIENCES TROUS NOIRS STABILITE CONVECTION THERMIQUE CARTIER CARTER GUYON MARCHAL P/79/310 IHES 11/1979 A4 15 f. FR TEXTE PREPUBLICATION P_79_310.pdf 1979 A Rigorous mathematical foundation of
functional integration INTEGRATION DE FONCTIONS PHYSIQUE THEORIQUE FORMES QUADRATIQUES VOLUME THEOREME THEORIE DES TRELLIS MESURES GAUSSIENNES CALCUL INTEGRAL APPLICATIONS CARTIER DEWITT-MORETTE M/97
/86 IHES 11/1997 A4 39 f. EN TEXTE PREPUBLICATION M_97_86.pdf 1997 Characterizing volume forms INTEGRATION DE FONCTIONS VOLUME FORMES THEORIE QUANTIQUE DES CHAMPS GEOMETRIE DIFFERENTIELLE CARTIER
BERG DEWITT-MORETTE WURM M/01/17 IHES 04/2001 A4 10 f. EN TEXTE PREPUBLICATION P_01_17.pdf 2001 Phase transitions in anisotropic lattice spin systems SPIN PHYSIQUE NUCLEAIRE ANISOTROPIE
FERROMAGNETISME FROHLICH LIEB P/78/199 IHES 01/1978 A4 35 f. EN TEXTE PREPUBLICATION P_78_199.pdf 1978 On the statistical mechanics of classical Coulomb - and dipole gases MECANIQUE STATISTIQUE GAZ
PHYSIQUE NUCLEAIRE FROHLICH SPENCER P/80/10 IHES 1980 A4 65 f. EN TEXTE PREPUBLICATION P_80_10.pdf 1980 The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb Gas
SPIN PHYSIQUE NUCLEAIRE GAZ DENSITE TRANSFORMATIONS ELECTROSTATIQUE FROHLICH SPENCER P/81/25 IHES 1981 A4 77 f. EN TEXTE PREPUBLICATION P_81_25.pdf 1981 On the Absence of spontaneous symmetry
breaking and of crytalline ordering in two-dimensional systems SYMETRIE BRISEE CRISTALLOGRAPHIE MATHEMATIQUE MECANIQUE STATISTIQUE DIMENSIONS FROHLICH PFISTER P/81/31 IHES 1981 A4 22 f. EN TEXTE
PREPUBLICATION P_81_31.pdf 1981 1- Spontaneously broken and dynamically enhanced global and local-symmetries 2 - Continuum (scaling) limits of lattice fields theories (triviality of lg q in d 1=y4
dimensions 3 -Results and problems near the interface between statistical mechanics and quantum field theory SYMETRIE BRISEE THEORIE DES CHAMPS METRIQUE MECANIQUE STATISTIQUE THEORIE QUANTIQUE DES
CHAMPS FROHLICH P/81/56 IHES 1981 A4 26 f. EN TEXTE PREPUBLICATION P_81_56.pdf 1981 Some Recent rigorous results in the theory of phase transitions and critical phenomena TRANSITIONS DE PHASES
PHENOMENES CRITIQUES THEORIE QUANTIQUE DES CHAMPS GROUPE DE RENORMALISATION MODELE D'ISING FROHLICH SPENCER P/82/23 IHES 1982 A4 22 f. EN TEXTE PREPUBLICATION P_82_23.pdf 1982 Return to equilibrium
GROUPE DE RENORMALISATION EQUILIBRE THERMODYNAMIQUE RAYONNEMENT DU CORPS NOIR KMS FROHLICH BACH SIGAL P/00/44 IHES 06/2000 A4 53 f. EN TEXTE PREPUBLICATION P_00_44.pdf 2000 Mathematical slices of
molecular biology BIOMATHEMATIQUES BIOLOGIE MOLECULAIRE MATHEMATIQUES GENETIQUE CYTOLOGIE GROMOV CARBONE M/01/03 IHES A4 44 f. EN TEXTE PREPUBLICATION M_01_03.pdf 2001 Functional labels and syntactic
entropy on DNA strings and proteins ADN SEQUANCAGE DES ACIDES NUCLEIQUES GENES PROTEINES MACROMOLECULES HYPERGRAPHES EVOLUTION ENTROPIE GROMOV CARBONE M/01/54 IHES A4 12 f. EN TEXTE M_01_54.pdf 2001
Random walk in random groups GROUPES INFINIS ENTROPIE ESPACES METRIQUES GROMOV M/02/03 IHES A4 35 f. EN TEXTE PREPUBLICATION M_02_03.pdf 2002 Isoperimetry of waists and concentration of maps
INEGALITES ISOPERIMETRIQUES ENSEMBLES CONVEXES GROMOV M/02/04 IHES A4 22 f. EN TEXTE PREPUBLICATION M_02_04.pdf 2002 Symmetry in condensed matter physics SYMETRIE MATIERE CONDENSEE MICHEL P/81/58
IHES 11/1981 A4 8 f. EN TEXTE PREPUBLICATION P_81_58.pdf 1981 Introduction to spontaneous symmetry breaking : some examples PHYSIQUE SYMETRIE BRISEE MICHEL P/85/27 IHES 04/1984 A4 14 f. EN TEXTE
PREPUBLICATION P_85_27.pdf 1985 The Present Status of CP, T and CPT Invariance PHYSIQUE NUCLEAIRE PARTICULES SYMETRIE INVARIANTS Introduction : Last time I came to Sweden, cars were drivent on the
left side of the roads. This week most cars I saw were driven on the right side. I am very grateful to the Nobel Foundation for this opportunity to observe this P violation. I may have found the
cause of this P violation by looking at buses or trucks with the same inscription on both sides ; they are not symmetrically written. I have also made a couter-experiment to verify this theory. In
Tokyo I found an equal probability for cars to be driven on the left side or the right side of the streets and I did observe that identical texts were placed symmetrically on buses and trucks : the
first character is near the front part on both sides so the same text must be read from left to right on one side and from right to left on the other side. Since I have never been yet in a car
undergoing a spontaneous CP transformation, I will be able to give you today my report on CP, T and CPT invariance. MICHEL P/68/31 IHES [06/1968] A4 18 f. EN TEXTE PREPUBLICATION P_68_31.pdf 1968 The
States of classical statistical mechanics MECANIQUE STATISTIQUE Abstract : a state of an infinite system in classical statistical mechanics is usually described by its correlation functions. We
discuss here other description in particular as 1) a state on a B*-algebras, 2) a collection of density distributions, 3) a field theory 4) a mesure on a space of configurations of infitely many
particles. We consier the relations between these various descriptions and prove under very general conditions an integral representation of a state as superposition of extremal invariant states
corresponding to pure thermodynamical phases. RUELLE P/66/02 IHES 1966 A4 31 f. EN TEXTE PREPUBLICATION P_66_02.pdf 1966 A Variational formulation of equilibrium statistical mechanics and the Gibbs
phase rule ENTROPIE RESEAUX PHYSIQUE STATISTIQUE SYSTEMES COMPLEXES Abstract : It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem
involving a mean entropy of states introduced earlier [2]. As an application, a version of the Gibbs phase rule is proved. RUELLE P/67/X012 IHES 1967 A4 6 f. EN TEXTE PREPUBLICATION P_67_X012.pdf
1967 Statistical mechanics of a one-dimensional lattice gas MECANIQUE STATISTIQUE RESEAUX GAZ Abstract : We study the statistical mechanics of an infinite one-dimensional classical lattice gas.
Extending a result of Van Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is
invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be a K-system. RUELLE P/67/X016 IHES 1967 A4 10 f. EN TEXTE PREPUBLICATION
P_67_X016.pdf 1967 Symmetry breakdown in statistical mechanics SYMETRIE BRISEE MECANIQUE STATISTIQUE CONGRES ET CONFERENCES Lecture given at the Ecole d'Eté de Physique Théorique. Cargèse, Corsica,
1969. Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical quantum lattice systems.
RUELLE P/69/30 IHES 1969 A4 24 f. EN TEXTE PREPUBLICATION P_69_30.pdf 1969 The Ergodic theory of axiom a flows THEORIE ERGODIQUE ENTROPIE AXIOMES RUELLE BOWEN P/74/78 IHES 03/1974 A4 38 f. EN TEXTE
PREPUBLICATION P_74_78.pdf 1974 On Manifolds of phase coexistence PHYSIQUE THEORIQUE PHYSIQUE MATHEMATIQUE Abstract : Using a theorem on convex functions due to Israel, it is shown that a point of
coexistence of n+1n+1 phases cannot be isolated in the space of interactions, but lies on some infinite dimensional manifold. RUELLE P/75/128 IHES 12/1975 A4 13 f. EN TEXTE PREPUBLICATION
P_75_128.pdf 1975 A Heuristic theory of phase transitions RESEAUX SYSTEMES COMPLEXES ESPACES DE BANACH TRANSITIONS DE PHASE Abstract : Let Z be a suitable Banach space of interactions for a lattice
spin system. If n+1 thermodynamic phases coexist for ?0 ?Z, it is shown that a manifold of codimension n of coexistence of (at least) n+1 phases passes through ?0. There are also n+1 manifolds of
codimension n?1 of coexistence of (at least) n phases; these have a common boundary along the manifold of coexistence of n+1 phases. And so on for coexistence of fewer phases. This theorem is proved
under a technical condition (R) which says that the pressure is a differentiable function of the interaction at ?0 when restricted to some codimensionn affine subspace of Z. The condition (R) has not
been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a
heuristic proof of the Gibbs phase rule. RUELLE P/76/149 IHES 10/1976 A4 25 f. EN TEXTE PREPUBLICATION P_76_149.pdf 1976 Integral representation of measures associated with a foliation CALCUL
INTEGRAL TOPOLOGIE DIFFERENTIELLE FEUILLETAGES THEORIE GEOMETRIQUE DES FONCTIONS RUELLE PM/77/181 IHES 05/1977 A4 6 f. EN TEXTE PREPUBLICATION PM_77_181.pdf 1977 Sensitive dependence on initial
conditions and turbulent behavior of dynamical systems SYSTEMES DYNAMIQUES DYNAMIQUE DIFFERENTIABLE Abstract : The asymptotic behavior of differentiable dynamical systems is analyzed. We discuss its
descriptoin by asymptotic measures and the turbulent behavior with senditive dependence on initial condition. RUELLE P/77/190 IHES 10/1977 A4 9 f. EN TEXTE PREPUBLICATION P_77_190.pdf 1977 Do there
exist turbulent crystals ? CRISTAUX TURBULENCE Abstract : We discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium
states at non-zero temperature. Turbulent crystals would not be invariant under translation, but would differ from other crystals by the fuzziness of some diffraction peaks. Turbulent crystals could
appear by breakdown of long range order in quasiperiodic crystals with two independent modulations. RUELLE P/82/02 IHES 01/1982 A4 6 f. EN TEXTE PREPUBLICATION P_82_02.pdf 1982 On the Ergodic theory
at infinity of an arbitrary discrete group of hyperbolic motions THEORIE ERGODIQUE GROUPES DISCRETS MOUVEMENT HYPERBOLES SULLIVAN M/78/229 IHES 06/1978 A4 18 f. EN TEXTE PREPUBLICATION M_78_229.pdf
1978 The Density at infinity of a discrete group of hyperbolic motions THEORIE ERGODIQUE THEORIE DE LA MESURE TRANSFORMATIONS SURFACES DE RIEMANN GROUPES DE KLEIN Dedicated to Rufus Bowen SULLIVAN M/
79/261 IHES 03/1979 A4 27 f. EN TEXTE PREPUBLICATION M_79_261.pdf 1979 Dynamical entropy of C* algebras and Von Neumann algebras ALGEBRES DE VON NEUMANN C*-ALGEBRES CONNES NARNHOFER THIRRING M/87/05
IHES 02/1987 A4 25 f. EN TEXTE PREPUBLICATION M_87_05.pdf 1987 Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theoies ALGEBRES DE VON NEUMANN
AUTOMORPHISMES THEORIE QUANTIQUE THERMODYNAMIQUE TEMPS GRAVITATION CONNES ROVELLI M/94/36 IHES 06/1994 A4 14 f. EN TEXTE PREPUBLICATION M_94_36.pdf 1994
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How do I run diversity index calculation on individual replicates for a single column of sample data
Suppose I have the following data in a single column (I've placed it in a row to save space but we alwasy work with it in a column):
Each number represents a differents species, and the number of each number represents the number of individuals of that species identified in the sample set (so in the above set there were two
individuals of species "7").
I want to take repeatedly sample 10 individuals from the above set, AND THEN run Simpson's/Shannon's diversity index individually on each replicate, AND THEN determine a mean diversity based on the
diversity of all individual replicates.
I'm a major beginner at SAS, from what I can tell I have proprietary software 9.3 with enhanced analytical products in the 12.1-12.2 region. I've figured out how to do the repeated sampling using the
proc surveyselect data = arrival method = urs sampsize = 10
rep = 10 out = my_data;
Can someone help me with the rest of the steps please?
p.s. I also don't know if my method of analysis is statistically viable, it is my attempt to compare the genetic diversity of two populations for each of which we have a different number of samples.
The different sample sizes prevent direct comparison of genetic diversity, so I thought we could subsample the larger set down to the size of the smaller set. If you have a big sexy statistical brain
I would appreciate comments on this idea as well.
06-29-2016 11:17 AM
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Maltese Ordinal Numbers - OrdinalNumbers.com
Maltese Ordinal Numbers
Maltese Ordinal Numbers – There are a myriad of sets that can be enumerated by using ordinal numbers as a tool. These numbers can be utilized as a tool to generalize ordinal numbers.
The foundational idea of math is that of the ordinal. It is a number that indicates where an object is in a list. The ordinal number is a number between 1 and 20. While ordinal numbers have many
functions, they are most commonly utilized to represent the sequence of the items in the list.
To represent ordinal numbers, you can make use of numbers, charts, and words. They may also be used to explain the way in which pieces are arranged.
The vast majority of ordinal numbers fall in one or any of these categories. Transfinite ordinals are represented by lowercase Greek letters, whereas finite ordinals are represented with Arabic
Based on the Axiom of Choice, any set that is well-organized should contain at minimum one ordinal. For instance, the very first person to complete a class would get the highest grade. The winner of
the contest was the student who had the highest grade.
Combinational ordinal numbers
Compounded ordinal numbers are numbers that have multiple digits. They are made by multiplying ordinal number by its final number. These numbers are most commonly used for ranking and dating
purposes. They don’t have an exclusive ending for the final digit like cardinal numbers do.
To identify the order in which elements are placed within the collection ordinal numbers are utilized. They also serve to identify the names of elements in collections. Regular numbers are available
in both regular and suppletive forms.
By prefixing a cardinal numbers by the suffix -u, regular ordinals can be created. Next, the number is typed in a word, and a hyphen follows it. There are many suffixes to choose from.
Suppletive ordinals are derived from prefixing words with the suffix -u. The suffix can be employed to count. It’s also wider than the normal one.
Limits of Ordinal
Limits for ordinal numbers are ordinal numbers that aren’t zero. Limit ordinal numbers have the disadvantage of not having the possibility of having a maximum element. They may be created by joining
non-empty sets without maximum elements.
Additionally, transcendent rules of recursion utilize limited ordinal numbers. Based on the von Neumann model, every infinite cardinal also has an order limit.
An ordinal numbers with a limit are equal to the sum of all ordinals lower than it. Limit ordinal amounts can be enumerated using mathematics and be expressed as a series or natural numbers.
The ordinal numbers serve to organize the data. They give an explanation about the location of an object’s numerical coordinates. They are often utilized in set theory and arithmetic. Despite sharing
the same form, they aren’t in the same classification as natural numbers.
The von Neumann model uses a well-ordered set. It is assumed that fyyfy represents one of the subfunctions g’ of a function that is described as a singular operation. Given that g’ meets the criteria
that g’ be an ordinal limit if fy is the lone subfunction (i, ii).
The Church Kleene oral is a limit-ordering order that works in a similar way. The Church-Kleene oral is a limit or limit ordinal that can be a well-ordered collection, that is comprised of smaller
Examples of ordinal numbers in stories
Ordinal numbers are commonly utilized to establish the hierarchy between objects and entities. They are essential for organizing, counting, as well as for ranking reasons. They can be utilized to
describe the position of objects in addition to providing a sequence of items.
Ordinal numbers are typically indicated with the letter “th”. Sometimes, however the letter “nd” can be substituted. Book titles typically contain ordinal numbers.
Even though ordinal figures are typically used in list format it is possible to write them down as words. They also can appear in the form of acronyms and numbers. These numbers are easier to
comprehend than cardinal numbers, however.
There are three kinds of ordinal numbers. Through games, practice, and other exercises, you may be able to learn more about the different kinds of numbers. It is essential to know about them in order
to enhance your math skills. Coloring exercises are a fun, easy and enjoyable way to improve. Make sure you check your work using the handy marking sheet.
Gallery of Maltese Ordinal Numbers
Award Ribbons Ordinal Numbers Clipart 10 Free Cliparts Download
Award Ribbons Ordinal Numbers Clipart 10 Free Cliparts Download
Ferrante Worksheets Year 2 Malti Askworksheet Il Malti Ghat Tfal L
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Degenerate energy levels - Wikipedia
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical
system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy
(or simply the degeneracy) of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue.^[1]^
:48 When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired.
In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy
states. These degenerate states at the same level all have an equal probability of being filled. The number of such states gives the degeneracy of a particular energy level.
Degenerate states in a quantum system
The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear
Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N×N matrix,
X a non-zero vector, and λ is a scalar, such that ${\displaystyle AX=\lambda X}$, then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ.
Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of C^n, which is called the eigenspace of λ. An eigenvalue λ which corresponds to two
or more different linearly independent eigenvectors is said to be degenerate, i.e., ${\displaystyle AX_{1}=\lambda X_{1}}$ and ${\displaystyle AX_{2}=\lambda X_{2}}$, where ${\displaystyle X_{1}}$
and ${\displaystyle X_{2}}$ are linearly independent eigenvectors. The dimension of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or
infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional.
The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give
the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its
eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any
linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue. This clearly follows from the fact that the
eigenspace of the energy value eigenvalue λ is a subspace (being the kernel of the Hamiltonian minus λ times the identity), hence is closed under linear combinations.
In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to
each energy eigenvalue. However, if the Hamiltonian ${\displaystyle {\hat {H}}}$ has a degenerate eigenvalue ${\displaystyle E_{n}}$ of degree g[n], the eigenstates associated with it form a vector
subspace of dimension g[n]. In such a case, several final states can be possibly associated with the same result ${\displaystyle E_{n}}$, all of which are linear combinations of the g[n] orthonormal
eigenvectors ${\displaystyle |E_{n,i}\rangle }$.
In this case, the probability that the energy value measured for a system in the state ${\displaystyle |\psi \rangle }$ will yield the value ${\displaystyle E_{n}}$ is given by the sum of the
probabilities of finding the system in each of the states in this basis, i.e., ${\displaystyle P(E_{n})=\sum _{i=1}^{g_{n}}|\langle E_{n,i}|\psi \rangle |^{2}}$
This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual
understanding of more complex systems.
In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function ${\displaystyle |\psi \rangle }$ moving in a
one-dimensional potential ${\displaystyle V(x)}$, the time-independent Schrödinger equation can be written as ${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+V\psi =E\psi }$
Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy ${\displaystyle E}$ at most, so that the degree of degeneracy never exceeds two. It can be
proven that in one dimension, there are no degenerate bound states for normalizable wave functions. A sufficient condition on a piecewise continuous potential ${\displaystyle V}$ and the energy ${\
displaystyle E}$ is the existence of two real numbers ${\displaystyle M,x_{0}}$ with ${\displaystyle Meq 0}$ such that ${\displaystyle \forall x>x_{0}}$ we have ${\displaystyle V(x)-E\geq M^{2}}$.^[3
] In particular, ${\displaystyle V}$ is bounded below in this criterion.
Proof of the above theorem.
Considering a one-dimensional quantum system in a potential ${\displaystyle V(x)}$ with degenerate states ${\displaystyle |\psi _{1}\rangle }$ and ${\displaystyle |\psi _{2}\rangle }$
corresponding to the same energy eigenvalue ${\displaystyle E}$, writing the time-independent Schrödinger equation for the system:
{\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{1}}{dx^{2}}}+V\psi _{1}&=E\psi _{1}\\-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi _{2}}{dx^{2}}}+V\psi _{2}&=E\psi _{2}\
end{aligned}}} Multiplying the first equation by ${\displaystyle \psi _{2}}$ and the second by ${\displaystyle \psi _{1}}$ and subtracting one from the other, we get: ${\displaystyle \psi _{1}{\
frac {d^{2}}{dx^{2}}}\psi _{2}-\psi _{2}{\frac {d^{2}}{dx^{2}}}\psi _{1}=0}$ Integrating both sides ${\displaystyle \psi _{1}{\frac {d\psi _{2}}{dx}}-\psi _{2}{\frac {d\psi _{1}}{dx}}={\mbox
{constant}}}$ In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: ${\displaystyle \
psi _{1}(x)=c\psi _{2}(x)}$ where ${\displaystyle c}$ is, in general, a complex constant. For bound state eigenfunctions (which tend to zero as ${\displaystyle x\to \infty }$), and assuming ${\
displaystyle V}$ and ${\displaystyle E}$ satisfy the condition given above, it can be shown^[3] that also the first derivative of the wave function approaches zero in the limit ${\displaystyle x\
to \infty }$, so that the above constant is zero and we have no degeneracy.
Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made
of monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon
etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of Particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical
models for several real world systems.
Consider a free particle in a plane of dimensions ${\displaystyle L_{x}}$ and ${\displaystyle L_{y}}$ in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with
wave function ${\displaystyle |\psi \rangle }$ can be written as ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}\psi }{{\partial x}^{2}}}+{\frac {\partial ^{2}\psi }{{\partial y}
^{2}}}\right)=E\psi }$ The permitted energy values are ${\displaystyle E_{n_{x},n_{y}}={\frac {\pi ^{2}\hbar ^{2}}{2m}}\left({\frac {n_{x}^{2}}{L_{x}^{2}}}+{\frac {n_{y}^{2}}{L_{y}^{2}}}\right)}$ The
normalized wave function is ${\displaystyle \psi _{n_{x},n_{y}}(x,y)={\frac {2}{\sqrt {L_{x}L_{y}}}}\sin \left({\frac {n_{x}\pi x}{L_{x}}}\right)\sin \left({\frac {n_{y}\pi y}{L_{y}}}\right)}$ where
${\displaystyle n_{x},n_{y}=1,2,3,\dots }$
So, quantum numbers ${\displaystyle n_{x}}$ and ${\displaystyle n_{y}}$ are required to describe the energy eigenvalues and the lowest energy of the system is given by ${\displaystyle E_{1,1}=\pi ^
{2}{\frac {\hbar ^{2}}{2m}}\left({\frac {1}{L_{x}^{2}}}+{\frac {1}{L_{y}^{2}}}\right)}$
For some commensurate ratios of the two lengths ${\displaystyle L_{x}}$ and ${\displaystyle L_{y}}$, certain pairs of states are degenerate. If ${\displaystyle L_{x}/L_{y}=p/q}$, where p and q are
integers, the states ${\displaystyle (n_{x},n_{y})}$ and ${\displaystyle (pn_{y}/q,qn_{x}/p)}$ have the same energy and so are degenerate to each other.
In this case, the dimensions of the box ${\displaystyle L_{x}=L_{y}=L}$ and the energy eigenvalues are given by ${\displaystyle E_{n_{x},n_{y}}={\frac {\pi ^{2}\hbar ^{2}}{2mL^{2}}}(n_{x}^{2}+n_{y}^
Since ${\displaystyle n_{x}}$ and ${\displaystyle n_{y}}$ can be interchanged without changing the energy, each energy level has a degeneracy of at least two when ${\displaystyle n_{x}}$ and ${\
displaystyle n_{y}}$ are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states
(n[x] = 7, n[y] = 1), (n[x] = 1, n[y] = 7) and (n[x] = n[y] = 5) all have ${\displaystyle E=50{\frac {\pi ^{2}\hbar ^{2}}{2mL^{2}}}}$ and constitute a degenerate set.
Degrees of degeneracy of different energy levels for a particle in a square box:
${\displaystyle n_{x}}$ ${\displaystyle n_{y}}$ ${\displaystyle E\left({\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\right)}$ Degeneracy
... ... ... ...
... ... ... ...
... ... ... ...
... ... ... ...
... ... ... ...
... ... ... ...
In this case, the dimensions of the box ${\displaystyle L_{x}=L_{y}=L_{z}=L}$ and the energy eigenvalues depend on three quantum numbers. ${\displaystyle E_{n_{x},n_{y},n_{z}}={\frac {\pi ^{2}\hbar ^
Since ${\displaystyle n_{x}}$, ${\displaystyle n_{y}}$ and ${\displaystyle n_{z}}$ can be interchanged without changing the energy, each energy level has a degeneracy of at least three when the three
quantum numbers are not all equal.
If two operators ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$ commute, i.e., ${\displaystyle [{\hat {A}},{\hat {B}}]=0}$, then for every eigenvector ${\displaystyle |\psi \rangle }$
of ${\displaystyle {\hat {A}}}$, ${\displaystyle {\hat {B}}|\psi \rangle }$ is also an eigenvector of ${\displaystyle {\hat {A}}}$ with the same eigenvalue. However, if this eigenvalue, say ${\
displaystyle \lambda }$, is degenerate, it can be said that ${\displaystyle {\hat {B}}|\psi \rangle }$ belongs to the eigenspace ${\displaystyle E_{\lambda }}$ of ${\displaystyle {\hat {A}}}$, which
is said to be globally invariant under the action of ${\displaystyle {\hat {B}}}$.
For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However, ${\displaystyle \lambda }$ is a degenerate
eigenvalue of ${\displaystyle {\hat {A}}}$, then it is an eigensubspace of ${\displaystyle {\hat {A}}}$ that is invariant under the action of ${\displaystyle {\hat {B}}}$, so the representation of $
{\displaystyle {\hat {B}}}$ in the eigenbasis of ${\displaystyle {\hat {A}}}$ is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of ${\displaystyle {\hat {A}}}$ are not,
in general, eigenvectors of ${\displaystyle {\hat {B}}}$. However, it is always possible to choose, in every degenerate eigensubspace of ${\displaystyle {\hat {A}}}$, a basis of eigenvectors common
to ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$.
If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of ${\displaystyle {\hat {A}}}$ are degenerate,
specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable ${\displaystyle {\hat {B}}}$, which commutes with ${\displaystyle {\hat {A}}}$, it is possible
to construct an orthonormal basis of eigenvectors common to ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$, which is unique, for each of the possible pairs of eigenvalues {a,b}, then $
{\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$ are said to form a complete set of commuting observables. However, if a unique set of eigenvectors can still not be specified, for at
least one of the pairs of eigenvalues, a third observable ${\displaystyle {\hat {C}}}$, which commutes with both ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$ can be found such that
the three form a complete set of commuting observables.
It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator
that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.
The parity operator is defined by its action in the ${\displaystyle |r\rangle }$ representation of changing r to −r, i.e. ${\displaystyle \langle r|P|\psi \rangle =\psi (-r)}$ The eigenvalues of P
can be shown to be limited to ${\displaystyle \pm 1}$, which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while
that with eigenvalue −1 is said to be odd.
Now, an even operator ${\displaystyle {\hat {A}}}$ is one that satisfies, ${\displaystyle {\tilde {A}}=P{\hat {A}}P}$ ${\displaystyle [P,{\hat {A}}]=0}$ while an odd operator ${\displaystyle {\hat
{B}}}$ is one that satisfies ${\displaystyle P{\hat {B}}+{\hat {B}}P=0}$ Since the square of the momentum operator ${\displaystyle {\hat {p}}^{2}}$ is even, if the potential V(r) is even, the
Hamiltonian ${\displaystyle {\hat {H}}}$ is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore
it is possible to look for the eigenstates of ${\displaystyle {\hat {H}}}$ among even and odd states. However, if one of the energy eigenstates has no definite parity, it can be asserted that the
corresponding eigenvalue is degenerate, and ${\displaystyle P|\psi \rangle }$ is an eigenvector of ${\displaystyle {\hat {H}}}$ with the same eigenvalue as ${\displaystyle |\psi \rangle }$.
The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find
the energy levels and degeneracies without solving the Schrödinger equation, hence reducing effort.
Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian
is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that ${\displaystyle H'=SHS^{-1}=SHS^{\dagger }}$, since S is unitary. If the Hamiltonian
remains unchanged under the transformation operation S, we have {\displaystyle {\begin{aligned}SHS^{\dagger }&=H\\[1ex]SHS^{-1}&=H\\[1ex]SH&=HS\\[1ex][S,H]&=0\end{aligned}}} Now, if ${\displaystyle |
\alpha \rangle }$ is an energy eigenstate, ${\displaystyle H|\alpha \rangle =E|\alpha \rangle }$ where E is the corresponding energy eigenvalue. ${\displaystyle HS|\alpha \rangle =SH|\alpha \rangle =
SE|\alpha \rangle =ES|\alpha \rangle }$ which means that ${\displaystyle S|\alpha \rangle }$ is also an energy eigenstate with the same eigenvalue E. If the two states ${\displaystyle |\alpha \rangle
}$ and ${\displaystyle S|\alpha \rangle }$ are linearly independent (i.e. physically distinct), they are therefore degenerate.
In cases where S is characterized by a continuous parameter ${\displaystyle \epsilon }$, all states of the form ${\displaystyle S(\epsilon )|\alpha \rangle }$ have the same energy eigenvalue.
The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the
algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a
particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a
n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.
Degeneracies in a quantum system can be systematic or accidental in nature.
This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a
certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.
It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in
the system.^[4] It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An
accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.
For a particle in a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of angular momentum due to
rotational invariance.
For a particle moving on a cone under the influence of 1/r and r^2 potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of
an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.
A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in
this case are the Landau levels which are infinitely degenerate.
In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum ${\
displaystyle {\hat {L}}^{2}}$, its component along the z-direction, ${\displaystyle {\hat {L}}_{z}}$, total spin angular momentum ${\displaystyle {\hat {S}}^{2}}$ and its z-component ${\displaystyle
{\hat {S}}_{z}}$. The quantum numbers corresponding to these operators are ${\displaystyle \ell }$, ${\displaystyle m_{\ell }}$, ${\displaystyle s}$ (always 1/2 for an electron) and ${\displaystyle
m_{s}}$ respectively.
The energy levels in the hydrogen atom depend only on the principal quantum number n. For a given n, all the states corresponding to ${\displaystyle \ell =0,\ldots ,n-1}$ have the same energy and are
degenerate. Similarly for given values of n and ℓ, the ${\displaystyle (2\ell +1)}$, states with ${\displaystyle m_{\ell }=-\ell ,\ldots ,\ell }$ are degenerate. The degree of degeneracy of the
energy level E[n] is therefore ${\displaystyle \sum _{\ell \mathop {=} 0}^{n-1}(2\ell +1)=n^{2},}$ which is doubled if the spin degeneracy is included.^[1]^:267f
The degeneracy with respect to ${\displaystyle m_{\ell }}$ is an essential degeneracy which is present for any central potential, and arises from the absence of a preferred spatial direction. The
degeneracy with respect to ${\displaystyle \ell }$ is often described as an accidental degeneracy, but it can be explained in terms of special symmetries of the Schrödinger equation which are only
valid for the hydrogen atom in which the potential energy is given by Coulomb's law.^[1]^:267f
It is a spinless particle of mass m moving in three-dimensional space, subject to a central force whose absolute value is proportional to the distance of the particle from the centre of force. ${\
displaystyle F=-kr}$ It is said to be isotropic since the potential ${\displaystyle V(r)}$ acting on it is rotationally invariant, i.e., ${\displaystyle V(r)={\tfrac {1}{2}}m\omega ^{2}r^{2}}$ where
${\displaystyle \omega }$ is the angular frequency given by ${\textstyle {\sqrt {k/m}}}$.
Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a
system is given by- ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^
{2}}}\right)+{\frac {1}{2}}{m\omega ^{2}\left(x^{2}+y^{2}+z^{2}\right)\psi }=E\psi }$
So, the energy eigenvalues are ${\displaystyle E_{n_{x},n_{y},n_{z}}=\left(n_{x}+n_{y}+n_{z}+{\tfrac {3}{2}}\right)\hbar \omega }$ or, ${\displaystyle E_{n}=\left(n+{\tfrac {3}{2}}\right)\hbar \omega
}$ where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets ${\displaystyle \{n_{x},n_{y},n_{z}\}}$ satisfying ${\
displaystyle n_{x}+n_{y}+n_{z}=n}$ The degeneracy of the ${\displaystyle n}$-th state can be found by considering the distribution of ${\displaystyle n}$ quanta across ${\displaystyle n_{x}}$, ${\
displaystyle n_{y}}$ and ${\displaystyle n_{z}}$. Having 0 in ${\displaystyle n_{x}}$ gives ${\displaystyle n+1}$ possibilities for distribution across ${\displaystyle n_{y}}$ and ${\displaystyle n_
{z}}$. Having 1 quanta in ${\displaystyle n_{x}}$ gives ${\displaystyle n}$ possibilities across ${\displaystyle n_{y}}$ and ${\displaystyle n_{z}}$ and so on. This leads to the general result of ${\
displaystyle n-n_{x}+1}$ and summing over all ${\displaystyle n}$ leads to the degeneracy of the ${\displaystyle n}$-th state, ${\displaystyle \sum _{n_{x}=0}^{n}(n-n_{x}+1)={\frac {(n+1)(n+2)}{2}}}$
For the ground state ${\displaystyle n=0}$, the degeneracy is ${\displaystyle 1}$ so the state is non-degenerate. For all higher states, the degeneracy is greater than 1 so the state is degenerate.
The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is
essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.
Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that
can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H[0] for the unperturbed
system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not
every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to
choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.
Lifting of degeneracy by first-order degenerate perturbation theory.
Consider an unperturbed Hamiltonian ${\displaystyle {\hat {H_{0}}}}$ and perturbation ${\displaystyle {\hat {V}}}$, so that the perturbed Hamiltonian
${\displaystyle {\hat {H}}={\hat {H_{0}}}+{\hat {V}}}$ The perturbed eigenstate, for no degeneracy, is given by- ${\displaystyle |\psi _{0}\rangle =|n_{0}\rangle +\sum _{keq 0}V_{k0}/(E_{0}-E_{k})|n_
{k}\rangle }$ The perturbed energy eigenket as well as higher order energy shifts diverge when ${\displaystyle E_{0}=E_{k}}$, i.e., in the presence of degeneracy in energy levels. Assuming ${\
displaystyle {\hat {H_{0}}}}$ possesses N degenerate eigenstates ${\displaystyle |m\rangle }$ with the same energy eigenvalue E, and also in general some non-degenerate eigenstates. A perturbed
eigenstate ${\displaystyle |\psi _{j}\rangle }$ can be written as a linear expansion in the unperturbed degenerate eigenstates as- ${\displaystyle |\psi _{j}\rangle =\sum _{i}|m_{i}\rangle \langle m_
{i}|\psi _{j}\rangle =\sum _{i}c_{ji}|m_{i}\rangle }$ ${\displaystyle [{\hat {H_{0}}}+{\hat {V}}]\psi _{j}\rangle =[{\hat {H_{0}}}+{\hat {V}}]\sum _{i}c_{ji}|m_{i}\rangle =E_{j}\sum _{i}c_{ji}|m_{i}\
rangle }$ where ${\displaystyle E_{j}}$ refer to the perturbed energy eigenvalues. Since ${\displaystyle E}$ is a degenerate eigenvalue of ${\displaystyle {\hat {H_{0}}}}$, ${\displaystyle \sum _{i}
c_{ji}{\hat {V}}|m_{i}\rangle =(E_{j}-E)\sum _{i}c_{ji}|m_{i}\rangle =\Delta E_{j}\sum _{i}c_{ji}|m_{i}\rangle }$ Premultiplying by another unperturbed degenerate eigenket ${\displaystyle \langle m_
{k}|}$ gives- ${\displaystyle \sum _{i}c_{ji}[\langle m_{k}|{\hat {V}}|m_{i}\rangle -\delta _{ik}(E_{j}-E)]=0}$ This is an eigenvalue problem, and writing ${\displaystyle V_{ik}=\langle m_{i}|{\hat
{V}}|m_{k}\rangle }$, we have- ${\displaystyle {\begin{vmatrix}V_{11}-\Delta E_{j}&V_{12}&\dots &V_{1N}\\V_{21}&V_{22}-\Delta E_{j}&\dots &V_{2N}\\\vdots &\vdots &\ddots &\vdots \\V_{N1}&V_{N2}&\dots
&V_{NN}-\Delta E_{j}\end{vmatrix}}.}$
The N eigenvalues obtained by solving this equation give the shifts in the degenerate energy level due to the applied perturbation, while the eigenvectors give the perturbed states in the unperturbed
degenerate basis ${\displaystyle |m\rangle }$. To choose the good eigenstates from the beginning, it is useful to find an operator ${\displaystyle {\hat {V}}}$ which commutes with the original
Hamiltonian ${\displaystyle {\hat {H_{0}}}}$ and has simultaneous eigenstates with it.
Some important examples of physical situations where degenerate energy levels of a quantum system are split by the application of an external perturbation are given below.
A two-level system essentially refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for
such a system are performed on a two-dimensional subspace of the state space.
If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.
If ${\displaystyle E_{1}}$ and ${\displaystyle E_{2}}$ are the energy levels of the system, such that ${\displaystyle E_{1}=E_{2}=E}$, and the perturbation ${\displaystyle W}$ is represented in the
two-dimensional subspace as the following 2×2 matrix ${\displaystyle \mathbf {W} ={\begin{bmatrix}0&W_{12}\\[1ex]W_{12}^{*}&0\end{bmatrix}}.}$ then the perturbed energies are {\displaystyle {\begin
Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent
property of the system include:
• Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
• Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
• H^+
[2] molecule, in which the electron may be localized around either of the two nuclei.
The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin–orbit coupling result in breaking the degeneracy in energy levels
for different values of l corresponding to a single principal quantum number n.
The perturbation Hamiltonian due to relativistic correction is given by ${\displaystyle H_{r}=-p^{4}/8m^{3}c^{2}}$ where ${\displaystyle p}$ is the momentum operator and ${\displaystyle m}$ is the
mass of the electron. The first-order relativistic energy correction in the ${\displaystyle |nlm\rangle }$ basis is given by ${\displaystyle E_{r}=\left(-1/8m^{3}c^{2}\right)\left\langle n\ell m\
right|p^{4}\left|n\ell m\right\rangle }$
Now ${\displaystyle p^{4}=4m^{2}(H^{0}+e^{2}/r)^{2}}$ {\displaystyle {\begin{aligned}E_{r}&=-{\frac {1}{2mc^{2}}}\left[E_{n}^{2}+2E_{n}e^{2}\left\langle {\frac {1}{r}}\right\rangle +e^{4}\left\langle
{\frac {1}{r^{2}}}\right\rangle \right]\\&=-{\frac {1}{2}}mc^{2}\alpha ^{4}\left[-3/(4n^{4})+1/{n^{3}(\ell +1/2)}\right]\end{aligned}}} where ${\displaystyle \alpha }$ is the fine structure constant.
The spin–orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The
interaction Hamiltonian is ${\displaystyle H_{so}=-{\frac {e}{mc}}{\frac {\mathbf {m} \cdot \mathbf {L} }{r^{3}}}={\frac {e^{2}}{m^{2}c^{2}r^{3}}}\mathbf {S} \cdot \mathbf {L} }$ which may be written
as ${\displaystyle H_{so}={\frac {e^{2}}{4m^{2}c^{2}r^{3}}}\left[J^{2}-L^{2}-S^{2}\right]}$
The first order energy correction in the ${\displaystyle |j,m,\ell ,1/2\rangle }$ basis where the perturbation Hamiltonian is diagonal, is given by ${\displaystyle E_{so}={\frac {\hbar ^{2}e^{2}}{4m^
{2}c^{2}}}{\frac {j(j+1)-\ell (\ell +1)-{\frac {3}{4}}}{a_{0}^{3}n^{3}\ell (\ell +{\frac {1}{2}})(\ell +1)}}}$ where ${\displaystyle a_{0}}$ is the Bohr radius. The total fine-structure energy shift
is given by ${\displaystyle E_{fs}=-{\frac {mc^{2}\alpha ^{4}}{2n^{3}}}\left[1/(j+1/2)-3/4n\right]}$ for ${\textstyle j=\ell \pm {\tfrac {1}{2}}}$.
The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment ${\displaystyle {\vec {m}}}$ of the atom with the applied
field is known as the Zeeman effect.
