content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Players take turns to select adjacent pieces of the
circular frame. The last
player who is able
to select an
adjacent pair
The word adjacent
means 'next to'. In this
case an adjacent piece of
the circular frame is one that is next
to the piece you selected for the first of each pair of counters you place.
Click on a link below to explore other parts of the Transum web site:
Fun Maths Home Transum Software Maths Map Times Tables Strategy Games Go Maths
Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your | {"url":"https://www.transum.org/Software/Fun_Maths/Games/Twins.asp","timestamp":"2024-11-08T05:59:19Z","content_type":"text/html","content_length":"24183","record_id":"<urn:uuid:ede9e040-4f7d-4bac-a792-649029fb7c7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00089.warc.gz"} |
Top Ten Lists of Common (Student) Math Errors!
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math
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If you enjoy this post, you may want to read a newer post on developing Ratio Sense for Middle Schoolers.]
In truth, we could drop the 'student' from the title and simply enumerate common math errors or, even better, fallacies. The latter term is more the flavor of this post, rather than simply careless
student errors. This is because fallacies imply that there is a plausibility to some of these errors, i.e., they are somewhat natural errors to make if one doesn't fully grasp the ideas. Experienced
math educators anticipate these errors and caution students about them during the lesson, preferable to commenting on these errors on their tests! In fact, it is possible that students can learn
considerable mathematics by being asked to comment on these and explain each error.
IMO, helping students understand the underlying concept in each of these common mistakes is an essential part of teaching and learning mathematics. This is my approach in this post, rather than a "
Can you believe I saw a student do this once!" attitude.
There are many such lists easily found by searching the web although most refer to common college errors (in reality, they include many precollegiate errors). Here is one of the best from a wonderful
professor at Vanderbilt University. It is thorough, contains excellent discussion and categorizes the errors. Rather than copy from these sources, I wrote a few off the top of my head. I will also
include Eric's which he posted in a recent comment to the 16/64 = 1/4 post.
Eric's Excellent List (and I know he has a few hundred more):
1. √(a²+b²) = a+b
2. (-x)ⁿ = - xⁿ
3. (fg)ʹ = fʹgʹ
These are wonderful. #1 and #3 could be classified as 'everything distributes' errors, although in verbal form, it could be interpreted as an 'everything commutes' error:
"The derivative of a product is the product of the derivatives" error.
Similarly for #1:
"The radical of a sum is the sum of the radicals" error.
#2 is an order of operations type of error and I included this in my list in a particular form.
Here's my initial offering for Grades 7-12. Feel free to bring your own list to the table for other grade levels. Common errors in calculus and beyond are fair game as well. I will try to classify
some of these...
Radical Errors
√49 = ±7 type; similarly 16^1/2 = ±4
√(n^2) = n
Operation Errors (Exponents)
-4^2 = 16
a^n = b^n ↔ a = b
x^2/3 = 16 → x = 16^3/2 or 64
[One possible correct method: x^2/3 = 16 → x^2 = 16^3 → x = ±√(16^3) = ±64]
Fraction Errors - Algebraic or Arithmetic (limited to the top 10000 errors please!)
(12x+7)/(4x+9) = (3+7)/9
[What name would you give this one? 'Cancelling error', 'Cancelling terms not factors error'?]
Rather than continue my list, I welcome offerings from our readers. A fairly thorough compilation of these could become a book! Perhaps, in a serious vein, a small monograph that could be helpful to
both students and math educators...
12 comments:
In particular, the error
(f/g)' = f'/g'
seems to increase after the students get exposed to L'Hospital's rule.
I'm not a math teacher but dividing
and multiplying inequalities with
negative values is often a source of
-2 > 2x --> 1 > -x
Another one:
Cancel (x-2) from both sides, and get x=-2.
So is the error of using l'Hôpital's rule without checking that the numerator and denominator have limits 0 (or ∞).
for one who is not a math teacher, you found a 'classic!'
tc, eric--
We could have an entire section devoted to L'Hopital's and limits in general!
A classic: (1+1/n)^n → 1
Ah, those fun indeterminate forms!
Keep 'em coming folks.
This list should be endless...
Simplifying errors with radicals:
((6)^.5)/2= 3^.5
Luv it!
Sometimes, students simply don't simplify far enough. They will leave
an answer with √1 in their answer! And, of course, √0 is undefined.
Who is keeping track of these!
canceling part of a sum. Do it in my class, and all partial credit on that question is voided.
How about √(75) = 3√5 ?