Taking into consideration the orbital and spin angular momenta, ${\displaystyle \mathbf {L} }$ and ${\displaystyle \mathbf {S} }$, respectively, of a single electron in the Hydrogen atom, the
perturbation Hamiltonian is given by ${\displaystyle {\hat {V}}=-(\mathbf {m} _{\ell }+\mathbf {m} _{s})\cdot \mathbf {B} }$ where ${\displaystyle \mathbf {m} _{\ell }=-e\mathbf {L} /2m}$ and ${\
displaystyle \mathbf {m} _{s}=-e\mathbf {S} /m}$. Thus, ${\displaystyle {\hat {V}}={\frac {e}{2m}}(\mathbf {L} +2\mathbf {S} )\cdot \mathbf {B} }$ Now, in case of the weak-field Zeeman effect, when
the applied field is weak compared to the internal field, the spin–orbit coupling dominates and ${\textstyle \mathbf {L} }$ and ${\textstyle \mathbf {S} }$ are not separately conserved. The good
quantum numbers are n, ℓ, j and m[j], and in this basis, the first order energy correction can be shown to be given by ${\displaystyle E_{z}=-\mu _{B}g_{j}Bm_{j},}$ where ${\displaystyle \mu _{B}={e\
hbar }/2m}$ is called the Bohr Magneton. Thus, depending on the value of ${\displaystyle m_{j}}$, each degenerate energy level splits into several levels.
Lifting of degeneracy by an external magnetic field
In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n, l, m[l], and m[s]. Here,
L[z] and S[z] are conserved, so the perturbation Hamiltonian is given by- ${\displaystyle {\hat {V}}=eB(L_{z}+2S_{z})/2m}$ assuming the magnetic field to be along the z-direction. So, ${\displaystyle
{\hat {V}}=eB(m_{\ell }+2m_{s})/2m}$ For each value of m[ℓ], there are two possible values of m[s], ${\displaystyle \pm 1/2}$.
The splitting of the energy levels of an atom or molecule when subjected to an external electric field is known as the Stark effect.
For the hydrogen atom, the perturbation Hamiltonian is ${\displaystyle {\hat {H}}_{s}=-|e|Ez}$ if the electric field is chosen along the z-direction.
The energy corrections due to the applied field are given by the expectation value of ${\displaystyle {\hat {H}}_{s}}$ in the ${\displaystyle |n\ell m\rangle }$ basis. It can be shown by the
selection rules that ${\displaystyle \langle n\ell m_{\ell }|z|n_{1}\ell _{1}m_{\ell 1}\rangle eq 0}$ when ${\displaystyle \ell =\ell _{1}\pm 1}$ and ${\displaystyle m_{\ell }=m_{\ell 1}}$.
The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states ${\displaystyle |2,0,0\
rangle }$ and ${\displaystyle |2,1,0\rangle }$, both corresponding to n = 2, is given by ${\displaystyle \Delta E_{2,1,m_{\ell }}=\pm |e|\hbar ^{2}/(m_{e}e^{2})E}$.
• Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck. Quantum Mechanics. Vol. 1. Hermann. ISBN 978-2-7056-8392-4.
• Shankar, Ramamurti (2013). Principles of Quantum Mechanics. Springer. ISBN 978-1-4615-7675-4.
• Larson, Ron; Falvo, David C. (30 March 2009). Elementary Linear Algebra, Enhanced Edition. Cengage Learning. pp. 8–. ISBN 978-1-305-17240-1.
• Hobson; Riley (27 August 2004). Mathematical Methods For Physics And Engineering (Clpe) 2Ed. Cambridge University Press. ISBN 978-0-521-61296-8.
• Hemmer (2005). Kvantemekanikk: P.C. Hemmer. Tapir akademisk forlag. Tillegg 3: supplement to sections 3.1, 3.3, and 3.5. ISBN 978-82-519-2028-5.
• Quantum degeneracy in two dimensional systems, Debnarayan Jana, Dept. of Physics, University College of Science and Technology
• Al-Hashimi, Munir (2008). Accidental Symmetry in Quantum Physics.
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Alternative Dice
Here is an fun proof I saw during a stay at MSRI (Mathematical Sciences Research Institute) – unfortunately, I can not remember the reference (or none was given). If you happen to know the originator
of this work, please let me know and I will attribute it appropriately. In any event, it is a “fun” problem, so I typed up the following summary:
Prop: There exists precisely one pair of numerically non-symmetric six-sided dice with no blank sides such that the sum-roll probability distribution is equivalent to normal symmetric six-sided dice.
(Where by sum-roll it is meant the probability of two dice rolling a sum of x, etc.)
Proof: Let a die be represented by a Polynomial in the following fashion. Powers represent the number on a side of a die, and the coefficient of a specific power represents the number of sides with
that number of dots. For example, a normal symmetric die has the following representation:
$P(x) = x + x^2 + x^3 + x^4 + x^5 + x^6$
And a four sided die with three sevens and one four would be:
$P(x) = x^4 + 3*x^7$
Notice that with this representation, $P(1)$ is the number of sides on the die. Also notice that this representation encapsulates the probability of a specific roll, i.e., the probability of a die $P
(x)$ rolling a “n” is:
$\dfrac{\text{coefficient of }x^n}{P(1)}$
Finally, the action of rolling two dice is equivalent to forming the product of the polynomial representations. For example, when two normal symmetric six-sided dice are rolled, the probability of
rolling a one (in sum) is zero, the probability of rolling a two is 1/36, the probability of rolling a seven is 1/6, etc. Notice that if we form the product
$(x + x^2 + x^3 + x^4 + x^5 + x^6)^2$
Then the coefficient of $x^1$ is zero, the coefficient of $x^2$ is one, and the coefficient of $x^7$ is six (0/36, 1/36, and 6/36).
Now, to answer the original question, we find two polynomials, $T(x)$ and $S(x)$ such that:
1) $T(x)*S(x) = (x + x^2 + x^3 + x^4 + x^5 + x^6)^2$
2) $T(1) = S(1) = 6$
3) $T(x) eq S(x)$
4) $T(0) = S(0) = 0$
Condition #1 gives us the same sum-roll distribution as normal symmetric dice.
Condition #2 gives us two six-sided dice.
Condition #3 assures us of the non-symmetry of $T(x)$ and $S(x)$.
Condition #4 is equivalent to saying that the polynomials have a zero coefficient for the $x^0$ term, which is the same as saying that the dice have no blank sides.
So, to proceed, we simply need to factor the polynomial in condition #1. Here it is:
$x^2 * (x^2 + 1)^2 * (x^2 - x + 1)^2 * (x^2 + x + 1)^2$
Which I’ll write as:
$a(x)^2 * b(x)^2 * c(x)^2 * d(x)^2$
$a(x) = x$
$b(x) = x^2 + 1$
$c(x) = x^2 - x + 1$
$d(x) = x^2 + x + 1$
Notice that $a(1)=1$ so multiplying by $a(x)$ will not change the number of sides on the die. Also notice that if we put both $a(x)$ terms in one of $T(x)$ or $S(x)$ then the other die would have a
blank side (since the product of any of the remaining terms $b(x)$, $c(x)$ or $d(x)$ would incur a monomial). Thus, both $T(x)$ and $S(x)$ must have $a(x)$ as a factor.
Now, notice that $b(1)=2$ and $d(1)=3$, whose product is six. Also notice that $c(1)=1$. Thus, if we want $T(x)$ and $S(x)$ to represent six-sided dice then they must both have $b(x)$ and $d(x)$ as
factors, since multiplying by $c(x)$ will not change the number of sides. So far we have
the following:
$T(x) = a(x) * b(x) * d(x) * ??$
$S(x) = a(x) * b(x) * d(x) * ??$
The only factor we have left is the $c(x)^2$. If we put a $c(x)$ in both of $T(x)$ and $S(x)$ then we will have created symmetric dice since $T(x)$ will be equal to $S(x)$ (in violation of condition
#3). The only other option is to put the entire $c(x)^2$ term into either $T(x)$ or $S(x)$ By symmetry, it is irrelevant which polynomial gets the factor, so we’ll try:
$T(x) = a(x) * b(x) * c(x)^2 * d(x)$
$S(x) = a(x) * b(x) * d(x)$
Now let’s check our work! Condition #1 is satisfied since $T(x)*S(x) = a(x)^2 * b(x)^2 * c(x)^2 * d(x)^2$. How about condition #2? Well, $T(1) = 6$ and $S(1) = 6$ so that looks good. Clearly, $T(x) !
= S(x)$ so condition #3 is ok. Finally, neither term has a monic, so $T(0)=S(0)=0$ and neither die has a blank side.
So, now you have all the information you need to find these dice. I didn’t want to spoil it by writing out the actual sides, so I’ll leave you the final step.
Alternative Dice — 1 Comment
1. I saw this pair of dice around 2006. I don’t remember a proof, but stumbled across yours as I was inspired to try to find the dice today.
As stated, a(x)^2 * b(x)^2 * c(x)^2 * d(x)^2 would yield a 14th degree polynomial, which would imply that one could roll a 14 on a normal pair of dice.
It seems that your b(x) is incorrect. If b(x)=x+1, things work out for a normal pair of dice.
The error was easy to miss since when x=1, x+1 = x^2 +1 = 2.
The proof using polynomials is clever. Definitely a different way to look at things. Yay math!
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Multiplying Polynomials Box Method Worksheet Answer Key
Multiplying Polynomials Box Method Worksheet Answer Key
Answer key multiplying binomials sheet 1 find the product using box method. Blackline master and color coded answer key included.
Algebra 2 Interactive Notebook Pages Galore Math Interactive Notebook Teaching Algebra Algebra 2
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Multiplying polynomials box method worksheet answer key. Free polynomials multiplication calculator multiply polynomials step by step this website uses cookies to ensure you get the best experience.
Box method multiplication displaying top 8 worksheets found for this concept. This activity goes hand in hand with the multiplying polynomials powerpoint that guides students through 4 examples of
multiplying polynomials using the box method.
2x 8 and 6x 2 solution. Box method and foil. R worksheet by kuta software llc kuta software infinite pre algebra name multiplying binomials date period.
Multiply the binomials worksheet 1. Use the box method to multiply the polynomials. This guide includes a free video lesson and multiplying binomials worksheet.
Multiply the following polynomials using box method. G h g g h g. Multiply the binomials worksheet 3.
In this method we need to draw a box which contains some rows and columns. Multiply the binomials worksheet 2. Here we use.
Multiply polynomials opener combine the like terms to simplify the expression. Box method examples 1 2. The lesson begins by reviewing vocabulary and then goes into an example of a monomial by a
My interactive note pages include all or some of the following. Here the number of rows is up to the number of terms of the first polynomial and the number of columns is up to the number of terms of
the second polynomial. R q2j0 u192l gk xu ltga9 1saoyf atjw va urueg blcl7ce 2 k wajl ul8 cr sihgghdt zsb 4rde cs6earavpezdb j r vmhatd 4e u dwfi itqh 8 bi 5n4f wixnzi lt gem bprrze j 1a 0ldg
3×2 11 3x 9 5×2 x 1 multiplying polynomials unlike adding and subtracting polynomials you are able to multiply unlike terms to create new terms. These worksheets are used in grade 10 math. This step
by step guide to multiplying binomials will show you how to use the box method area model and foil method foil math strategies for multiplying binomials and multiplying polynomials.
Some of the worksheets for this concept are name date digit multiplication box method work out the name 10 date digit multiplication box method work out multiplying binomials 1 multiplying
polynomials 1 multiplication boxes multiplying polynomials multiplying binomials date period multiplying polynomials. This is a guided color coded notebook page for the interactive math notebook
introducing polynomials. Use the foil method to complete the binomial worksheets.
2x x 2 2. Includes notes on multiplying polynomials. Multiply binomials worksheets with answers on the second page.
Factoring Polynomials Free Worksheet Factoring Polynomials Maths Algebra School Algebra
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Multiply Binomials Using Box Method Box Method Multiplication Printable Math Worksheets Algebra Worksheets
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IQ test for 9-12 year olds series--179wg
• 1/10 What is the value of the 4 in this: 174976?
• 2/10 How do you spell the word pronounced bee-you- tee- full?
• 3/10 Beth is 11 years old. Sandra is Alice's age older than Beth. Alice is half the age Beth will be next year. How old is Sandra?
• 4/10 Yasmine wants to sit next to Ali bit not next to Annabelle. Ali wants to sit next to Yasmine and Annabelle but not next to Emma.
Emma doesn't care who she sits next to. What is the best couch formation?
• 5/10 Lola has 45 cakes. She wants to sell 9 for every 6 her friends eat. What fraction of the cakes is she selling?
• 6/10 Which of these does not correctly describe the word article?
• 7/10 If Emmet is twice Isobelle's age, Isobelle is treble Jeni's age, Jeni is a fifth of Johnny's age and Johnny is 10, how old is Emmet?
• 8/10 Kira weighs 8 stone. Layla weighs 7 stone. How many pounds different are their weights?
• 9/10 Which of these is a hyphen?
• 10/10 What is 2b+b?
Click the button below to get your answer
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Raspberry Pi
Product ID: 2779
What's better than our 2.8" 320x240 and 3.5" 480x320 PiTFT Pluses? How about a custom-designed enclosure for said PiTFTs, in a handsome midnight-blue? Crafted out of eleven unique layers, each
laser-cut from colourful high-quality cast acrylic and once stacked they securely contain a Raspberry Pi 2, Model B+, Pi 3, or Pi 3 B+ while leaving the primary ports...
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counting sort vs quick sort
Explanation for the article: http://www.geeksforgeeks.org/counting-sort/This video is contributed by Arjun Tyagi. Counting sort is a sorting algorithm that sorts the elements of an array by counting
the number of occurrences of each unique element in the array and sorting them according to the keys that are small integers. So, the restriction of the counting sort is that the input should only
contain integers and they should lie in the range of 0 to k, for some integer k. Quick Sort is also a cache friendly sorting algorithm as it has good locality of reference when used for arrays. As
opposed to bubble sort and quicksort, counting sort is not comparison based, since it enumerates occurrences of contained values. Some algorithms (selection, bubble, heapsort) work by moving elements
to their final position, one at a time. Heap Sort vs Merge Sort vs Insertion Sort vs Radix Sort vs Counting Sort vs Quick Sort I had written about sorting algorithms (Tag: Sorting ) with details
about what to look out for along with their code snippets but I wanted a do a quick comparison of all the algos together to see how do they perform when the same set of input is provided to them. 1.
Those algorithms, that does not require any extra space is called in-place sorting algorithm. Let’s look at an illustrative example: Each invocation of the Counting Sort subroutine preserves the
order from the previous invocations. Tim-sort is a sorting algorithm derived from insertion sort and merge sort. These techniques are considered as comparison based sort because in these techniques
the values are compared, and placed into sorted position in ifferent phases. This sorting technique is effective when the difference between different keys are not so big, otherwise, it can increase
the space complexity. Cycle sort is a comparison sorting algorithm which forces array to be factored into the number of cycles where each of them can be rotated to produce a sorted array. Quick sort
is an in-place sorting algorithm. 0. bix 55. Set the first index of the array to left and loc variable. Auxiliary Space : Mergesort uses extra space, quicksort requires little space and exhibits good
cache locality. Counting sort runs in time, making it asymptotically faster than comparison-based sorting algorithms like quicksort or merge sort. Tim-sort. Quick Sort. For example, if you choose
8-bits wide digits when sorting 32-bit integers, presuming counting sort is used for each radix, it means 256 counting slots or 4 passes through the array to count and 4 passes to sort. Counting sort
assumes that each of the elements is an integer in the range 1 to k, for some integer k.When k = O(n), the Counting-sort runs in O(n) time. Examples: Input : arr = {4, 3, 5, 1, 2} Output : 11
Explanation We have to make 11 comparisons when we apply quick sort to the array. I had written about sorting algorithms (Tag: Sorting) with details about what to look out for along with their code
snippets but I wanted a do a quick comparison of all the algos together to see how do they perform when the same set of input is provided to them. Selection Sort Complexity is O(n^2). These
techniques are considered as comparison based sort because in these techniques the values are compared, and placed into sorted position in ifferent phases. Here are some key points of counting sort
algorithm – Counting Sort is a linear sorting algorithm. The lower bound for Comparison based sorting algorithm (Merge Sort, Heap Sort, Quick-Sort .. etc) is Ω(nLogn), i.e., they cannot do better
than nLogn.. Pictorial presentation - Quick Sort algorithm: Sample Solution: Sample C Code: ; It uses a key element (pivot) for partitioning the elements. It is also known as “partition exchange
sort”. The quick sort is internal sorting method where the data that is to be sorted is adjusted at a time in main memory. Merge sorts are also practical for physical objects, particularly as two
hands can be used, one for each list to merge, while other algorithms, such as heap sort or quick sort, are poorly suited for human use. Merge Sort with inversion counting, just like regular Merge
Sort, is O(n log(n)) time. I ensured that they all have the same set of procedures during their run. Radix sort's efficiency = O(c.n) where c = highest number of digits among the input key set. Space
complexity : O(max) Therefore, larger the range of elements, larger is the space complexity. Twitter Facebook Google+ LinkedIn UPDATE : Check this more general comparison ( Bubble Sort Vs Selection
sort Vs Insertion Sort Vs Merge Sort Vs Merge Sort Vs Quick Sort ) Before the stats, You must already know what is Merge sort, Selection Sort, Insertion Sort, Arrays, how to get current time. It
means keys are not compared with each other. But merge sort is out-place sorting technique. Some sorting techniques are comparison based sort, some are non-comparison based sorting technique. On the
other hand, the quick sort doesn’t require much space for extra storage. You sort an array of size N, put 1 item in place, and continue sorting an array of size N – 1 (heapsort is slightly
different). 45 VIEWS. On the other hand, the quick sort doesn’t require much space for extra storage. Radix sort is different from Merge and Quick sort in that it is not a comparison sort. Is
counting sort as defined above a stable sort? ; Counting Sort is stable sort as relative order of elements with equal values is maintained. Some algorithms (selection, bubble, heapsort) work by
moving elements to their final position, one at a time. Finally, sort values based on keys and make… They are provided for all algorithms on the right-most column. These sorting algorithms are
usually implemented recursively, use Divide and Conquer problem solving paradigm, and run in O(N log N) time for Merge Sort and O(N log N) time in expectation for Randomized Quick Sort. The code is
written in such a way that it can be easily translated into other languages (e.g., each implementation should be quite efficient in C++). Detailed tutorial on Quick Sort to improve your understanding
of {{ track }}. Task. If not, how could the given code be changed so that it is stable? In this tutorial, you will understand the working of counting sort with working code in C, C++, Java, and
Python. Here we will see time complexity of these techniques. Note: Quick sort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total
order) is defined. Counting sort is a linear time sorting algorithm that sort in O(n+k) time when elements are in the range from 1 to k.. What if the elements are in the range from 1 to n 2? This is
a bit misleading: 1) "at least order of number of bits" should actually be "at most". Radix Sort Overview. Store the count of each element at their respective index in count array For example: If the
count of element “4” occurs 2 times then 2 is stored These are non-comparison based sort because here two elements are not compared while sorting. In this tutorial I am sharing counting sort program
in C. Steps that I am doing to sort the elements are given below. Weaknesses: Restricted inputs. This is a bit misleading: 1) "at least order of number of bits" should actually be "at most". Bucket
sort may be used for many of the same tasks as counting sort, with a similar time analysis; however, compared to counting sort, bucket sort requires linked lists, dynamic arrays or a large amount of
preallocated memory to hold the sets of items within each bucket, whereas counting sort instead stores a single number (the count of items) per bucket. This algorithm follows divide and conquer
approach. Comparison of Searching methods in Data Structures, Principles of Recursion in Data Structures, Bernoulli Distribution in Data Structures, Geometric Distribution in Data Structures.
Counting sort is a sorting technique based on keys between a specific range.. It is an adaptive sorting algorithm which needs O(n log n) comparisons to sort an array of n elements. As you can see,
now Bucket Sort works faster than Quick Sort. It works by counting the number of objects having distinct key values (kind of hashing). Quick sort = 16 * 4 = 64 time units. Counting sort utilizes the
knowledge of the smallest and the largest element in the array (structure). Selection Sort, Bubble Sort, Insertion Sort, Merge Sort, Heap Sort, QuickSort, Radix Sort, Counting Sort, Bucket Sort,
ShellSort, Comb Sort, Pigeonhole Sort. Merge sort is more efficient than quick sort. Counting sort runs in time, making it asymptotically faster than comparison-based sorting algorithms like
quicksort or merge sort. This corresponds to theory, but let’s check how Bucket Sort behaves with larger collections. Each iteration having the same input, Each algo being timed the exact same way as
another. In-place sorting means no additional storage space is needed to perform sorting. This sorting technique is based on the frequency/count of each element to be sorted and works using the
following algorithm-Input: Unsorted array A[] of n elements; Output: Sorted arrayB[] Merge sort is more efficient than quick sort. n = number of keys in input key set. It counts the number of keys
whose key values are same. This sorting technique is efficient when difference between different keys are not so big, otherwise it can increase the space complexity. Some of them are Radix sort,
Bucket sort, count sort. It counts the number of items for distinct key value, use these keys to determine position or indexing on the array and store respective counts for each key. Comparison Based
Soring techniques are bubble sort, selection sort, insertion sort, Merge sort, quicksort, heap sort etc. Description. Some of the algorithms being tested were: Created a simple base class for all
algorithms: AlgoStopwatch, Provide a function called doSort() that would allow derived classes to implement their algorithm, Ensures that every algorithm has a name and description - to help us
distinguish, Another class to help manage the testing of all the algorithms: AlgoDemo, All instances are created here for the algorithms, The input array is provided by this class to all algorithms.
Looking at the numbers below, it may be hard to compare the actual values. Other algorithms, such as library sort, a variant of insertion sort … Assume 16 numbers to be sorted with 6 digits each:
Radix sort = 16 * 6 = 96 time units. Quick sort's best case = O(n. log n) where n = number of keys in input key set. Quick sort is an internal algorithm which is based on divide and conquer strategy.
The basic idea of Counting sort is to determine, for each input elements x, the number of elements less than x.This information can be used to place directly into its correct position. Tim-sort. ; It
is not an in-place sorting algorithm as it requires extra additional space O(k). There are 200+ sorting techniques. Update: For merge sort, you need to do some "merging," which needs extra array(s)
to store the data before merging; but in quick sort… We don’t have to understand how it works, but that Counting Sort is stable. Counting sort is an efficient algorithm for sorting an array of
elements that each have a nonnegative integer key, for example, an array, sometimes called a list, of positive integers could have keys that are just the value of the integer as the key, or a list of
words could have keys assigned to them by some scheme mapping the alphabet to integers (to sort in alphabetical order, for instance). Quick sort and counting sort Read n values into array and Sort
using Quick Sort. I have now put together all of them in a single project on GitHub. The quick sort is internal sorting method where the data that … Counting sort algorithm is a sorting algorithm
which do not involve comparison between elements of an array. Counting Sort Algorithm is an efficient sorting algorithm that can be used for sorting elements within a specific range. Quick sort is
the widely used sorting algorithm that makes n log n comparisons in average case for sorting of an array of n elements. Merge sort requires additional memory space to store the auxiliary arrays. n =
number of keys in input key set. Implement the Counting sort.This is a way of sorting integers when the minimum and maximum value are known. Merge sort requires additional memory space to store the
auxiliary arrays. 1) Bubble sort 2) Bucket sort 3) Cocktail sort 4) Comb sort 5) Counting sort 6) Heap sort 7) Insertion sort 8) Merge sort 9) Quicksort 10) Radix sort 11) Selection sort 12) Shell
sort. In QuickSort, ideal situation is when median is always chosen as pivot as this results in minimum time.In this article, Merge Sort Tree is used to find median for different ranges in QuickSort
and number of comparisons are analyzed. You sort an array of size N, put 1 item in place, and continue sorting an array of size N – 1 (heapsort is slightly different). Counting sort is able to look
at each element in the list exactly once, and with no comparisons generate a sorted list. From the above mentioned techniques, the insertion sort is online sorting technique. Attention reader! Learn:
Counting Sort in Data Structure using C++ Example: Counting sort is used for small integers it is an algorithm with a complexity of O(n+k) as worst case. Don’t stop learning now. Een voorwaarde
daarvoor is dat de kleinste en de grootste voorkomende waarde bekend zijn, en dat de te sorteren … Sorts are most commonly in numerical or a form of alphabetical (called lexicographical) order, and
can be in ascending (A-Z, 0-9) or descending (Z-A, 9-0) order. Hence I started working on a simple implementation for each one of them. It counts the number of keys whose key values are same. Merge
sort requires a temporary array to merge the sorted arrays and hence it is not in-place giving Quick sort the advantage of space. 3 - Quick sort has smaller constant factors in it's running time than
other efficient sorting algorithms. Implement the Counting sort.This is a way of sorting integers when the minimum and maximum value are known. Counting sort is a stable sorting technique, which is
used to sort objects according the keys that are small numbers. Sorting algorithms are a set of instructions that take an array or list as an input and arrange the items into a particular order.
Counting sort algorithm is based on keys in a specific range. thnx Hence I decided to normalize them by calculating how much time will be required to sort 100 numbers using the same rate as the
actual numbers. ### [Insertion Sort](http://codersdigest.wordpress.com/2012/09/18/insertion-sort/), ### [Heap Sort 1](http://codersdigest.wordpress.com/2012/10/17/heap-sort/), ### [Heap Sort 2](http:
//codersdigest.wordpress.com/2012/10/17/heap-sort/), ### [Heap Sort 3](http://codersdigest.wordpress.com/2012/10/17/heap-sort/), ### [QuickSort](http://codersdigest.wordpress.com/2012/09/22/
quick-sort/), ### [Counting Sort](http://codersdigest.wordpress.com/2012/09/11/counting-sort/), ### [Radix Sort](http://codersdigest.wordpress.com/2012/09/13/radix-sort/). February 25, 2018 6:12 AM.
The algorithm processes the array in the following way. Some sorting algorithms are in-place sorting algorithms, and some are out-place sorting algorithms. Merge Sort with inversion counting, just
like regular Merge Sort, is O(n log(n)) time. Quick Sort and its Randomized version (which only has one change). Sorting techniques can also be compared using some other parameters. Then doing some
arithmetic to calculate the position of each object in the output sequence. Then doing some arithmetic to calculate the position of each object in the output sequence. Such as quicksort, heapsort
algorithms are in-place. Time complexity of Counting Sort is O(n+k), where n is the size of the sorted array and k is the range of key values. Here we will see some sorting methods. Selection Sort
Complexity is O(n^2). 2 - Quick sort is easier to implement than other efficient sorting algorithms. With our inversion counting algorithm dialed in, we can go back to our recommendation engine
hypothetical. Radix Sort Overview. Java quick sort and counting sort. Now we will see the difference between them based on different type of analysis. We will see few of them. Counting sort is a
stable sorting technique, which is used to sort objects according to the keys that are small numbers. It can be easily avoided with … Counting sort (ultra sort, math sort) is an efficient sorting
algorithm with asymptotic complexity, which was devised by Harold Seward in 1954. Weaknesses: Restricted inputs. Merge sorts are also practical for physical objects, particularly as two hands can be
used, one for each list to merge, while other algorithms, such as heap sort or quick sort, are poorly suited for human use. This is because non-comparison sorts are generally implemented with few
restrictions like counting sort has a restriction on its input which we are going to study further. Here we will see time complexity of these techniques. void […] Sadly this algorithm can only be run
on discrete data types. Radix sort is different from Merge and Quick sort in that it is not a comparison sort. Summary: Radix sort's efficiency = O(d.n) where d = highest number of digits among the
input key set. For example, if you choose 8-bits wide digits when sorting 32-bit integers, presuming counting sort is used for each radix, it means 256 counting slots or 4 passes through the array to
count and 4 passes to sort. Explanation for the article: http://www.geeksforgeeks.org/counting-sort/This video is contributed by Arjun Tyagi. Instead, Radix sort takes advantage of the bases of each
number to group them by their size. The techniques are slightly different. Counting sort only works when the range of potential items in the input is known ahead of time. some sorting algorithms are
non-comparison based algorithm. With our inversion counting algorithm dialed in, we can go back to our recommendation engine hypothetical. Task. It was designed to perform in an optimal way on
different kind of real world data. Some of the items I wanted to ensure was: Same number of iterations. Counting sort also called an integer sorting algorithm. It is theoretically optimal in the
sense that it reduces the number of writes to the original array. Counting Sort . These sorting algorithms are usually implemented recursively, use Divide and Conquer problem solving paradigm, and
run in O(N log N) time for Merge Sort and O(N log N) time in expectation for Randomized Quick Sort. Counting sort only works when the range of potential items in the input is known ahead of time.
Comparison Based Soring techniques are bubble sort, selection sort, insertion sort, Merge sort, quicksort, heap sort etc. It works by counting the number of objects having distinct key values (kind
of hashing). Heap Sort vs Merge Sort vs Insertion Sort vs Radix Sort vs Counting Sort vs Quick Sort I had written about sorting algorithms (Tag: Sorting ) with details about what to look out for
along with their code snippets but I wanted a do a quick comparison of all the algos together to see how do they perform when the same set of input is provided to them. Hi, Can anyone tell me if
'counting sort' can be made to sort in desending order? Comparison Based Soring techniques are bubble sort, selection sort, insertion sort, Merge sort, quicksort, heap sort etc. Counting sort (ultra
sort, math sort) is an efficient sorting algorithm with asymptotic complexity, which was devised by Harold Seward in 1954.As opposed to bubble sort and quicksort, counting sort is not comparison
based, since it enumerates occurrences of contained values.. Counting sort is a sorting technique based on keys between a specific range.. If the algorithm accepts new element while the sorting
process is going on, that is called the online sorting algorithm. First of all I am reading n elements in array a[]. Quick Sort and its Randomized version (which only has one change). These
techniques are considered as comparison based sort because in these techniques the values are compared, and placed into sorted position in ifferent phases. Tim-sort is a sorting algorithm derived
from insertion sort and merge sort. Worst Cases : The worst case of quicksort O(n 2) can be avoided by using randomized quicksort. Here we will see time complexity of these techniques. Similarity:
These are not comparison sort. Some algorithms are online and some are offline. As usual the code for the project is available here: It can be run using Visual Studio without any changes. It is an
adaptive sorting algorithm which needs O(n log n) comparisons to sort an array of n elements. Please write comments if you find anything incorrect, or you want to share more information about the
topic discussed above. Also try practice problems to test & improve your skill level. Counting sort is an efficient algorithm for sorting an array of elements that each have a nonnegative integer
key, for example, an array, sometimes called a list, of positive integers could have keys that are just the value of the integer as the key, or a list of words could have keys assigned to them by
some scheme mapping the alphabet to integers (to sort in alphabetical order, for instance). Instead, Radix sort takes advantage of the bases of each number to group them by their size. What about the
other sorting algorithms that were discussed previously (selection sort, insertion sort, merge sort, and quick sort) -- were the versions of those algorithms defined in … void […] Quick Sort
Algorithm Merge Sort Algorithm turgay Posted in C# .NET , Sorting Algorithms C# , counting sort algorithm , counting sort implementation , implementation , sorting algorithm 1 Comment Counting Sort
is a stable integer sorting algorithm. In this: The array of elements is divided into parts repeatedly until it is not possible to divide it further. Other algorithms, such as library sort, a variant
of insertion sort … Counting Sort Algorithm. The worst case is possible in randomized version also, but worst case doesn’t occur for a particular pattern (like sorted array) and randomized Quick Sort
works well in practice. This time, I was really surprised with the results: Bucket Sort was slower than Quick Sort -- It was designed to perform in an optimal way on different kind of real world
data. Counting sort, soms ook count-sort genoemd, is een extreem simpel sorteeralgoritme, dat alleen kan worden gebruikt voor gehele getallen en daarmee vergelijkbare objecten.Juist door de beperkte
toepassingsmogelijkheden, kan het een zeer efficiënte manier van sorteren zijn. Twitter Facebook Google+ LinkedIn UPDATE : Check this more general comparison ( Bubble Sort Vs Selection sort Vs
Insertion Sort Vs Merge Sort Vs Merge Sort Vs Quick Sort ) Before the stats, You must already know what is Merge sort, Selection Sort, Insertion Sort, Arrays, how to get current time. Refer : Radix
sort for a discussion of efficiency of Radix sort and other comparison sort algorithms. I increased the number of the array’s elements to 300,000 and profiled the application again. Another class to
help manage the testing of all the algorithms: AlgoDemo Reduces the number of keys in a specific range a [ ] corresponds theory! Comparison between elements of an array of elements is divided into
parts repeatedly until it is also a friendly. It enumerates occurrences of contained values back to our recommendation engine hypothetical space, quicksort, sort..., C++, Java, and some are
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algorithms are in-place sorting algorithm do! An adaptive sorting algorithm which is used to sort objects according the keys that are small numbers ’. Requires little space and exhibits good cache
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requires little space and exhibits good cache locality the knowledge of the bases of number... As another like regular merge sort, selection sort, count sort above stable... For arrays of analysis
invocation of the items I wanted to ensure was: same number of objects having key... It works by counting the number of objects having distinct key values are same problems to test & improve skill.
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sort is easier to implement than other efficient sorting algorithms )! Big, otherwise it can increase the space complexity are non-comparison based sort because two. Thnx counting sort is easier to
implement than other efficient sorting algorithms, and Python same number of having... Efficiency = O ( d.n ) where d = highest number of objects having distinct values. K ) with 6 digits each: Radix
sort 's best case O!
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e Math
Lesson 12
What’s the Story?
Warm-up: Estimation Exploration: How many People? (10 minutes)
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. For this picture, it is hard to tell how many people
there are, so a wide range of responses can be considered “about right.” Students will also use this image in the cool-down, and there is an option for them to use the picture to generate ideas
for story problems.
• Groups of 2
• “How many people are in the picture?”
• “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
Student Facing
How many people are in the picture?
Record an estimate that is:
│ too low │ about right │ too high │
│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│
Activity Synthesis
• “How did you make an estimate that was too low?” (I saw 10 to 20 kids running in front and then a whole lot of people behind them, so I made a low estimate of 50.)
• “How did you make an estimate that was too high?” (It is hard to tell how far back the people go, so I just said 1,000.)
• “Based on this discussion, does anyone want to revise their estimate?”
Activity 1: What’s the Story? (15 minutes)
The purpose of this activity is to write story problems for equations with an unknown value. There is a pair of addition and subtraction equations and in each pair one of them has the starting value
unknown. Students may write Add To, Take From, Put Together/Take Apart, or Compare problems. When students contextualize the equations and make connections between the stories their peers share and
the equations, they reason abstractly and quantitatively (MP2).
• Groups of 2
• Display list of topics for story problems from the previous lesson.
• Split the class into two groups, A and B. The students in group A will work with the equations labeled A and the students in group B will work with the equations labeled B.
• “You will write stories for the 2 equations in A or the 2 equations in B. Consider using the same context for both of your stories. It might make it easier for others to make sense of your
stories if they are about the same thing.”
• 5 minutes: independent work time
• “Share your stories with your partner.”
• 5 minutes: group work time
Student Facing
Your teacher will assign you A or B. For each of your equations, write a story problem that fits the equation.
A Equations
\(23 + \underline{\hspace{1 cm}} = 37\)
\(\underline{\hspace{1 cm}} +9 = 45\)
B Equations
\(73-\underline{\hspace{1 cm}} = 28\)
\(\underline{\hspace{1 cm}} -15 = 18\)
Activity Synthesis
• Display text: “There are 23 baseballs in the gym and 37 baseballs on the playground.”
• “This is Andre’s story. Does the equation \(23 + \underline{\hspace{1 cm}} = 37\) represent Andre’s story?” (No, Andre’s story doesn’t have a question or anything happening. It might match, but
it needs more information or a question.)
• “How would you improve Andre’s story?” (He could add a question like, “How many more baseballs are on the playground than in the gym?” He could keep it about baseballs, but add some things that
happen. He might say someone took the 23 baseballs to the playground and now there are 37 baseballs on the playground and ask how many were already on the playground.)
• Record student suggestions for revising Andre’s story.
• “Does the equation \(23 + \underline{\hspace{1 cm}} = 37\) represent any of these stories now?” (It represents the question about how many more baseballs are on the playground.)
Activity 2: Make Math Stories (20 minutes)
The purpose of this activity is for students to write math stories. Several options are available for fueling their imagination, including:
• looking at pre-selected images such as the one used in the Estimation Exploration in this lesson or those in the optional blackline master
• looking at images in magazines or newspapers
• looking around the classroom
• going for a walk around the school or community
Whichever source is used for ideas, students write a story problem that connects to mathematical ideas they have found. If students come up with a context, but are not able to count or estimate the
quantities they see, display a set of numbers (such as 11, 25, 38, 56, 77, 93) that students can use to write their story problem.
When students write math stories based on images or things in their environment, and eventually answer those questions, they model with mathematics (MP4).
MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Action and Expression: Develop Expression and Communication. Provide students with alternatives to writing on paper. Students can share their learning by drawing or creating a picture of their story
problem, or verbally by creating a video that tells their story.
Supports accessibility for: Attention, Organization, Language
Required Materials
Materials to Gather
Materials to Copy
Required Preparation
• Gather a see-through container with a collection of connecting cubes (or other math tool or object that might generate different math questions) to display in the launch.
• (Optional) Provide a copy of the blackline master for each group of 2 students.
• Groups of 2
• Display the see-through container with connecting cubes (or other math tool).
• “What are some math questions you can ask about the connecting cubes?” (How many connecting cubes are there altogether? How many green connecting cubes are there? Are there more blue connecting
cubes or yellow connecting cubes? How many more?)
• 1 minute: independent think time
• 1 minute: partner share time
• Share student responses, highlighting in each case how it could be made into a story problem (for example, “How many connecting cubes are there?” could be answered if we knew how many there are
of each color.)
• “Now, we are going to look for mathematical ideas in _____. Your goal is to take notes about what you see. Focus on things that can be counted, so that you can write a story problem about it.”
• “If you have time, you can count or estimate what you see. If not, write a story without exact quantities, and I will give you numbers that you can use in your story.”
• 8 minutes: math walk
• “Write a story problem and then share with your partner.”
• 8 minutes: partner work time
• Monitor for students who write different types of stories: single-step, two-step, Add To, Take From, Put Together/Take Apart, and Compare.
Activity Synthesis
• Invite selected students to share their stories.
Lesson Synthesis
“What is your favorite story that you heard today? Why?”
“Tomorrow you will make a poster to share your story and a solution, and then you’ll look at all of your classmates’ stories.”
Cool-down: What Could the Question Be? (5 minutes)
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Lesson 5
Squares and Circles
Problem 1
Match each quadratic expression with an equivalent expression in factored form.
Problem 2
An equation of a circle is \( x^2 - 8x + 16 + y^2 + 10y + 25 = 81\).
1. What is the radius of the circle?
2. What is the center of the circle?
Problem 3
Write 3 perfect square trinomials. Then rewrite them as squared binomials.
Problem 4
Write an equation of the circle that has a diameter with endpoints \((12,3)\) and \((\text-18,3)\).
Problem 5
1. Graph the circle \((x-2)^2+(y-1)^2=25\).
2. For each point, determine if it is on the circle. If not, decide whether it is inside the circle or outside of the circle.
1. \((4,0)\)
2. \((\text-3,3)\)
3. \((\text-2,\text-2)\)
3. How can you use distance calculations to decide if a point is inside, on, or outside a circle?
Problem 6
The triangle whose vertices are \((2,5), (3,1),\) and \((4,2)\) is transformed by the rule \((x,y) \rightarrow (x-2,y+4)\). Is the image similar or congruent to the original figure?
The image is congruent to the original triangle.
The image is similar but not congruent to the original triangle.
The image is neither similar nor congruent to the original triangle.
Problem 7
Technology required. A triangular prism has height 6 units. The base of the prism is shown in the image. What is the volume of the prism? Round your answer to the nearest tenth.
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Euclidean distance | EngatiEuclidean distance | Engati
What is Euclidean distance?
In about 300 B.C.E., the Greek mathematician Euclid was examining the relationships between angles and distances. Euclidean geometry is still widely used and taught even today and applies to spaces
of two or three dimensions, but it can be generalized to higher-order dimensions. At this point, the Euclidean distance is pretty much the most common use of distance. In most situations in which
people are talking about distance, they are referring to the Euclidean distance. It examines the root of square distances between the co-ordinates of a pair of objects. To derive the Euclidean
distance, you would have to compute the square root of the sum of the squares of the differences between corresponding values.
Euclidean space is a two- or three-dimensional space to which the axioms and postulates of Euclidean geometry apply. Euclidean distance refers to the distance between two points in Euclidean space.
By making use of the Pythagorean formula for distance, Euclidean space (or even any inner product space) would become a metric space. The associated norm is referred to as the Euclidean norm, which
is defined as the distance of each vector from the origin. One of the important properties of the Euclidean norm, relative to other norms, is that it remains unchanged under arbitrary rotations of
space around the origin. According to Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is more or less Euclidean. The Euclidean norm
is the only norm that posesses this property.
Earlier, this metric was also known as the Pythagorean metric. Due to the fact that it is possible for you to find the Euclidean distance by making use of the coordinate points and the Pythagoras
theorem, it is also sometimes referred to as the Pythagorean distance.
Source: Wikipedia
Which are the three prime Euclidean terms?
1. Euclidean distance is the distance from each cell in a raster to the closest source.
2. Euclidean allocation helps identify the cells which should be allocated to a source depending on proximity.
3. Euclidean direction shows us the direction from each cell to the nearest source.
How is Euclidean distance applied in machine learning?
In machine learning, it is most commonly used to understand and measure how similar observations are to each other. With vision AI, euclidean distance can be used for the camera to infer specific
movements based on distance changes. This is applicable in many fields including AI exercise motion capture and analysis.
How does Euclidean distance work?