But of course the most common student error is the magically disappearing minus sign...
good one, jonathan!
I attempted to prevent this common transposition by requiring that students write the extra step:
√25 ⋅ √3, but no instructional technique works with all students all the time! (It did work some of the time with some students...)
We could probably devote an entire book to negatives, which is why I often told my algebra students:
SUCCESS IN ALGEBRA = THINKING 'NEGATIVELY'!
∫_a^b f(g(x))g'(x) dx = ∫_a^b f(u) du.
Isn't the function f(x) = sqrt(x) defined at x=0??, i.e., sqrt(x) = 0.
Yes, √0 = 0. I meant that students think it's undefined! Sorry for my lack of clarity. I was listing common errors but I should have made that clearer so one doesn't take it as a fact. | {"url":"https://mathnotations.blogspot.com/2008/02/top-ten-lists-of-common-student-math.html","timestamp":"2024-11-08T15:32:18Z","content_type":"application/xhtml+xml","content_length":"188334","record_id":"<urn:uuid:93d2cafc-7219-4b1b-b3c4-32165a20ba53>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00456.warc.gz"} |
What is Apr on a Loan - Knittystash.com
What is Apr on a Loan
APR, or Annual Percentage Rate, is a comprehensive measure that reflects the true cost of a loan. It includes not only the interest rate but also additional fees and charges associated with the loan.
By considering all the costs over the life of the loan, APR provides a more accurate estimate of the total cost of borrowing. This makes it easier for borrowers to compare different loan options and
make informed decisions. APR is essential for understanding the actual cost of a loan and should always be considered when evaluating credit options.
Annual Percentage Rate (APR) Basics
The Annual Percentage Rate (APR) is a measure of the cost of borrowing money. It is expressed as a percentage and includes all fees and charges associated with the loan, such as origination fees,
closing costs, and interest.
APR is important because it allows you to compare the cost of different loans. A higher APR means that you will pay more interest over the life of the loan.
• How APR is calculated: APR is calculated by dividing the total finance charges by the amount borrowed and multiplying by 100.
• What APR includes: APR includes all fees and charges associated with the loan, such as origination fees, closing costs, and interest.
• Why APR is important: APR is important because it allows you to compare the cost of different loans. A higher APR means that you will pay more interest over the life of the loan.
Loan Type Typical APR Range
Personal loans 6% – 36%
Credit cards 12% – 29%
Mortgages 3% – 6%
APR on a Loan
Annual Percentage Rate (APR) is the yearly interest rate charged on a loan. It represents the total cost of borrowing money, including not just the stated interest rate but also any additional fees
or charges associated with the loan.
Calculating APR
APR is calculated using a complex formula that takes into account the following factors:
• Loan amount
• Loan term
• Interest rate
• Loan fees and charges
Using APR
APR is a valuable tool for comparing loan offers and making informed decisions. By comparing the APR of different loans, you can easily determine which loan offers the lowest overall cost of
Here’s how to use APR:
• Compare APRs from different lenders.
• Choose the loan with the lowest APR.
• Be aware that other factors, such as loan terms and fees, may also affect the overall cost of the loan.
As a general rule, a lower APR means a lower overall cost of borrowing. However, it’s important to consider all the details of a loan before making a decision.
Example of APR
│ Loan │Amount │ Term │Interest Rate │ APR │
│Personal Loan │$10,000│2 years│6.50% │6.99%│
│Auto Loan │$25,000│5 years│4.25% │4.54%│
In the table above, the personal loan has a slightly higher APR (6.99%) compared to the auto loan (4.54%). This is because the personal loan has a shorter term and higher loan fees.
APR vs. Interest Rate
When you borrow money, you’ll likely encounter two terms: annual percentage rate (APR) and interest rate. While these terms are often used interchangeably, they’re not the same thing.
**APR** is the cost of borrowing money expressed as a yearly percentage. It includes the interest rate plus any fees or charges associated with the loan. APR is a more comprehensive measure of the
cost of borrowing than the interest rate alone because it takes into account all of the costs associated with the loan.
**Interest rate** is the percentage of the loan amount that you’re charged for borrowing the money. It’s usually expressed as a yearly percentage. Interest rate is the cost of borrowing money without
taking into account any fees or charges.