The distance from each cell to each source cell is found by calculating the Hypotenuse with x_max and y_max as the other two sides of the triangle.
This method helps us find the actual distance, rather than the cell distance. If the shortest distance to a source is less than the specified maximum distance, we assign the value to the cell
location on the output raster.
How to calculate Euclidean distance (formula)?
Let’s say that (x1, x2) and (y1, y2) exist in a two-dimensional space. If a line segment formed between these two points, it could be the hypotenuse of a right-angled triangle.
Then the other two sides of the right-angled triangle would be |x1 — y1| and |x2 — y2|.
The Euclidean distance between (x1, x2) and (y1, y2) could be considered to be the length of the hypotenuse of the right-angled triangle. Since it is nothing but the straight line distance between
two given points, we can make use of the Pythagorean Theorem.
The distance between (x1, x2) and (y1, y2) would be (x1 — y1)2+(x2 — y2)2.
Now, if points (x1, x2, x3) and (y1, y2, y3) were in a three-dimensional space, the Euclidean distance between them would be (x1 — y1)2+(x2 — y2)2+(x3 — y3)2.
Taking this further, to calculate the Euclidean distance between, (x1, x2,..., xn) and (y1, y2,..., yn) in an n-dimensional space, the euclidean distance formula would be (x1 — y1)2+(x2 — y2)2+...+
(xn — yn)2
What is the Euclidean Squared Distance Metric?
The Euclidean squared distance metric makes use of the same equation as the Euclidean distance metric, but it does not take the square root. Because of this, clustering can be performed at a faster
pace with the Euclidean Squared Distance Metric than it can be carried out with the regular Euclidean distance.
Even if you replace the Euclidean distance with the Euclidean squared distance metric, the output of Jarvis-Patrick clustering and of K-Means clustering will not be affected. But if you do this, the
output of hierarchical clustering will be very likely to change.
Since squared Euclidean distance does not satisfy the triangle inequality, it does not form a metric space. Instead, it is a smooth, strictly convex function of the two points, unlike the distance,
which is non-smooth (near pairs of equal points) and convex but not strictly convex.
The Euclidean squared distance is preferred in optimization theory because it enables convex analysis to be used. Due to the fact that squaring happens to be a monotonic function of non-negative
values, minimizing squared distance is equivalent to minimizing the Euclidean distance. Therefore the optimization problem is equivalent in terms of either, but it is easier to solve by making use of
squared distance.
The collection of all squared distances between pairs of points from a finite set could be stored in a Euclidean distance matrix and can be used in that form in distance geometry.
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Design of RF Energy Harvesting Antenna using Optimization Techniques
Volume 09, Issue 03 (March 2020)
Design of RF Energy Harvesting Antenna using Optimization Techniques
DOI : 10.17577/IJERTV9IS030458
Download Full-Text PDF Cite this Publication
S. Vijay Gokul , M. Suba Lakshmi , T. Swetha, 2020, Design of RF Energy Harvesting Antenna using Optimization Techniques, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 09,
Issue 03 (March 2020),
• Open Access
• Authors : S. Vijay Gokul , M. Suba Lakshmi , T. Swetha
• Paper ID : IJERTV9IS030458
• Volume & Issue : Volume 09, Issue 03 (March 2020)
• Published (First Online): 03-04-2020
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Design of RF Energy Harvesting Antenna using Optimization Techniques
Mr. S. Vijay Gokul
Ms. M. Suba Lakshmi , Ms. T. Swetha Dept of Electronics and Communication Engg, Mepco Schlenk Engineering
Assistant Professor, College, Sivakasi,
Dept of Electronics and Communication Engg, Mepco Schlenk Engineering College, Sivakasi, Tamilnadu, India.
Tamilnadu, India.
Abstract: This paper deals with the design, analysis and Optimization of Radio Frequency (RF) energy harvesting antenna for Wireless Local Area Network (WLAN) sources
.The designed antenna is simulated using ANSYS HFSS(High Frequency Structure Simulator) and FR4 epoxy material as a substrate with a dielectric constant of 4.4 with a loss tangent of 0.02. It is a
rectangular microstrip patch antenna with H&E slots and consists of a radiating element with 50 ohm microstrip inset feed. Moreover, the combined configuration of proposed antenna with H&E shaped
slots in design provides maximum efficiency.More simulation iterations are performed in order to maximize the Gain of ISM band operating at a frequency of 2.45GHz which provides a return loss of
-24.77 dB, Voltage Standing Wave Ratio (VSWR) of
1. and Gain of 6.58 dB.The optimization of the antenna is carried out by employing Genetic Algorithm (GA). Antenna optimization and gain enhancement by applying genetic algorithm is implemented
with the help of MATLAB and ANSYS Optimetrics tool. After applying optimization algorithm the performance of the antenna has been improved with a Gain of 7dB. The design methodology of
microstrip patch antenna and optimizations are presented.
Keywords: Microstrip patch antenna; ISM band; RF energy harvesting; Genetic Algorithm (GA); Gain and Return loss.
1. INTRODUCTION
The energy harvesting is the method through which energy is derived from external sources like: solar power, thermal energy, wind energy and kinetic energy (eg: ambient energy) which are
captured and fed to small electronic wireless operating devices like wireless sensor networks etc., [1]. Lot of RF energy gets wasted due to non reception of the device. In order to make
use of renewable energy resources (Radio Frequency) to drive low power appliances and make it battery free RF energy harvesting antenna is mainly used. This enhances the usability and
reliability of
operate them separately [4].However it is observed that a standard microstrip patch antennas should possess very narrow operating bandwidth. Many procedures have been introduced to solve
this problem to some extent they are listed as: use a substrate of high dielectric permittivity [5], use of defected ground structures at the ground plane [6], use of metamaterials at the
ground plane [7] and addition of slots on the patches [8] and use of patches with H & E Shape instead of other shapes and optimization of patch shapes and gain by introducing one of the
optimization algorithms like genetic algorithm for better antenna performance. These challenges are efficient in order to attain broader bandwidth and to achieve more gain and directivity
for energy harvesting application [9-12]. In this paper RF energy is harvested using microstrip patch antenna and the performance is improved by applying genetic algorithm optimization
technique [GA].
2. ANTENNA DESIGN
Microstrip patch antennas are more popular because of low cost and ease of fabrication. Microstrip patch antenna consists of dielectric material, radiating element and the ground plane.
The rectangular shape is widely considered to realize the microstrip antenna. The simulation tool used for antenna design is High Frequency Structure Simulator version 2017(HFSS 17.2).
The simple microstrip patch antenna is designed using the substrate FR4_epoxy because of its low cost and ease of fabrication. It has the dielectric constant value of 4.4 and the loss
tangent is 0.02 and the resonating frequency is 2.45GHz. The detailed parameters are given in the Table 1. The formulae to determine the patch dimensions are as follows:
the device [2]. The idea of energy harvesting is not innovative rather it came hundred years ago. The method
2 r 1
of energy harvesting is to extract energy from the 1
environment in order to produce electricity is called energy
harvesting or energy scavenging [3]. Usually the energy harvesters will not provide sufficient amount of power for
1. fr
feeding electronic devices, the reason for this is mainly because there are no such technologies have developed in order to extract more and more RF energy. But this technology can provide enough
amount amount of energy sufficient of operating low power devicesin order to
Using the above equations (1) and (2) we have calculated the width and length of the patch.
From the above equations, eff
, 0 and L are the
1. SIMULATION RESULTS AND DISCUSSION
effective dielectric constant , permeability of free space and the extension length respectively. The effective dielectric
and the extension length
L can be
computed through the following equations respectively as.
r 1 r 1 12W 2
W 0.
L 0.412h
eff 0.3 h
eff 0.258 W
The proposed microstrip patch antenna is designed with an input impedance of 50.The parametric values are calculated from the given specific equations and the values are mentioned in the
given Table 1. With those calculated numerical values the simple microstrip patch antenna with H & E shaped slots are designed as shown in the Figure 1. For an antenna to radiate, it
should attain a return loss of more than -10dB. In order to further improve the return loss for better performance two slots are introduced on the patch. Thus in this proposed design H&E
shaped slots are introduced in the antenna. Here two patches are introduced (i.e. Patch 1 and Patch 2). Patch 1 is fed by microstrip inset feed and due to mutual conductance the feed is
automatically fed to Patch 2 through Patch 1.This enhances the better performance of the antenna with improved return loss and vswr values. In order to improve the gain and directivity,
defected ground structures (DGS) i.e., metamaterials are introduced on the ground in antenna design. This improves the gain, directivity and overall efficiency of the antenna. The design
specifications are mentioned in the below Table 1.
S.no Parameters Values
4 2.45GHz
Operating Frequency Substrate dielectric constant Substrate Thickness Substrate Width
5 4.4
Substrate Length Patch 1 Width
6 1.575 mm 53 mm 49 mm 31 mm 28 mm 31 mm
Patch 1 Length
7 1 mm
Patch 2 Width
8 4.9 mm
Patch 2 Length Feed Width Feed Length Input Impedance
9 17.5 mm 50
S.no Parameters Values
4 2.45GHz
Operating Frequency Substrate dielectric constant Substrate Thickness Substrate Width
5 4.4
Substrate Length Patch 1 Width
6 1.575 mm 53 mm 49 mm 31 mm 28 mm 31 mm
Patch 1 Length
7 1 mm
Patch 2 Width
8 4.9 mm
Patch 2 Length Feed Width Feed Length Input Impedance
9 17.5 mm 50
Table 1: Design Specifications for Energy Harvesting Antenna
Figure 1: H&E Shaped Microstrip Patch Antenna (Top View) in HFSS Simulator
Where, Ls, Ws = Length and Width of the substrate Lp, Wp = Length and Width of the patch Lf, Wf=Length and Width of the feed
1. DEFECTEDGROUNDSTRUCTURES
Defected ground structure (i.e., metamaterials) is introduced in the ground in order to improve the gain, directivity and overall efficiency of the antenna for better performance of
the antenna in energy harvesting capability which was shown in Figure 2.
Figure 2: DGS on Ground in the proposed design
2. SIMULATION RESULTS
1. Return loss
The simulated return loss (S11) parameterfor the given microstrip patch antenna before optimization is -29.25 dB which is shown in the Figure 3.
(d) Gain
Figure 5. Radiation Pattern
2. VSWR
Figure 3: Return loss (S11)
The observed Gain for given H&E shaped microstrip patch antenna before optimization is 5dB which is given in the below Figure 6.
The Voltage Standing Wave Ratio (VSWR) obtained before optimization from the given design is 1.07 which is shown in the Figure 4.
3. Radiation Pattern
Figure 4: VSWR
1. Directivity
Figure 6: Gain in 3D view
The radiation pattern obtained in this designed antenna is omnidirectional, whis is shown in the Figure 5.
The observed Directivity for the given microstrip patch antenna before optimization is 6dB which is shown in the Figure 7.
Figure 7: Directivity in 3D view
2. Current Distribution
The observed current distribution of simulated H&E shaped microstrip patch antenna before optimization is shown in the Figure 8.
Figure 8. Current distribution
2. GENETIC ALGORITHM
Genetic Algorithm (GA) is one of the powerful optimization techniques used in the wide area of electromagnetics.It is different from other optimization techniques. Holland and De Jong
introduced the hypothesis. It contains the optimization search aproaches and concepts besed on natural selection and evolution. The functional block diagram of GA is represented in the
Figure 9.
Figure 9. Functional Block Diagram of Genetic Algorithm
Initially Genetic Algorithm is performed by creation of initial population. Each individuals in the populations are coded as string of bits they are represented as chromosomes. They are
created randomly. The fitness of each individual is determined by the cost function or objective function. A good chromosome is always determined by the best value of the objective
function. The objective function is used to calculate the fitness of these individuals.By mating these individuals new generation will be formed. The individuals who have higher fitness
values are chosen for reproduction. Crossover and mutations are generally used for global search of the objective function or the cost function.The best individuals will be obtained in
the next generation without any change. This process will be repeated continuously untill end of the evolution is reached.In this paper,Genetic Algorithm is used for has been used to
enhance the performance of microstrip patch antenna (MPA) by obtaining high gain and directivity and reduced size at the given resonating frequency by optimizing the substrate dimensions
and patch dimensions. The antenna design is simulated in Ansys HFSS 2017 software and the genetic algorithm optimization is done through MATLAB software. The resultant results from the
MATLAB simulation is taken and applied in the HFSS environment to perform further simulations.
A.GENETIC ALGORITHMIMPLEMENTATION
The Optimization steps for obtaining the best results using GA in MATLAB are summarized as below:
Step 1:Study the parameters that need to be optimized using GA from HFSS antenna design.
Step 2:Design the variable behavior for mapping.
Step 3: Formulate the constraints in Matlab.
Step 4:Model the objective function equation and Genetic algorithm program for the design parameters.
Step 5:Setup the bound values ( i.e, Upper and lower limits
) for the desired parameters.
Step 6: Run optimization process and Analyse the results.
Step 7: Select the best optimal antenna design values
Step 8:Apply thebest values which are obtained from the MATLAB in HFSS antenna design.
Step 9:Run and Analyse the results.
Table 2: GA Parameters Used for Antenna Design
GA Parameters Values
Population Size Population Type Crossover Fraction Crossover Mutation Bit String 0.2
Total No of Iterations Single Point Crossover 0.01
1. RESULTANT MICROSTRIP PATCH ANTENNA (AFTER OPTIMIZATION)
The resultant microstrip H & E shaped patch antenna after optimization using genetic algorithm for energy harvesting is shown in the Figure 10 and 11.
Figure 10 : ProposedMPA in Top View
Figure 11 : Proposed MPA in Bottom View
2. PROPOSED ANTENNA SPECIFICATIONS Table 3 : Design Specifications After Optimization
S.no Parameters Values
Operating Frequency Substrate dielectric constant 2.45GHz
Substrate Thickness Substrate Width Substrate Length Patch 1 Width 4.4
Patch 1 Length 0.25 mm 40 mm 48 mm 28 mm 31 mm 31 mm 1.96mm
Patch 2 Width 0.25 mm
Patch 2 Length Feed Width Feed Length Input Impedance 4.9 mm 50
3. RESULTS AND DISCUSSIONS (AFTER OPTIMIZATION)
1. Return loss
The simulated results for the given microstrip H & E shaped microstrip patch antenna after optimization is –
24.77 dB which is shown in the Figure 12.
(d) Directivity
Figure 14: Gain in 3D view
2. VSWR
Figure 12: Return loss (S11)
The observed Directivity for the given microstrip patch antenna after optimization is 7.0dB which is shown in the Figure 15.
The Voltage Standing Wave Ratio (VSWR) obtained after optimization from the given design is 1.12 which is shown in the Figure 13.
Figure 15: Directivity in 3D view
Parameters Gain Directivity Returnloss VSWR
H & E Shaped Microstrip Patch Antenna ( Before Optimization ) 5dB 6dB -29.25dB 1.07
H & E Shaped Microstrip Patch Antenna ( After Optimization ) 6.5dB 7dB -24.77dB 1.12
Parameters Gain Directivity Returnloss VSWR
H & E Shaped Microstrip Patch Antenna ( Before Optimization ) 5dB td> 1.07
H & E Shaped Microstrip Patch Antenna ( After Optimization ) 6.5dB 7dB -24.77dB 1.12
3. Gain
Figure 13: VSWR
Table 4: Comparison between the Proposed RF Energy Harvesting Antenna (Before & After Optimization)
The observed Gain for given H&E shaped microstrip patch antenna after optimization is 6.58 dB which is given in the below Figure 14.
3. CONCLUSION
Thus H & E shaped microstrip patch antenna for energy harvesting in one of the well known simulators known as Ansys High Frequency Structure Simulator (HFSS) and the results will be further improved
by using one of the optimization techniques i.e., Genetic Algorithm (GA) in well known software called MATLAB 2019a and then the optimized results will be fed into HFSS inorder to obtain the high
gain and antenna efficiency. The simulation results shows that the antenna will radiate at 2.45GHz frequency which will be included under ISM band operating range. The aim of this work is to improve
gain of the antenna inorder to improve the antenna performance for RF energy harvesting capability.This proposed design exhibits high gain of 6.5 dB with a return loss of -24.77 dB and vswr of
1. and directivity of 7 dB. The efficiency of the proposed design is reached upto 95% and thus the it can provide far field radiation and makes the antenna size as much more compact for easy
handling capability. Hence the antenna can be used as the front end section of the energy harvesting system. Therefore the antenna can be suitable for RF energy harvesting
applications.Finally it is concluded that the proposed energy harvesting antenna can be used for providing power on demand for short range sensing applications.
VI . REFERENCES
1. Muhammad Salman Iqbal, Tariq Jameel Khanzada,Faisal A. Dahri, Asif Ali, Mukhtiar Ali, Abdul Wahab Khokhar, Analysis and Maximizing Energy Harvesting from RF Signals using T- Shaped Micro
strip Patch Antenna , (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 10, No. 1, 2019
2. Kayhan Çelik, Erol Kurt, A novel meander line integrated E- shaped rectenna for energy harvesting applications, Int J RF Microwave Computer Aided Eng.2019
3. Jaget Singh,B.S. Sohi, Kanav Badhan,Slit loaded H- Shaped Microstrip Patch Antenna for 2.4 GHz, Int Journal of Applied Engineering Research ISSN 0973-4562 Volume13 , Number 18 (2018)
4. Bilal S. Taha , Hamzah M. Marhoon , Ahmed A.Naser , Simulating of RF energy harvesting micro-strip patch antenna over 2.45 GHZ ,International Journal of Engineering & Technology, 7 (4)
(2018) 5484-5488
5. Rajdevinder Kaur idhu, Jagpal Sing Ubhi, Alpana Aggarwal, A Survey Study of Different RF Energy Sources for RF Energy Harvesting International Conference on Automation Computational and
Technology Management (ICACTM)
6. Navpreet Kaur, Nipun Sharma and Naveen Kumar, RF Energy Harvesting and Storage System of Rectenna Indian Journal of Science and Technology, Vol 11(25), DOI: 10.17485/ijst/2018/v11i25/
114309, July 2018
7. Sukhveer Kaur, Sushil Kakkar, Shweta Rani, Design and Analysis of Micro strip Patch Antenna for RF Energy Harvesting,International Journal of Electrical Electronics & Computer Science
Engineering Volume 5, Issue 2 April, 2018
8. Li Zhu, Jiawei Zhang , Wanyang Han, Leijun Xu, XueBai,A novel RF energy harvesting cube based on air dielectric antenna arrays, Int J RF Microwave Computer Aided Eng.2018
9. Mahima Arrawatia, Maryam Shojaei Baghini, and Girish Kumar, Differential Micro strip Antenna for RF Energy Harvesting IEEE Transactions on Antennas and Propagation, vol. 63, no. 4,
pp.15811588, April 2015
10. Raj Gaurav Mishra, Ranjan Mishra, Piyush Kucchnal, N.Prasanthi Kumari, Analysis of the microstrip patch antenna design using genetic algorithm based optimization for wide-band
applications International Journal of Pure and Applied Mathematics Volume 118No.112018,841-849
11. Raj Gaurav Mishra, Ranjan Mishra, Piyush Kucchnal, N. Prasanthi Kumari, Optimization and analysis of high gain wideband microstrip patch antenna using genetic algorithm International
Journal of Engineering & Technology, 7 (1.5) (2018) 176-179
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9.4 Rare Events, the Sample, and the Decision and Conclusion
Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you
should consider when working out a hypothesis test.
Rare Events
Rare Events
The thinking process in hypothesis testing can be summarized as follows: You want to test whether or not a particular property of the population is true. You make an assumption about the true
population mean for numerical data or the true population proportion for categorical data. This assumption is the null hypothesis. Then you gather sample data that is representative of the
population. From this sample data you compute the sample mean (or the sample proportion). If the value that you observe is very unlikely to occur (a rare event) if the null hypothesis is true, then
you wonder why this is happening. A plausible explanation is that the null hypothesis is false.
For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be
blindfolded. There are 200 plastic bubbles in the basket, and Didi and Ali have been told that there is only one with a $100 bill. Didi is the first person to reach into the basket and pull out a
bubble. Her bubble contains a $100 bill. The probability of this happening is $12001200$ = 0.005. Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are
more $100 bills in the basket. A rare event has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket.
Using the Sample to Test the Null Hypothesis
Using the Sample to Test the Null Hypothesis
After you collect data and obtain the test statistic (the sample mean, sample proportion, or other test statistic), you can determine the probability of obtaining that test statistic when the null
hypothesis is true. This probability is called the p-value.
When the p-value is very small, it means that the observed test statistic is very unlikely to happen if the null hypothesis is true. This gives significant evidence to suggest that the null
hypothesis is false, and to reject it in favor of the alternative hypothesis. In practice, to reject the null hypothesis we want the p-value to be smaller than 0.05 (5 percent) or sometimes even
smaller than 0.01 (1 percent).