Comparison of APR and Interest Rate
The following table compares APR and interest rate:
Feature APR Interest Rate
Definition Cost of borrowing money expressed as a yearly percentage, including fees and charges Percentage of the loan amount that you’re charged for borrowing the money
Includes Interest rate, fees, and charges Interest rate only
Purpose Provides a more comprehensive measure of the cost of borrowing Indicates the cost of borrowing money without taking into account fees and charges
APR’s Impact on Loan Repayments
Annual Percentage Rate (APR) is a comprehensive measure of the cost of a loan, including interest rates and any additional fees or charges. Understanding the impact of APR on loan repayments is
crucial for informed financial decision-making.
• Increased Monthly Payments: Higher APRs result in higher monthly loan payments, as the interest accrues at a faster rate.
• Longer Repayment Period: Loans with higher APRs may require longer repayment periods to keep monthly payments manageable.
• Increased Total Interest Paid: Over the life of the loan, higher APRs lead to paying more interest charges, increasing the overall cost of borrowing.
• Reduced Borrowing Power: Lenders consider APR when assessing loan applications. Higher APRs may lower the maximum loan amount you qualify for.
To illustrate the impact of APR, consider the following table:
Loan Amount APR Monthly Payment Total Interest Paid
$10,000 5% $200 $1,000
$10,000 10% $224 $2,240
In this example, a 5% APR difference on a $10,000 loan results in a $24 monthly payment increase and a $1,240 increase in total interest paid over the loan’s life.
When comparing loans, choosing the loan with the lowest APR will minimize your monthly payments, reduce the total interest paid, and allow you to pay off your debt faster.
Thanks for sticking with me, loan-seeker! By now, you should have a rock-solid understanding of APR. Remember, it’s the VIP (Very Important Percentage) that determines how much extra you’ll be paying
on your loan. Keep it low, and your finances will sing like a choir of angels. I’ll be here, ready to tackle any more money mysteries you throw my way. So bookmark me, share me with your mates, and
drop by again whenever the financial fog rolls in. Cheers to savvy borrowing! | {"url":"https://knittystash.com/what-is-apr-on-a-loan/","timestamp":"2024-11-04T09:25:56Z","content_type":"text/html","content_length":"115310","record_id":"<urn:uuid:41ffce4d-7e24-4786-b150-e02b3611d8cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00491.warc.gz"} |
openvms and xterm
2024-04-17 00:19:50 UTC
So, a really basic one here.
What's the current best practice to match term types between vms and ssh
clients? OpenVms doesn't see to understand a termtype of xterm, and I'm
not sure if it recognises termtypes from an (inbound session) openssh
config file.
It would probably be good for VSI to spend a bit of time documenting
this, maybe creating some client configs to try. | {"url":"https://comp.os.vms.narkive.com/XXUAm72v/openvms-and-xterm","timestamp":"2024-11-12T06:04:40Z","content_type":"text/html","content_length":"315167","record_id":"<urn:uuid:1062d5b6-2a36-4e65-8fac-c2161138fb61>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00819.warc.gz"} |
What Is Calculus And How Does It Help Students? | Hire Someone To Take My Proctored Exam
In math instruction, pre-calculus is typically a unit, or an entire course, which consists of algebra and calculus at a very high level that is intended to prepare students for the complex study of
calculus in higher studies. The terms calculus and algebra are used interchangeably to denote the same subject, the application of algebraic techniques to quantitative problems, where the solution to
the former involves the integration of the variables and the latter the use of trigonometric and other formulas. Different schools tend to differentiate between algebra and calculus as two different
subjects of the curriculum, while others simply treat them as a single course.
Calculus serves as an important tool in the analytical and numerical skills of every student who plans on taking calculus in the future. It is taught for its innate value in the development of
mathematical skills in students from pre-primary school through high school, but students also take this course to supplement their knowledge with a fundamental understanding of nature’s workings.
Some students will opt to learn calculus for its own sake and to sharpen their critical and deductive reasoning, while others will choose it as the foundation of a more complicated course of study
such as a calculus degree program.
Calculus enables the student to determine the relationship between different quantities and to formulate equations for predicting and solving for the results. This means that in addition to knowing
how many decimals are in a number, students can calculate their mass or weight by using calculus principles. Calculus is also instrumental in discovering relationships between different physical
quantities such as velocity, force, etc., and the relationship between these physical quantities and their relative proportions. Students who want to pursue further studies in their chosen field of
study can even use calculus to find the relationship among various fields of study and between the various properties of physical objects.
Calculus can be applied in a variety of contexts, including scientific investigation, engineering, statistics, and computer science. Many of the formulas involved in the study of physics and
chemistry are derived from calculus, as are most of the formulas involved in the study of mechanics and fluid mechanics. Many calculus course materials contain an introduction to the concepts of
algebraic equations and calculus itself.
Algebraic problems which can be solved using calculus include those involving the definition of constant values and solutions to linear equations. Calculus also solves equations concerning the ratio
problems, the power series, exponents of unknown functions, and various forms of polynomial equations.
Various types of formulas used in the study of algebra are the product rule, the chain rule, and the binomial rule. Calculus also provides tools for solving quadratic equations and geometric
problems. It is also used to determine the value of a particular variable, to find the roots of a function and to evaluate an unknown function by finding the greatest common divisor.
There are different types of pre-calculus courses that can be found on the Internet. Some of these are offered by individual colleges, while others are available online for free. Many sites will
offer both a full course and a part-course on the same subject. One of the more advanced types of pre-calculus courses is the calculus certificate program, which gives students a thorough
introduction to the concepts of the subject. Students who complete this type of course will gain a basic knowledge of algebraic problems and understand how to use algebraic formulas.
Other college courses include the Calculus I course, which will give students an overview of the subject and introduce them to the theory, before they venture into more advanced courses. The Calculus
II course is similar, except that it will provide students a more in-depth look at calculus, so that they can apply it to their personal lives. Calculus III courses will prepare students for college
level calculus. These courses will take a more hands-on approach to the subject and introduce students to advanced calculus concepts, such as the Taylor series and the complex numbers. | {"url":"https://crackmyproctoredexam.com/what-is-calculus-and-how-does-it-help-students/","timestamp":"2024-11-03T12:02:17Z","content_type":"text/html","content_length":"107043","record_id":"<urn:uuid:c6fb2de0-915c-4841-8490-8565526e1e17>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00482.warc.gz"} |
Standard and Variable Deviations : Basics Of Statistics For Machine Learning Engineers - Edureify-Blog
Standard and Variable Deviations : Basics Of Statistics For Machine Learning Engineers
The requirements and expectations of the businesses are expanding along with the global digitization process. To land a decent job, you must develop the best knowledge and abilities in the technology
industry, such as JAVA, Python, Ruby, C++, C#, etc. You may seize possibilities in the developing world and advance your career with Edureify, the greatest AI learning software.
Numerous online and cost-effective certified coding classes are available from Eduriefy. The lessons are filled with useful elements, and the subject-matter specialists have a lot of coding-related
Standard and Variance Deviation
In this post, we’ll examine the measurement of data variability in more detail. You’ll master the fundamentals of standard deviation and variance in the classes that follow. This piece follows the
last post in the series, where we learned how to quantify and depict data distribution.
Variance in Machine Learning
Variance is defined as the average of the squared deviations from the mean.
Let’s first look at some data set, where we have a list of 12 incomes, to better grasp what it represents.
There is an extreme value (an outlier) of 100,000 that raises the mean to 40,200 and widens the range to 175,000, however, the majority of the values are concentrated between 15,000 and 35,000.
Now, referring to the preceding concept, let’s compute the variance. The square of each point’s deviation from the mean will be added, multiplied by the number of values in the collection, and then
Commonly, the variance is denoted by the Greek letter Sigma squared (2). The following equation can be used to determine the variance.
Where n is the total number of terms in the set, is the mean, and x stands for each term in the set. You will learn more about this in the Bootcamp coding courses at Edureify.
Variance Example:-
Variance is another number that expresses how to extend (arrange) out the values
For example, if you take the square root of the variance, you get the standard variation!
The other way is, if you multiply the standard deviation by itself, you get the variance!
Find the mean:
(90+98+100+89+86+95+97) / 7 = 93.57
Find the difference from the mean for each value:
90 – 93.57 = -3.57
98 – 93.57 = 4.43
100 – 93.57 = 6.43
89 – 93.57 = -4.57
86 – 93.57 = -7.57
95 – 93.57 = 1.43
97 – 93.57 = 3.43
Find the square value:
(-3.57)2 = 12.7449
(4.43)2 = 19.62
(6.43)2 = 41.344
(-4.57)2 = 20.884
(-7.57)2 = 57.3049
(1.43)2 = 2.044
(3.43)2 = 11.7649
Find the variance of the average number of these squared differences:
(12.7449+19.62+41.344+20.884+57.3049+2.044+11.7649) / 7 = 23.672.
Standard Deviation in Machine Learning
Finding the standard deviation is rather simple after determining the variance. It is the variance’s square root. Recall that the variance is represented by the number 2? The symbol for the standard
deviation is denoted by π.