Example 9.9
Suppose a baker claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a
hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5
cm and the distribution of heights is normal.
The null hypothesis could be H[0]: μ ≤ 15. The alternate hypothesis is H[a]: μ > 15.
The words is more than translates as a ">" so "μ > 15" goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.
Since σ is known (σ = 0.5 cm), the distribution for the population is known to be normal with mean μ = 15 and standard deviation $σ n = 0.5 10 =0.16 σ n = 0.5 10 =0.16$.
Suppose the null hypothesis is true (which is that the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis
test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p-value is the
probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.
The p-value, then, is the probability that a sample mean is the same or greater than 17 cm when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution
for means. In Figure 9.2 below, the p-value is the area under the normal curve to the right of 17. Using a normal distribution table or a calculator, we can compute that this probability is
practically zero.
p-value = P($x ¯ x ¯$ > 17), which is approximately zero.
Because the p-value is almost 0, we conclude that obtaining a sample height of 17 cm or higher from 10 loaves of bread is very unlikely if the true mean height is 15 cm. We reject the null hypothesis
and conclude that there is sufficient evidence to claim that the true population mean height of the baker’s loaves of bread is higher than 15 cm.
Try It 9.9
A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.
H[0]: μ ≤ 12
H[a]: μ > 12
-value is 0.0013.
Draw a graph that shows the
Decision and Conclusion
Decision and Conclusion
A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the p-value and a preset or preconceived α, also called the level of significance of the test.
A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.
When you make a decision to reject or not reject H[0], do as follows:
• If p-value$<α p−value<α$, reject H[0]. p-value The results of the sample data are significant. There is sufficient evidence to conclude that H[0] is an incorrect belief and that the alternative
hypothesis, H[a], may be correct.
• If p-value $≥α p−value≥α$, do not reject H[0]. The results of the sample data are not significant.There is not sufficient evidence to conclude that the alternative hypothesis, H[a], may be
• When you do not reject H[0], it does not mean that you should believe that H[0] is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt
about the truthfulness of H[0].
Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem.
Example 9.10
When using the p-value to evaluate a hypothesis test, you might find it useful to use the following memory device:
If the p-value is low, the null must go.
If the p-value is high, the null must fly.
This memory aid relates a p-value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p-value higher than the established alpha (the p is high) as
not rejecting the null hypothesis.
Fill in the blanks.
Reject the null hypothesis when ______________________________________.
The results of the sample data _____________________________________.
Do not reject the null hypothesis when __________________________________________.
The results of the sample data ____________________________________________.
Solution 9.10
Reject the null hypothesis when the p-value is less than the established alpha value. The results of the sample data support the alternative hypothesis.
Do not reject the null hypothesis when the p-value is greater or equal to the established alpha value. The results of the sample data do not support the alternative hypothesis.
Try It 9.10
It’s a Boy Genetics Labs, a genetics company, claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:
H[0]: p = 0.50, H[a]: p > 0.50
α = 0.01
p-value = 0.025
Interpret the results and state a conclusion in simple, nontechnical terms.
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When It Rains It Pours
#23: Covariance and Convexity in Financial Models.
This week, we will explore the mechanics and pitfalls of financial modeling through the lens of a private equity leveraged buyout (LBO). Why do models often fail to predict common outcomes?
The Paper LBO
Close your eyes and picture the scene.
It’s 9:30AM on a Thursday morning. You, a 23 year-old sleep-deprived investment banking analyst, told your desk you would be out for the morning getting a cavity filled - an unoriginal excuse - but
instead, you are sitting in a glass-paneled office overlooking Midtown Manhattan. The interviewer sitting across from you, a mid-level private equity professional, asks you to take out a blank sheet
of paper.
You know what’s coming, you’ve practiced for it. She begins:
Consider a company with the following characteristics:
• $50 million in revenue next year growing at 10% per year
• 20% EBITDA margins
• $4 million of capital expenditures (capex) and depreciation & amortization (D&A) per year, growing in line with revenue
• No working capital
• 40% tax rate
Now, assume you can buy the company at 10x forward EBITDA, using 5x leverage at a 5% interest rate, and sell the company in 5 years at the same multiple. What is your return?
After frantically scribbling down the prompt, you get to work.
First, build a sources and uses table, to determine the purchase price and how the transaction will be funded1. Second, project each of the line items from revenue down to net cash flow2. Then,
calculate the net proceeds you will receive in year 5 based on the exit multiple and net debt outstanding3. Finally, compare your net proceeds to your initial investment to calculate the return4.
By the end of the exercise, your paper looks like this:
You can now answer the interviewer’s question. You invest $50 million of equity to buy the company and receive $125.5 million at exit - a 2.5x multiple of invested capital (MOIC). A 2.5x return on
capital over five years is a 20% internal rate of return (IRR) - a relationship that you’ve memorized in advance.
This exercise, known as a “paper LBO”, is ubiquitous in private equity interviews because it captures the business model in a nutshell. To answer the question, you must be able to mechanically lay
out an LBO model, calculate net cash flows, complete some mental math, and have an understanding of investment returns.
Yet, like any model, even a paper LBO is only as good as its assumptions.
Components of Return
A typical follow up question to a paper LBO is to disaggregate the drivers of the $75.5 million gain on the investment. The textbook response says there are three elements of return:
1. Growth: In our example, EBITDA grew from $10 million to $16.1 million, or $6.1 million over the hold period. When applying the 10x EBITDA multiple to this growth, we see that growth accounts for
$61.1 million of the gain on investment.
2. Cash Flow: The company generated $14.5 million of cash flow during the hold period.
3. Exit Multiple: Because we assume that we enter and exit the investment at a 10x multiple, there is no gain or loss associated with the exit multiple.
Based on this breakdown, we see that the growth assumption drives nearly all of the investment return, while actual cash generated during the hold period is only a minor contributor. Our assumption
of exit/entry parity has no bearing on return.
Knowing that the investment hinges heavily on the growth assumptions, it makes sense to sensitize this variable to see how deviations from expectations impact returns. When holding all other
variables constant and changing the assumed revenue growth, we see that revenue growth has a near linear impact on investment returns.
While this analysis shows that investment returns are highly dependent on revenue growth, it also suggests a surprisingly resilient investment profile. If you were to totally whiff on your
assumptions, and the company does not grow at all, you would still have a marginally profitable investment (driven by cash flow during the period).
In reality, this is almost certainly not the case. If you underwrite a private equity investment to a 20% return assuming 10% compounded growth, and the company does not grow at all, you will likely
lose a lot of money.
Covariance and Convexity
The simple breakdown of LBO returns into growth, cashflow, and exit multiple is a useful mathematic bridge but creates a false sense of independence between the variables. In reality, all of the
variables are correlated and cannot be separated in such a simplistic sense. For one, revenue growth rate will impact the cash flow generated during the hold period. More important is the impact on
the exit multiple.
The universal convention in LBO modeling is to assume parity in exit and entry multiples. This is often considered “conservative” in the sense that you don’t assume multiple expansion in your base
case. It is also true that if a company was to continue growing at your initial growth assumptions, then a subsequent buyer with the same cost of capital should pay the same multiple for the
But, while multiple parity is a justifiable base case assumption, it cannot be assumed to be static. Rather, the exit multiple must flex along with the actual performance of the business. If the
business only grows at 5% instead of 10%, this will reduce the EBITDA growth and cash flow during the hold period but also cause multiple contraction at exit. Alternatively, if the business
outperforms, a subsequent buyer should be willing to pay a higher multiple.
If we assume a subsequent buyer has the same cost of capital as the initial buyer (i.e. requires the same 20% return), we can create a dynamic exit multiple assumption that incorporates the actual
growth of the business during the hold period5.
There is a non-linear relationship between the company’s growth rate and the multiple the next buyer could pay to achieve a 20% return, assuming the buyer extrapolated the growth rate going forward.
Now, we can re-run our previous revenue growth sensitivity to include both its impact to the cash flows during the hold period as well as its implications for exit valuation.
By incorporating the covariance of revenue growth and exit multiples, the dynamic sensitivity shows a convex line with far more upside and downside. The two curves intersect on the base case
assumption. If the company outperforms on growth, multiple expansion juices returns. If the company underperforms though, multiple compression leads to much lower returns. When it rains, it pours.
Unlike the static approach, which shows a small positive return in the zero-growth case, the dynamic exit multiple shows that company must achieve at least a 5% growth rate in order for the
investment to be profitable. If the company has no growth, the MOIC falls to just 0.35x, or a loss of 65% of principal.
These two conventions paint starkly different risk and reward profiles. The static multiple implies a range-bound return profile with little risk of principal loss, while the dynamic multiple shows
the investment’s true colors - a highly levered investment contingent on strong future growth.
Investment decision are analyzed and iterated ad nauseum. A deal team will present pages of detailed model outputs, backed by endless assumptions, inputs, and calculations from overflowing excel
files. Yet, the end result is always the same - the Base Case looks good, the Upside Case is a bit better, and the Worst Case Scenario is not all that bad.
Never will you see a 10-bagger or a complete donut in a memo, yet both these outcomes happen regularly. One key failure is the tendency to think about assumptions as independent (and often
counterbalancing), when in reality they row in the same direction, compounding moves in good and bad directions. Better models incorporate covariance and convexity and the volatility each brings.
As always, thank you for reading. If you enjoy The Last Bear Standing, tell a friend! And please, let me know your thoughts in the comments - I respond to all of them.
You buy the company for $100 million (10 x $10 million in EBITDA), funded with $50 million in debt (5 x $10 million in EBITDA) and the remaining $50 million with equity.
Multiply revenue by 20% margins to get EBITDA, then subtract capex, interest, and taxes to get cash flow. Because D&A is equal to capex in this example, the calculations of pre-tax income and cash
flow can be simplified. If the two figures were different, you would need to subtract D&A to calculate pre-tax income, then add back D&A and subtract capex in the cash flow.
Multiply Year 6 EBITDA of $16.1 by 10 to calculate the gross exit proceeds of $161.1 million. Then subtract your debt ($50 million), and add the cumulative cash flow from the investment period ($14.5
million) to calculate net proceeds.
Divide the $125.2 million exit proceeds by the $50 million investment to determine a multiple of invested capital of 2.5x. Memorizing a cheat sheet of MOIC vs. IRR conversions.
Assumes the subsequent buyer also purchases with 5.0x leverage at a 5% interest rate.
Thank you Bear! This is a great lesson!
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Thanks for throwing that one in. Appreciate a glimpse into the VC side of things!
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Finding a Certain Term in a Binomial Expansion
Question Video: Finding a Certain Term in a Binomial Expansion Mathematics • Third Year of Secondary School
Find the term independent of π ₯ in the expansion of (π ₯ + 1/π ₯)ΒΉΒ² β (π ₯ β 1/π ₯)ΒΉΒ².
Video Transcript
Find the term independent of π ₯ in the expansion of π ₯ plus one over π ₯ all to the power of 12 minus π ₯ minus one over π ₯ all to the power of 12.
We have an expression that needs expanding. And weβ re looking to find the term thatβ s independent of π ₯. Another way of describing this term is to think of it as the constant term. And this
will exist roughly in the middle of each of the expansions. And letβ s see why. The binomial expansion of π plus π to the power of π is given as π to the power of π plus π choose
one multiplied by π to the power of π minus one multiplied by π plus π choose two multiplied by π to the power of π minus two multiplied by π squared, all the way through to π
to the power of π .
So letβ s see what this looks like when we have the expansion of π ₯ plus one over π ₯ to the power of 12. Our π here is π ₯. Our π is one over π ₯. And π is equal to 12. So the first
term is quite simple in this expansion. Itβ s just π ₯ to the power of 12. Our second term is 12 choose one multiplied by π ₯ to the power of 11 multiplied by one over π ₯ to the power of one.
Then we have 12 choose two multiplied by π ₯ to the power of 10 multiplied by one over π ₯ squared and so on. Now, since the value of π is even, this means thereβ s going to exist a term for
which the power of π ₯ is equal to the power of one over π ₯. In fact, theyβ re both going to be six.
This is the seventh term of this sequence. Itβ s given by 12 choose six multiplied by π ₯ to the power of six multiplied by one over π ₯ again to the power of six. One over π ₯ all to the power
of six is one to the power of six over π ₯ to the power of six which is just one over π ₯ to the power of six. And then letβ s look at what happens. π ₯ to the power of six multiplied by one over
π ₯ to the power of six is the same as π ₯ to the power of six divided by π ₯ to the power of six which is simply one. And so the term independent of π ₯ in the expansion of π ₯ plus one over π
₯ all to the power of 12 is 12 choose six.
Weβ re not going to evaluate 12 choose six just yet. Letβ s repeat this process for the expansion of the second bracket. Our first few terms look extremely similar. But instead of one over π ₯, we
have negative one over π ₯ in our brackets. We are, however, still interested in the seventh term. Itβ s 12 choose six multiplied by π ₯ to the power of six multiplied by negative one over π ₯ to
the power of six. But in fact, negative one over π ₯ to the power of six is just one over π ₯ to the power of six. Remember, when a negative number is raised to a positive exponent or a positive
power, it becomes a positive number.
Once again, π ₯ to the power of six multiplied by one over π ₯ to the power of six is one. And we see that the seventh term in expansion of π ₯ minus one over π ₯ to the power of 12 and the
constant of the term independent of π ₯ is 12 choose six. Weβ re subtracting the expansion of π ₯ minus one over π ₯ to the power 12 from the expansion of π ₯ plus one over π ₯ to the power of
12. So to find the term independent of π ₯ in this expansion, we work out 12 choose six minus 12 choose six which is, of course, zero. The term independent of π ₯ is zero.
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Procedural Generation with Cellular Automata
In this post, we will explore the basics of cellular automata as a tool for procedural generation. Both cellular automata and procedural generation are topics with near-infinite depth, so we’ll only
scratch the surface; however, it’ll provide a base on which to build your knowledge.
What are Cellular Automata?
A cellular automaton is a grid of cells where every cell is in some state belonging to a finite number of states. A single number often represents these states, so as a result, a cellular automaton
is essentially a spreadsheet of numbers, which in itself is not particularly exciting. However, cellular automata can model sophisticated, complex behaviours by attaching simple rulesets. One famous
example is Conway’s Game of Life, a deep topic in itself. Most CA rulesets have each cell check its neighbours and change its state based on what it finds. A single step of the automaton usually
involves every cell applying the ruleset one time, resulting in a new state for the entire grid. Although CAs are used to model physics, biology, fluids, particles, etc. In this post, we’ll start
with a simple implementation.
The plan
This post will build a relatively simple CA and demonstrate how it generates cave-like structures. First, we’ll fill a cell grid with randomly distributed live cells. Then, each cell will check how
many of its neighbours are alive for each step. If the number passes the threshold, that cell will come alive; otherwise, it will die. After several steps, the grid will reach stasis, and we’ll have
our cave. Let’s get started!
Creating the grid
For simplicity, I used a CPU-based approach this time. As a result, our automaton will live inside a 2D int array. We’ll fill a texture, with each pixel representing a cell in the grid, and slap it
on a quad in front of the camera to display it.
Our automaton contains two possible states, on and off. So to initialize the grid, we go through every cell and choose a state. The easiest way to do this is to flip a coin, which gives a random
distribution. However, to make things a little more interesting, we’ll add a fill percentage to set the ratio of live cells to dead cells on initialization. That way, we can explore the different
structures that emerge from different ratios.
public class CellularAutomataGenerator : MonoBehaviour
int[,] _cellularAutomata;
[SerializeField] int _width;
[SerializeField] int _height;
float _fillPercent = 0.5f;
void ResetAutomata()
_cellularAutomata = new int[_width, _height];
for (int x = 0; x < _width; ++x)
for (int y = 0; y < _height; ++y)
_cellularAutomata[x, y] = Random.value > _fillPercent ? 0 : 1;
Here’s our method to reset the automaton to an initial random state. Next, let’s create a texture to draw the CA’s state.
Texture2D _caTexture;
void OnEnable()
_caTexture = new Texture2D(_width, _height);
_caTexture.filterMode = FilterMode.Point;
void UpdateTexture()
var pixels = _caTexture.GetPixels();
for (int i = 0; i < pixels.Length; ++i)
var value = _cellularAutomata[i % _width, i / _height];
pixels[i] = value * Color.white;
First, we create a texture that’s the same width and height as our cell grid. Then, iterate through each pixel and assign a colour. To determine the colour, look at the corresponding grid cell’s
value. If it’s 0, use black; otherwise, white. Finally, apply the pixels so that the texture updates.
Viewing the automaton
Now, let’s create a quad and place it in front of the camera to view the texture. Create the quad in the scene and place it so the camera can see it. Also, create a new material using the Unlit/
Texture shader and attach that to the quad’s MeshRenderer. Now we reference that material in the Inspector to attach the texture in the OnEnable method.
[SerializeField] Material _material;
void OnEnable()
_caTexture = new Texture2D(_width, _height);
_caTexture.filterMode = FilterMode.Point;
_material.SetTexture("_MainTex", _caTexture);
Now hit play to view the current cell grid. Great! Now let’s talk about the rules of our automaton.
Basic ruleset
When it comes to cellular automata, most rulesets involve having a cell check the state of its neighbouring cells. Then, it has the option to modify its state based on what it finds. We’ll implement
one of the simplest versions of that, which is if the number of live neighbours passes a threshold, the cell comes alive; otherwise, it dies. This ruleset is a simplified version of Conway’s Game of
Life because we remove the overpopulation clause. Aside from creating cave structures, you could also use it to simulate fire propagation or erosion by water. For example, if everything around you is
on fire, you should also be on fire.
Since we’re not concerned with performance right now, we’ll implement this as naively as possible. It’s worth noting that Cellular Automata are an excellent candidate for multithreading and easy to
compute on the GPU. We can compute millions or potentially billions of cells in real-time by doing so, but let’s save that for another time.
So with that, let’s look at the code.
int GetNeighbourCellCount(int x, int y)
int neighbourCellCount = 0;
if (x > 0)
neighbourCellCount += _cellularAutomata[x - 1, y];
if (y > 0)
neighbourCellCount += _cellularAutomata[x - 1, y - 1];
if (y > 0)
neighbourCellCount += _cellularAutomata[x, y - 1];
if (x < _width - 1)
neighbourCellCount += _cellularAutomata[x + 1, y - 1];
if (x < _width - 1)
neighbourCellCount += _cellularAutomata[x + 1, y];
if (y < _height - 1)
neighbourCellCount += _cellularAutomata[x + 1, y + 1];
if (y < _height - 1)
neighbourCellCount += _cellularAutomata[x, y + 1];
if (x > 0)
neighbourCellCount += _cellularAutomata[x - 1, y + 1];
return neighbourCellCount;
Most of the code here is checking if the neighbours are inside the bounds of the automaton. Aside from that, we add all the cells up and return the result.
Next, we’ll write our Step method. This method iterates through every cell, gathering the number of live neighbour cells to see if it meets our threshold. If the cell meets the requirement, it will
come to life; otherwise, it dies.
void Step()
int[,] caBuffer = new int[_width, _height];
for (int x = 0; x < _width; ++x)
for (int y = 0; y < _height; ++y)
int liveCellCount = _cellularAutomata[x, y] + GetNeighbourCellCount(x, y);
caBuffer[x, y] = liveCellCount > _liveNeighboursRequired ? 1 : 0;
for (int x = 0; x < _width; ++x)
for (int y = 0; y < _height; ++y)
_cellularAutomata[x, y] = caBuffer[x, y];
By the way, you may have noticed that we perform the cell computations in a separate buffer before copying it to the primary grid. This approach prevents changes to the grid from affecting the cells
we haven’t checked yet. Additionally, when multithreading the code, this avoids dependencies between threads and generally makes it trivial to write.
We now have everything we need to experiment with simple Cellular Automata. I hooked up some UI elements to start testing. I linked to a WebGL build and Github repository at the end of this post if
you’d like to try it out. Otherwise, here’s what I found.
Given the default parameters, the automaton stabilizes into a cave-looking structure within a handful of steps.
Generated Cave
Increasing the starting fill percent trends towards more live cells in the end. Given a high enough percentage, the cave structure turns into a large landmass that resembles a piece of torn
Generated landmass / parchment
If we generate a stable cave and then decrease the live neighbour threshold to three, the black pixels slowly overtake the white pixels. The result is something that looks more like an island than a
Generated island
Conversely, increasing the live neighbour threshold to five creates a rising tide effect. That is, the black pixels slowly creep up on the white islands. This approach could simulate tides coming in
and out or water levels rising over time.
Generated small islands
Next steps
Right now, our automaton exists as a 2D array and a texture. It’s fun to play with but not particularly useful in the context of a game. The next step is to use the data to generate an actual world,
which we’ll explore in the future. If you’d like to get started right away, some potential approaches are using the grid as a tilemap or using a technique like marching squares to generate a mesh.
Check out the project on GitHub or play it in your browser here. If you’d like to support my work, join my mailing list. That way, I’ll notify you whenever I write a new post.
3 thoughts on “Procedural Generation with Cellular Automata”
Glen Johnson
That’s some cool stuff. I will definitely play around with that code. Thanks!
Thanks! Let me know how it goes š
You didn’t include a screenshot or text documenting when you added the variable ‘_liveNeighboursRequired’
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How to Calculate The Debt Yield RatioHow to Calculate The Debt Yield Ratio
The debt yield is becoming an increasingly important ratio in commercial real estate lending. Traditionally, lenders have used the loan to value ratio and the debt service coverage ratio to
underwrite a commercial real estate loan. Now, the debt yield is used by some lenders as an additional underwriting ratio. However, since it’s not widely used by all lenders, it’s often
misunderstood. In this article, we’ll discuss the debt yield in detail, and we’ll also walk through some relevant examples.
What is The Debt Yield?
First, what exactly is the debt yield? Debt yield is defined as a property’s net operating income divided by the total loan amount.
For example, if a property’s net operating income is $100,000 and the total loan amount is $1,000,000, then the debt yield would simply be $100,000 / $1,000,000, or 10%.
The debt yield equation can also be re-arranged to solve for the Loan Amount:
For example, if a lender’s required debt yield is 10% and a property’s net operating income is $100,000, then the total loan amount using this approach would be $1,000,000.
What The Debt Yield Means
The debt yield provides a measure of risk that is independent of the interest rate, amortization period, and market value. Lower debt yields indicate higher leverage and therefore higher risk.
Conversely, higher debt yields indicate lower leverage and therefore lower risk. The debt yield is used to ensure a loan amount isn’t inflated due to low market cap rates, low-interest rates, or high
amortization periods. The debt yield is also used as a common metric to compare risk relative to other loans.
What’s a good debt yield? As always, this will depend on the property type, current economic conditions, strength of the tenants, strength of the guarantors, etc. However, according to the
Comptroller’s Handbook for Commercial Real Estate, a recommended minimum acceptable debt yield is 10%.
Debt Yield vs Loan to Value Ratio
The debt service coverage ratio and the loan to value ratio are the traditional methods used in commercial real estate loan underwriting. However, the problem with using only these two ratios is that
they are subject to manipulation. The debt yield, on the other hand, is a static measure that will not vary based on changing market valuations, interest rates and amortization periods.
The loan to value ratio is the total loan amount divided by the appraised value of the property. In this formula, the total loan amount is not subject to variation, but the estimated market value is.
This became apparent during the 2008 financial crises, when valuations rapidly declined and distressed properties became difficult to value. Since market value is volatile and only an estimate, the
loan to value ratio does not always provide an accurate measure of risk for a lender.
As you can see, the LTV ratio changes as the estimated market value changes (based on direct capitalization). While an appraisal may indicate a single probable market value, the reality is that the
probable market value falls within a range and is also volatile over time. The above range indicates a market cap rate between 4.50% and 5.50%, which produces loan to value ratios between 71% and
86%. With such potential variation, it’s hard to get a static measure of risk for this loan. The debt yield can provide us with this static measure, regardless of what the market value is. For the
loan above, it’s simply $95,000 / $1,500,000, or 6.33%.
Debt Yield vs Debt Service Coverage Ratio
The debt service coverage ratio is the net operating income divided by annual debt service. While it may appear that the total debt service is a static input into this formula, the DSCR can in fact
also be manipulated. This can be done by simply lowering the interest rate used in the loan calculation or by changing the amortization period for the proposed loan. For example, if a requested loan
amount doesn’t achieve a required 1.25x DSCR at a 20-year amortization, then a 25-year amortization could be used to increase the DSCR. This also increases the risk of the loan, but is not reflected
in the DSCR or LTV.
As you can see, the amortization period greatly affects whether the DSCR requirement can be achieved. Suppose that in order for our loan to be approved, it must achieve a 1.25x DSCR or higher. As
demonstrated by the chart above, this can be accomplished with a 25-year amortization period, but going down to a 20-year amortization breaks the DSCR requirement.
Assuming we go with the 25-year amortization and approve the loan, is this a good bet? Since the debt yield isn’t impacted by the amortization period, it can provide us with an objective measure of
risk for this loan with a single metric. In this case, the debt yield is simply $90,000 / $1,000,000, or 9.00%. If our internal policy required a minimum 10% debt yield, then this loan would not
likely be approved, even though we could achieve the required DSCR by changing the amortization period.
Just like the amortization period, the interest rate can also significantly change the debt service coverage ratio.
As shown above, the DSCR at a 7% interest rate is only 1.05x. Assuming the lender was not willing to negotiate on amortization but was willing to negotiate on the interest rate, then the DSCR
requirement could be improved by simply lowering the interest rate. At a 5% interest rate, the DSCR dramatically improves to 1.24x.
This also works in reverse. In a low-interest rate environment, abnormally low rates present future refinance risk if the rates return to a more normalized level at the end of the loan term. For
example, suppose a short-term loan was originally approved at 5%, but at the end of a 3-year term rates were now up to 7%. As you can see, this could present significant challenges when it comes to
refinancing the debt. The debt yield can provide a static measure of risk that is independent of the interest rate. As shown above, it is still 9% for this loan.
Market valuation, amortization period, and interest rates are in part driven by market conditions. So, what happens when the market inflates values and banks begin competing on loan terms such as
interest rate and amortization period? The loan request can still make it through underwriting, but will become much riskier if the market reverses course. The debt yield is a measure that doesn’t
rely on any of these variables and therefore can provide a standardized measure of risk.
Using Debt Yield To Measure Relative Risk
Suppose we have two different loan requests, and both require a 1.20x DSCR and an 80% LTV. How do we know which one is riskier?
As you can see, both loans have identical structures with a 1.20x DSCR and an 80% LTV ratio, except the first loan has a lower cap rate and a lower interest rate. With all of the above variables, it
can be hard to quickly compare the risk between these two loans. However, by using the debt yield, we can quickly get an objective measure of risk by only looking at NOI and the loan amount:
As you can see, the first loan has a lower debt yield and is therefore riskier according to this measure. Intuitively, this makes sense because both loans have the same NOI, except the total loan
amount for Loan 1 is $320,000 higher than Loan 2. In other words, for every dollar of loan proceeds, Loan 1 has just 7.6 cents of cash flow versus Loan 2 which has 10.04 cents of cash flow.
This means that there is a larger margin of safety with Loan 2, since it has higher cash flow for the same loan amount. Of course, underwriting and structuring a loan is much deeper than just a
single ratio. There are other factors that the debt yield can’t consider such as guarantor strength, supply and demand conditions, property condition, strength of tenants, etc. However, the debt
yield is a useful ratio to understand, and it’s being utilized by lenders more frequently since the financial crash in 2008.
The debt service coverage ratio and the loan to value ratio have traditionally been used (and will continue to be used) to underwrite commercial real estate loans. However, the debt yield can provide
an additional measure of credit risk that isn’t dependent on the market value, amortization period, or interest rate. These three factors are critical inputs into the DSCR and LTV ratios, but are
subject to manipulation and volatility. The debt yield on the other hand uses net operating income and total loan amount, which provides a static measure of credit risk, regardless of the market
value, amortization period, or interest rate. In this article, we looked at the debt yield calculation, discussed how it compares to the DSCR and the LTV ratios, and finally looked at an example of
how the debt yield can provide a relative measure of risk.
Source: How to Calculate The Debt Yield Ratio
Randolph is a Multifamily Investment Sales Broker with eXp Commercial servicing Multifamily Buyers and Sellers in the Greater Chicago Area.
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Split connectivity matrix into subpopulations
splitConnMat {ConnMatTools} R Documentation
Split connectivity matrix into subpopulations
This function tries to optimally split a given subpopulation into two smaller subpopulations.
tries = 5,
threshold = 1e-10,
alpha = 0.1,
maxit = 500
indices vector of indices of sites in a subpopulation
conn.mat a square connectivity matrix. This matrix has typically been normalized and made symmetric prior to using this function.
beta controls degree of splitting of connectivity matrix, with larger values generating more subpopulations.
tries how many times to restart the optimization algorithm. Defaults to 5.
threshold controls when to stop each "try". Defaults to 1e-10.
alpha controls rate of convergence to solution
maxit Maximum number of iterations to perform per "try".
List with one or two elements, each containing a vector of indices of sites in a subpopulations
David M. Kaplan dmkaplan2000@gmail.com
Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.
See Also
See also optimalSplitConnMat, recSplitConnMat, subpopsVectorToList
version 0.3.5
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Overview of the Student Statistics Subpackage
Calling Sequence Reference Tables
Description Getting Help with a Command in the Package
Quantities Example Worksheet
Hypothesis Testing Compatibility
Interactive exploration
See Also
Calling Sequence
command(arguments) Statistics/
• The Student:-Statistics subpackage is designed to help teachers present and students understand the basic material of a standard course RandomVariable
in statistics. There are three components to the subpackage: quantities (including visualization and formulas), hypothesis testing, and
interactive exploration. These components are described in the following sections.
• Each command in the Student:-Statistics subpackage can be accessed by using either the long form or the short form of the command name in
the command calling sequence.
The long form, Student:-Statistics:-command, is always available. The short form can be used after loading the package.
• Most of the commands and tutors in the Student:-Statistics package can be accessed through the context menu. These commands are
consolidated under the Student:-Statistics name.
• In this subpackage, we can work with random variables taken from various distributions:
BernoulliRandomVariable BetaRandomVariable BinomialRandomVariable CauchyRandomVariable ChiSquareRandomVariable
DiscreteUniformRandomVariable EmpiricalRandomVariable ExponentialRandomVariable FRatioRandomVariable GammaRandomVariable
GeometricRandomVariable HypergeometricRandomVariable LogNormalRandomVariable NegativeBinomialRandomVariable NormalRandomVariable
PoissonRandomVariable StudentTRandomVariable UniformRandomVariable
• For more information, see RandomVariable.
• We can also work with data samples. Data samples within this subpackage are of the form of list, Vector, or Matrix.
• Data stored in a list and data stored in a Vector are interpreted the same way. For example, [1,2,3] and <1,2,3> represent the same data
sample. For data stored in a Matrix, we treat the Matrix as a collection of data samples where each column of the Matrix represents an
individual list or Vector data sample.
• We can use the following commands to work with these random variables and data samples:
Correlation Covariance DataSummary Decile ExpectedValue InterquartileRange
Kurtosis Mean Median Mode Moment Percentile
Quantile Quartile Skewness StandardDeviation Variance
• These commands only work for random variables:
CDF MGF Probability PDF ProbabilityFunction Sample
• Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
• To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot)
where $X$ is a $\mathrm{BinomialRandomVariable}⁡\left(5,\frac{1}{2}\right)$.
• Commands able to generate such a visualization are:
CDF DataSummary Decile ExpectedValue InterquartileRange Mean
Median Mode PDF ProbabilityFunction Percentile Quantile
Quartile Sample StandardDeviation
• Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the
command on a random variable (or an expression involving random variables) and specify the optional parameter inert. For example, define
Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
• Commands that are able to return a formula are:
CDF Correlation Covariance ExpectedValue Kurtosis Mean
Moment MGF Probability PDF ProbabilityFunction Skewness
StandardDeviation Variance
Hypothesis Testing
• This section contains commands for some commonly used hypothesis tests.
• The command TestsGuide can be used to select an appropriate test from among the options Maple offers. It explains the assumptions the
test makes and the hypothesis it tests.
For more information, see HypothesisTest.
Interactive exploration
• To explore a random variable interactively, you can use the ExploreRV command. It takes a random variable, or an expression involving one
or more random variables, and inserts sliders and graphs into the current worksheet, so that you can see how many quantities and the PDF
and CDF change when you change the parameters of the distribution.
Reference Tables
• The Student Statistics package also contains two commands to generate and read values from statistical reference tables.
CriticalTable ProbabilityTable
Getting Help with a Command in the Package
• To display the help page for a particular Student:-Statistics command, see Getting Help with a Command in a Package.
Example Worksheet
For introductory examples, see Statistics Example Worksheet.
• The Student:-Statistics package was introduced in Maple 18.
• For more information on Maple 18 changes, see Updates in Maple 18.
Calling Sequence Reference Tables
Description Getting Help with a Command in the Package
Quantities Example Worksheet
Hypothesis Testing Compatibility
Interactive exploration
See Also
Calling Sequence
command(arguments) Statistics/
• The Student:-Statistics subpackage is designed to help teachers present and students understand the basic material of a standard course RandomVariable
in statistics. There are three components to the subpackage: quantities (including visualization and formulas), hypothesis testing, and
interactive exploration. These components are described in the following sections.
• Each command in the Student:-Statistics subpackage can be accessed by using either the long form or the short form of the command name in
the command calling sequence.
The long form, Student:-Statistics:-command, is always available. The short form can be used after loading the package.
• Most of the commands and tutors in the Student:-Statistics package can be accessed through the context menu. These commands are
consolidated under the Student:-Statistics name.
• In this subpackage, we can work with random variables taken from various distributions:
BernoulliRandomVariable BetaRandomVariable BinomialRandomVariable CauchyRandomVariable ChiSquareRandomVariable
DiscreteUniformRandomVariable EmpiricalRandomVariable ExponentialRandomVariable FRatioRandomVariable GammaRandomVariable
GeometricRandomVariable HypergeometricRandomVariable LogNormalRandomVariable NegativeBinomialRandomVariable NormalRandomVariable
PoissonRandomVariable StudentTRandomVariable UniformRandomVariable
• For more information, see RandomVariable.
• We can also work with data samples. Data samples within this subpackage are of the form of list, Vector, or Matrix.
• Data stored in a list and data stored in a Vector are interpreted the same way. For example, [1,2,3] and <1,2,3> represent the same data
sample. For data stored in a Matrix, we treat the Matrix as a collection of data samples where each column of the Matrix represents an
individual list or Vector data sample.
• We can use the following commands to work with these random variables and data samples:
Correlation Covariance DataSummary Decile ExpectedValue InterquartileRange
Kurtosis Mean Median Mode Moment Percentile
Quantile Quartile Skewness StandardDeviation Variance
• These commands only work for random variables:
CDF MGF Probability PDF ProbabilityFunction Sample
• Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
• To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot)
where $X$ is a $\mathrm{BinomialRandomVariable}⁡\left(5,\frac{1}{2}\right)$.
• Commands able to generate such a visualization are:
CDF DataSummary Decile ExpectedValue InterquartileRange Mean
Median Mode PDF ProbabilityFunction Percentile Quantile
Quartile Sample StandardDeviation
• Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the
command on a random variable (or an expression involving random variables) and specify the optional parameter inert. For example, define
Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
• Commands that are able to return a formula are:
CDF Correlation Covariance ExpectedValue Kurtosis Mean
Moment MGF Probability PDF ProbabilityFunction Skewness
StandardDeviation Variance
Hypothesis Testing
• This section contains commands for some commonly used hypothesis tests.
• The command TestsGuide can be used to select an appropriate test from among the options Maple offers. It explains the assumptions the
test makes and the hypothesis it tests.
For more information, see HypothesisTest.
Interactive exploration
• To explore a random variable interactively, you can use the ExploreRV command. It takes a random variable, or an expression involving one
or more random variables, and inserts sliders and graphs into the current worksheet, so that you can see how many quantities and the PDF
and CDF change when you change the parameters of the distribution.
Reference Tables
• The Student Statistics package also contains two commands to generate and read values from statistical reference tables.
CriticalTable ProbabilityTable
Getting Help with a Command in the Package
• To display the help page for a particular Student:-Statistics command, see Getting Help with a Command in a Package.
Example Worksheet
For introductory examples, see Statistics Example Worksheet.
• The Student:-Statistics package was introduced in Maple 18.
• For more information on Maple 18 changes, see Updates in Maple 18.
• The Student:-Statistics subpackage is designed to help teachers present and students understand the basic material of a standard course in statistics. There are three components to the subpackage:
quantities (including visualization and formulas), hypothesis testing, and interactive exploration. These components are described in the following sections.
• Each command in the Student:-Statistics subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
The long form, Student:-Statistics:-command, is always available. The short form can be used after loading the package.
• Most of the commands and tutors in the Student:-Statistics package can be accessed through the context menu. These commands are consolidated under the Student:-Statistics name.
• The Student:-Statistics subpackage is designed to help teachers present and students understand the basic material of a standard course in statistics. There are three components to the subpackage:
quantities (including visualization and formulas), hypothesis testing, and interactive exploration. These components are described in the following sections.
The Student:-Statistics subpackage is designed to help teachers present and students understand the basic material of a standard course in statistics. There are three components to the subpackage:
quantities (including visualization and formulas), hypothesis testing, and interactive exploration. These components are described in the following sections.
• Each command in the Student:-Statistics subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
Each command in the Student:-Statistics subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
The long form, Student:-Statistics:-command, is always available. The short form can be used after loading the package.
• Most of the commands and tutors in the Student:-Statistics package can be accessed through the context menu. These commands are consolidated under the Student:-Statistics name.
Most of the commands and tutors in the Student:-Statistics package can be accessed through the context menu. These commands are consolidated under the Student:-Statistics name.
• In this subpackage, we can work with random variables taken from various distributions:
BernoulliRandomVariable BetaRandomVariable BinomialRandomVariable CauchyRandomVariable ChiSquareRandomVariable
DiscreteUniformRandomVariable EmpiricalRandomVariable ExponentialRandomVariable FRatioRandomVariable GammaRandomVariable
GeometricRandomVariable HypergeometricRandomVariable LogNormalRandomVariable NegativeBinomialRandomVariable NormalRandomVariable
PoissonRandomVariable StudentTRandomVariable UniformRandomVariable
• For more information, see RandomVariable.
• We can also work with data samples. Data samples within this subpackage are of the form of list, Vector, or Matrix.
• Data stored in a list and data stored in a Vector are interpreted the same way. For example, [1,2,3] and <1,2,3> represent the same data sample. For data stored in a Matrix, we treat the Matrix as
a collection of data samples where each column of the Matrix represents an individual list or Vector data sample.
• We can use the following commands to work with these random variables and data samples:
Correlation Covariance DataSummary Decile ExpectedValue InterquartileRange
Kurtosis Mean Median Mode Moment Percentile
Quantile Quartile Skewness StandardDeviation Variance
• These commands only work for random variables:
CDF MGF Probability PDF ProbabilityFunction Sample
• Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
• To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot) where $X$ is a $\mathrm{BinomialRandomVariable}&
• Commands able to generate such a visualization are:
CDF DataSummary Decile ExpectedValue InterquartileRange Mean
Median Mode PDF ProbabilityFunction Percentile Quantile
Quartile Sample StandardDeviation
• Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the command on a random variable (or an expression involving
random variables) and specify the optional parameter inert. For example, define Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
• Commands that are able to return a formula are:
CDF Correlation Covariance ExpectedValue Kurtosis Mean
Moment MGF Probability PDF ProbabilityFunction Skewness
StandardDeviation Variance
• In this subpackage, we can work with random variables taken from various distributions:
In this subpackage, we can work with random variables taken from various distributions:
BernoulliRandomVariable BetaRandomVariable BinomialRandomVariable CauchyRandomVariable ChiSquareRandomVariable
DiscreteUniformRandomVariable EmpiricalRandomVariable ExponentialRandomVariable FRatioRandomVariable GammaRandomVariable
GeometricRandomVariable HypergeometricRandomVariable LogNormalRandomVariable NegativeBinomialRandomVariable NormalRandomVariable
PoissonRandomVariable StudentTRandomVariable UniformRandomVariable
• We can also work with data samples. Data samples within this subpackage are of the form of list, Vector, or Matrix.
We can also work with data samples. Data samples within this subpackage are of the form of list, Vector, or Matrix.
• Data stored in a list and data stored in a Vector are interpreted the same way. For example, [1,2,3] and <1,2,3> represent the same data sample. For data stored in a Matrix, we treat the Matrix as
a collection of data samples where each column of the Matrix represents an individual list or Vector data sample.
Data stored in a list and data stored in a Vector are interpreted the same way. For example, [1,2,3] and <1,2,3> represent the same data sample. For data stored in a Matrix, we treat the Matrix as a
collection of data samples where each column of the Matrix represents an individual list or Vector data sample.
• We can use the following commands to work with these random variables and data samples:
We can use the following commands to work with these random variables and data samples:
Correlation Covariance DataSummary Decile ExpectedValue InterquartileRange
Kurtosis Mean Median Mode Moment Percentile
Quantile Quartile Skewness StandardDeviation Variance
• Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
• To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot) where $X$ is a $\mathrm{BinomialRandomVariable}⁡
• Commands able to generate such a visualization are:
CDF DataSummary Decile ExpectedValue InterquartileRange Mean
Median Mode PDF ProbabilityFunction Percentile Quantile
Quartile Sample StandardDeviation
• Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
Some of the commands mentioned above are designed to give a visual demonstration to help users better understand the material.
• To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot) where $X$ is a $\mathrm{BinomialRandomVariable}⁡\
To return a plot from one of the commands listed below, specify the optional parameter output=plot, for example: CDF(X,2,output=plot) where $X$ is a $\mathrm{BinomialRandomVariable}⁡\
CDF DataSummary Decile ExpectedValue InterquartileRange Mean
Median Mode PDF ProbabilityFunction Percentile Quantile
Quartile Sample StandardDeviation
• Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the command on a random variable (or an expression involving random
variables) and specify the optional parameter inert. For example, define Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
• Commands that are able to return a formula are:
CDF Correlation Covariance ExpectedValue Kurtosis Mean
Moment MGF Probability PDF ProbabilityFunction Skewness
StandardDeviation Variance
• Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the command on a random variable (or an expression involving random
variables) and specify the optional parameter inert. For example, define Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
Some of the commands can give a formula for the quantity they compute, rather than the end result. To obtain such a formula, use the command on a random variable (or an expression involving random
variables) and specify the optional parameter inert. For example, define Y := PoissonRandomVariable(lambda) and then compute Variance(Y, inert).
• Commands that are able to return a formula are:
CDF Correlation Covariance ExpectedValue Kurtosis Mean
Moment MGF Probability PDF ProbabilityFunction Skewness
StandardDeviation Variance
Hypothesis Testing
• This section contains commands for some commonly used hypothesis tests.
• The command TestsGuide can be used to select an appropriate test from among the options Maple offers. It explains the assumptions the test makes and the hypothesis it tests.
For more information, see HypothesisTest.
• This section contains commands for some commonly used hypothesis tests.
This section contains commands for some commonly used hypothesis tests.
• The command TestsGuide can be used to select an appropriate test from among the options Maple offers. It explains the assumptions the test makes and the hypothesis it tests.
The command TestsGuide can be used to select an appropriate test from among the options Maple offers. It explains the assumptions the test makes and the hypothesis it tests.
Interactive exploration
• To explore a random variable interactively, you can use the ExploreRV command. It takes a random variable, or an expression involving one or more random variables, and inserts sliders and graphs
into the current worksheet, so that you can see how many quantities and the PDF and CDF change when you change the parameters of the distribution.
• To explore a random variable interactively, you can use the ExploreRV command. It takes a random variable, or an expression involving one or more random variables, and inserts sliders and graphs
into the current worksheet, so that you can see how many quantities and the PDF and CDF change when you change the parameters of the distribution.
To explore a random variable interactively, you can use the ExploreRV command. It takes a random variable, or an expression involving one or more random variables, and inserts sliders and graphs into
the current worksheet, so that you can see how many quantities and the PDF and CDF change when you change the parameters of the distribution.
Reference Tables
• The Student Statistics package also contains two commands to generate and read values from statistical reference tables.
CriticalTable ProbabilityTable
• The Student Statistics package also contains two commands to generate and read values from statistical reference tables.
The Student Statistics package also contains two commands to generate and read values from statistical reference tables.
Getting Help with a Command in the Package
• To display the help page for a particular Student:-Statistics command, see Getting Help with a Command in a Package.
• To display the help page for a particular Student:-Statistics command, see Getting Help with a Command in a Package.
To display the help page for a particular Student:-Statistics command, see Getting Help with a Command in a Package.
• The Student:-Statistics package was introduced in Maple 18.
• For more information on Maple 18 changes, see Updates in Maple 18.
• For more information on Maple 18 changes, see Updates in Maple 18.
For more information on Maple 18 changes, see Updates in Maple 18.
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PPT - Zeroing in on the Difference Between Two Numbers - Math Puzzle
Zeroing in on the Difference Between Two Numbers - Math Puzzle
This math puzzle involves finding the differences between numbers and placing them in a sequence, working towards all differences adding up to zero. The process is illustrated step by step until
reaching the final result. It's a fun and challenging exercise to test your mathematical skills.
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presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
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differential equations
[y1,,yN] = vpasolve(eqns,vars) numerically solves the system of equations eqns for the variables vars. This syntax assigns the solutions to the variables y1,,yN. If you do not specify vars, vpasolve
solves for the default variables determined by symvar.
What you are outlining in your question (parallel) are so-called coupled differential equations. x1 and x2 - or rather, their time derivatives - are functions of each other. The only way to solve
these kinds of equations is by solving them, as you said, in parallel. And that's accomplished in MATLAB by using e.g. ode45.
The MATLAB ODE solvers do not accept symbolic expressions as an input. Therefore, before you can use a MATLAB ODE solver to solve the system, you must convert that system to a MATLAB function.
Generate a MATLAB function from this system of first-order differential equations using matlabFunction with V as an input. Solve numerically a system of first-order differential equations; Attempt to
execute SCRIPT dsolve as a function: How to solve a system of differential equations where one equation contains the answer to another; Solving a System of ODEs; Solving coupled 2nd order
differential equations; Using ODE45 to solve two coupled second order ODEs The system. Consider the nonlinear system. dsolve can't solve this system. I need to use ode45 so I have to specify an
initial value.
If you want to gain confidence in solving real-world problems in MATLAB a MATLAB code which solves the advection partial differential equation (PDE) dudt A well-working numerical algorithm (method
of lines) was applied for solving the reactor model with Matlab 7.1 and the results followed experimental trends very well. The aim was to illustrate how these parabolic partial differential
equations over-determined system om equations. ⎡. ⎢. ⎢. ⎣. 1 We consider the differential equation y + xy + x2 y = 1 − x2.
This example shows you how to convert a second-order differential equation into a system of differential equations that can be solved using the numerical solver ode45 of MATLAB®.
Solve this system of linear first-order differential equations. d u d t = 3 u + 4 v, d v d t = − 4 u + 3 v. First, represent u and v by using syms to create the symbolic functions u (t) and v (t).
syms u (t) v (t) Define the equations using == and represent differentiation using the diff function.
and an equation-oriented approach to generate numerical results Delivers a systems, two-point boundary value problems and partial differential equations av E Bahceci · 2014 — stable numerical
solution using a high-order finite difference method. dispersive models since linear and non-linear partial differential equations Using Matlab the ary conditions for finite-difference schemes
solving hyperbolic systems:.
Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, systems of linear differential
equations, the properties of solutions t
Organize and share your learning with Class The laws of supply and demand help to determine what the market wants and how much. These laws are reflected in the prices paid in everyday life. These
prices are set using equations that determine how many items to make and whether to rais Customers don't care about a 'buyer's journey' or your internal org chart. They only care about their success
-- and that's what you should care the most about, too. Overview of all products Overview of HubSpot's free tools Marketing automa The key to happiness could be low expectations — at least, that is
the lesson from a new equation that researchers used to predict how happy someone would be in the future. In a new study, researchers found that it didn't matter so much whe Study of ordinary
differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, systems of linear differential equations, the
properties of solutions t Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients,
systems of linear differential equations, the properties of solutions t Numerical methods for ordinary differential equations are methods used to find numerical Numerical methods for solving
first-order IVPs often fall into one of two large categories: linear multistep methods, Quantized state systems deal with the large, complicated, and nonlinear systems of equations seen in practice.
Consider the nonlinear system. dsolve can't solve this system. I need to use ode45 so I have to specify an initial value. Solution using ode45. This is the three dimensional analogue of Section
14.3.3 in Differential Equations with MATLAB. Use MATLAB® to numerically solve ordinary differential equations.
Erik haag kalle zackari
This page contains two examples of solving stiff ordinary differential equations using ode15s.MATLAB® has four solvers designed for stiff ODEs. What you are outlining in your question (parallel) are
so-called coupled differential equations. x1 and x2 - or rather, their time derivatives - are functions of each other. The only way to solve these kinds of equations is by solving them, as you said,
in parallel.
Hero Images/Getty Images Early algebra requires working with polynomials and the four opera Learn how to use linear algebra and MATLAB to solve large systems of differential equations.
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The key to happiness could be low expectations — at least, that is the lesson from a new equation that researchers used to predict how happy someone would be in the future. In a new study,
researchers found that it didn't matter so much whe
d u d t = 3 u + 4 v, d v d t = − 4 u + 3 v. First, represent u and v by using syms to create the symbolic functions u (t) and v (t). syms u (t) v (t) Define the equations using == and represent
differentiation using the diff function. MATLAB: Numerically Solving a System of Differential Equations Using a First-Order Taylor Series Approximation. event function guidance MATLAB numerical
solutions ode's ode45 plotting second order ode system of differential equations system of second order differential equations taylor series • Matlab has several different functions (built-ins) for
the numerical solution of ODEs.
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Error sending composing message to Sigfox
Error sending composing message to Sigfox
I'm using this way of compose the message to send to Sigfox
s.send(struct.pack("<f", sbatt)+struct.pack("<f", stemp)+struct.pack("<B", shum)+struct.pack("<B", spress)+struct.pack("<f", captemp21)+struct.pack("<B", caphum21))
but I recieved this error:
OSError: [Errno 122] EMSGSIZ
What I'm doing wrong???
Thank's for your help guys
Hi @robert-hh
PErfect, now is sending the info and now I'm decoding the info in my server...all works perfect!! thanks for your help
@ecabanas See https://forum.pycom.io/topic/6564/how-build-sigfox-message/9
It depends on the range and the precision you need.
For instance for the battery level, if you only need one decimal:
□ Multiply by 10. You now have a number between 33 and 47
□ Subtract 33. You now have a number between 0 and 14.
□ Convert to integer to truncate any additional digits
□ You now have a number than fits in 4 bits!
If you need two decimals:
□ Multiply by 100. You now have a number between 330 and 470
□ Subtract 330. You now have a number between 0 and 140.
□ Convert to integer to truncate any additional digits
□ You now have a number than fits in 8 bits (one byte).
Same for the temperature, depending on how many significant digits you need, you could store it as:
□ 0 to 70 (add 20) -> fits in 7 bits,
□ or 0 to 700 (multiply by 10, add 200) -> fits in 10 bits
And so on.
Even numbers that are already integers can use less space. shum only need 7 bits. spress only needs 8 bits.
Multiples of 8 bits are easy to manipulate and store with ustruct.pack. If you want to optimise every single bit (which when you can only send 96 bits at a time can be really useful), you'll
either have to do some bit-shifting and ORing, or you can use something like the micropython-bitstring package.
• robert-hh last edited by robert-hh
@ecabanas For instance instead of
struct.pack("<f", sbatt)
you could write:
struct.pack("<h", int(sbatt * 100))
Obviously you would receive a value 100 times larger than the initial value, so you have to scale that back (divide by 100)
You would have to do than anyhow for pressure, which has initially the range of 900-1100. Since a byte has the range of 0-255, you would need something like:
struct.pack("<B", spress - 900)
Note, that you can put that all into a single struct instruction like:
struct.pack("<hhBhhB", int(sbatt * 100), int(stemp * 100), shum, spress, int(captemp21 * 100), caphum21)
@robert-hh said in Error sending composing message to Sigfox:
If you scale all of them by a factor
Hi @robert-hh
How do I do this? I'm sorry but I'm very new at this and I don't have the slightest idea about this subject
Thank you very much
@ecabanas Depending on the required resolution you can scale the floats to int, saving 2 bytes or 3 bytes depending on the resolution. If you scale all of them by a factor of 100, they would
still fit into a 16 bit int, with a resolution of 10mV or 0.01 degree, which may be sufficient.
Hi @jcaron
Thank's for your reply, this is the worst part of my code, and allways is a nightmare.
Any idea how to reduce them? I have no idea.
here it is the data perimeter:
sbatt: from 3,3 to 4,7 (float)
stemp: from -20 to 50 (float)
shum: from 0 to 100 (int)
spress: from 900 to 1100 (int)
captemp21: from -20 to 50 (float)
caphum21: from 0 to 100 (int)
@ecabanas SigFox messages are limited to 12 bytes. I believe yours is 15 bytes long.
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Baccarat Rules
Nov 15 2019
Baccarat Policies
Baccarat is played with 8 decks of cards in a shoe. Cards than are of a value less than ten are of their printed number whereas 10, J, Q, K are 0, and A are each equal to 1. Bets are placed on the
‘banker,’ the ‘player’ or for a tie (these aren’t actual players; they simply symbolize the 2 hands to be given out).
Two hands of 2 cards will now be given out to the ‘banker’ as well as ‘player’. The total for any hand is the sum total of the two cards, but the first digit is dropped. For e.g., a hand of 7 as well
as 5 gives a score of 2 (7plus5=twelve; drop the ‘1′).
A 3rd card can be dealt depending on the following codes:
- If the player or banker has a score of 8 or nine, both bettors stand.
- If the player has 5 or less, he/she hits. Players stand otherwise.
- If player stands, the banker hits of five or less. If the bettor hits, a chart is used in order to decide if the banker stands or hits.
Baccarat Odds
The greater of the two scores will be the winner. Successful bets on the banker pay nineteen to 20 (even money minus a five % commission. Commission is followed closely and paid out when you leave
the table so make sure to have $$$$$ left over before you leave). Winning bets on the player pay one to one. Winning bets for tie typically pays out at 8 to 1 and occasionally nine to one. (This is
not a good bet as ties occur less than one every 10 hands. abstain from wagering on a tie. Nevertheless odds are significantly better – 9 to 1 vs. 8 to 1)
When played correctly, baccarat offers relatively decent odds, aside from the tie bet of course.
Baccarat Strategy
As with just about all games, Baccarat has some well-known misconceptions. One of which is close to a roulette myth. The past is in no way a predictor of future outcomes. Monitoring of old results on
a chart is simply a waste of paper … a slap in the face for the tree that gave its life to be used as our stationary.
The most established and possibly most successful method is the one-three-two-six method. This tactic is employed to accentuate winnings and cutting back risk.
commence by betting 1 unit. If you win, add one more to the two on the table for a total of three on the 2nd bet. If you win you will have 6 on the table, take away four so you have two on the 3rd
wager. If you win the 3rd gamble, add 2 to the four on the table for a grand total of 6 on the 4th gamble.
If you don’t win on the initial bet, you suck up a loss of one. A win on the 1st bet followed by loss on the 2nd causes a loss of two. Wins on the 1st two with a loss on the third gives you a profit
of 2. And wins on the first three with a loss on the fourth mean you break even. Coming away with a win on all four bets leaves you with 12, a profit of ten. Therefore that you can fail to win the
second bet 5 times for every successful streak of 4 bets and still break even.
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Radius-dependent homogeneous strain in uncoalesced GaN nanowires
Jean-François Molinari, Roozbeh Rezakhani
The state of stress in plates, where one geometric dimensions is much smaller than the others, is often assumed to be of plane stress. This assumption is justified by the fact that the out-of-plane
stress components are zero on the free-surfaces of a plate ...
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Average for Nested Bar instead of Total, or another option?
Good morning. We'd like to show the average of the two bars in this card, and ideally it would be using a Nested Bar chart but open to other suggestions (currently using a Grouped Bar chart).
The Value is a beast mode, which takes into consideration the actual sale amount less the valuation, divided by the valuation: (CASE WHEN `Valuation` = 0 THEN 0 ELSE (SUM(`SaleAmount` - `Valuation`))
/ SUM(`Valuation`) END)
The Series consists ot the two (2) stores, the bars in the cart would be Store 1 and Store 2.
Any thoughts on how we can achieve this, i.e. an average bar, or line, to show the average of both Valuation to Sale Amount Variances by Store? Thanks
• Have you tried utilizing the Line+nested bar chart type?
In this example I've created a simple beast mode for an average of my values for A1, A2, and A3 divided by 3 and plugged that into the Y Axis dimension.
**Say "Thanks" by clicking the heart in the post that helped you.
**Please mark the post that solves your problem by clicking on "Accept as Solution"
• Thanks, @JasonAltenburg, believe that got us in the right direction!
Wat we've done is separate the "stores" out into their own columns (Store 1 & Store 2), and you can see from image 1 that only the row of data that applies to that store appears in the store
specific column that was created with a beast mode.
However, we're still not able to properly calculate the individual stores so their bar shows the correct variance seen in image 2 (top portion). The total is correct on the existing card (top)
and the new card (bottom), but you'll see the difference in the store variances and the new card (bottom) is incorrect.
Believe it may have something to do with the null values in the newly created columns, and have tried several beast modes to account for them but haven't been able to get it so they reconcile
with the individual store variances from the existing card (top).
This beast mode, which separates the stores into new columns, is where we also have to calcualte the correct percentage of 2.95% for each, and this is what we've come up with so far that brings
us closest to what we need but still not accurate.
WHEN `ParentStore` = 'Store 1' THEN (SUM(IFNULL(`Valuation To Sale Amount Variance`,0)) / COUNT(IFNULL(`Valuation To Sale Amount Variance`,0)))
Any additional assistance would be greatly appreciated!
• Any ideas on why we're unable to get the individual stores to reconcile? Is there anything additional that we could provide to help understand the ask better? Thank you in advance for any
assistance on this!
• Could you try this as your beastmode?
WHEN `ParentStore` = 'Store 1' THEN IFNULL(`Valuation To Sale Amount Variance`,0) end)
when `ParentStore`='Store 1' and ABS(ifnull(`Valuation To Sale Amount Variance`,0))>0 then 1 else 0 end)
I'm pretty sure that this expression will count the 0's as a value as well:
COUNT(IFNULL(`Valuation To Sale Amount Variance`,0)))
which is why the average was getting thrown off.
“There is a superhero in all of us, we just need the courage to put on the cape.” -Superman
• I did a little bit of testing to confirm my previous post:
I think it really depends on your expected result. I used three different beastmodes to test the outcomes of each.
1. count ifnull
This was testing the denominator in your original field. This does, indeed, count a value for every row even if there was a null in the dataset.
2. count()
This will include a count for any row with any value (exlcudes null values)
3. sum()
sum(case when ABS(IFNULL(`count`,0))>0 then 1 else 0 end)
This will tell you the number of rows with a value (excludes 0's and nulls)
“There is a superhero in all of us, we just need the courage to put on the cape.” -Superman
• @ST_-Superman-_would it be ok to DM you, maybe provide some of the raw data we're working with via Excel?
Still unable to get this to work as expected, not sure if it has to do with WHAT we're doing or maybe related to the information provided in this thread not being thorough enough to get the
correct feedback on a solution. Thanks!
This discussion has been closed.
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P(A ∩ B) - (Intro to Statistics) - Vocab, Definition, Explanations | Fiveable
P(A ∩ B)
from class:
Intro to Statistics
P(A ∩ B) represents the probability of the intersection of two events, A and B. It refers to the likelihood that both events A and B will occur simultaneously. This concept is crucial in
understanding the relationships between events and calculating probabilities in various statistical analyses.
congrats on reading the definition of P(A ∩ B). now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The probability of the intersection of two events, P(A ∩ B), is the probability that both events A and B will occur together.
2. P(A ∩ B) is calculated by multiplying the probability of event A, P(A), by the conditional probability of event B given that event A has occurred, P(B|A).
3. If events A and B are independent, then P(A ∩ B) = P(A) × P(B).
4. When events A and B are mutually exclusive, P(A ∩ B) = 0, as the occurrence of one event prevents the occurrence of the other.
5. The concept of P(A ∩ B) is fundamental in understanding and calculating probabilities in various statistical applications, such as decision-making, risk analysis, and hypothesis testing.
Review Questions
• Explain the relationship between P(A ∩ B) and independent events.
□ If events A and B are independent, then the probability of their intersection, P(A ∩ B), is equal to the product of their individual probabilities, P(A) and P(B). This is because the
occurrence of one event does not affect the probability of the other event when they are independent. In other words, for independent events, P(A ∩ B) = P(A) × P(B).
• Describe the relationship between P(A ∩ B) and mutually exclusive events.
□ When events A and B are mutually exclusive, meaning they cannot occur simultaneously, the probability of their intersection, P(A ∩ B), is equal to 0. This is because the occurrence of one
event completely prevents the occurrence of the other event. In the case of mutually exclusive events, the probability of their intersection is always zero, or P(A ∩ B) = 0.
• Analyze how the concept of P(A ∩ B) can be applied in statistical decision-making and hypothesis testing.
□ The understanding of P(A ∩ B) is crucial in statistical decision-making and hypothesis testing. For example, in hypothesis testing, the probability of the intersection of the null hypothesis
(H0) and the alternative hypothesis (H1), P(H0 ∩ H1), should be zero, as the two hypotheses are mutually exclusive. Additionally, in risk analysis and decision-making, the probability of the
intersection of multiple events, P(A ∩ B ∩ C), can be used to assess the likelihood of complex scenarios and inform strategic decisions.
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