There is a faster approach to discovering the variance as well. Please verify the following equation.
Obs: You’ll see that I set the argument to zero and that I used it. Avoid the trouble. Delta Degrees of Freedom, or ddof, is what Python Pandas gives us the variance normalized by n — ddof, and ddof
is defined as 1 by default. If you had set ddof to zero, then the standard variation will be equal to zero
The definition of standard deviation-The standard deviation is a measure of how the values are distributed.
A low standard deviation indicates that the majority of the data are within a small range of the mean value or average value.
A high standard deviation indicates that the data fall within a wide range.
Example: This time, we recorded the ages of older individuals, but we only counted seven of them.
age = [86,87,88,86,87,85,86]
0.9 is the standard deviation.
The standard deviation and variance form a great part of the online coding courses at Edureify. You can enroll and know more about it through Edureify, the best AI learning app. The various machine
learning principles including Azure learning, Machine learning algorithms, No Code Learning, and A- Z statistics of machine learning have already been discussed by Eduriefy in earlier articles. You
can refer to all of them for overall knowledge.
Some Frequently Asked Questions
Q:- What are standard deviation and variance in machine learning?
Ans:- Variance. Variance is another number that indicates how spread out the values are. If you take the square root of the variance, you get the standard deviation! Or the other way around, if you
multiply the standard deviation by itself, you get the variance!
Q:- What is the difference between variation and standard deviation?
Ans:- Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in distribution, but their units differ:
Standard deviation is expressed in the same units as the original values (e.g., minutes or meters).
Q:- Does standard deviation mean more variation?
A:- Standard deviation is the square root of the variance. The variance helps determine the data’s spread size when compared to the mean value. As the variance gets bigger, more variation in data
values occurs, and there may be a larger gap between one data value and another.
Q:- Why do we need a variance and standard deviation?
A:- Variance helps to find the distribution of data in a population from a mean, and standard deviation also helps to know the distribution of data in a population, but standard deviation gives more
clarity about the deviation of data from a mean.
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An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (7 ,5 ) to (8 ,2 ) and the triangle's area is 27 , what are the possible coordinates of the triangle's third corner? | HIX Tutor
An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,5 )# to #(8 ,2 )# and the triangle's area is #27 #, what are the possible coordinates of
the triangle's third corner?
Answer 1
The coordinates are $\left(23.7 , 8.9\right)$ and $\left(- 8.7 , - 1.9\right)$
The length of side #A=sqrt((7-8)^2+(5-2)^2)=sqrt10#
Let the height of the triangle be #=h#
The area of the triangle is
The altitude of the triangle is #h=(27*2)/sqrt10=54/sqrt10#
The mid-point of #A# is #(15/2,7/2)#
The gradient of #A# is #=(2-5)/(8-7)=-3#
The gradient of the altitude is #=1/3#
The equation of the altitude is
The intersection of this circle with the altitude will give the third corner.
We solve this quadratic equation
The points are #(23.7,9.65)# and #(-8.7,-1.9)#
graph{(y-1/3x-1)((x-7.5)^2+(y-3.5)^2-291.6)((x-7)^2+(y-5)^2-0.05)((x-8)^2+(y-2)^2-0.05)(y-5+3(x-7))=0 [-12, 28, -10, 10]}
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Answer 2
To find the possible coordinates of the triangle's third corner, we can use the formula for the area of a triangle, which is given by:
[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ]
Since the triangle is isosceles, we can consider side ( A ) as the base. Then, the length of side ( A ) can be found using the distance formula between the given points ((7, 5)) and ((8, 2)).
Once we have the length of side ( A ) and the area of the triangle, we can find the height of the triangle. With the height, we can then find the possible coordinates of the third corner by
considering the symmetry of the isosceles triangle.
Let's calculate:
1. Calculate the length of side ( A ): [ A = \sqrt{(8 - 7)^2 + (2 - 5)^2} = \sqrt{1 + 9} = \sqrt{10} ]
2. Use the area formula to find the height: [ \text{Area} = \frac{1}{2} \times A \times \text{height} ] [ 27 = \frac{1}{2} \times \sqrt{10} \times \text{height} ] [ \text{height} = \frac{2 \times
27}{\sqrt{10}} = \frac{54}{\sqrt{10}} ]
3. With the height, we can find the coordinates of the third corner. Since the triangle is isosceles, the third corner will be at the same distance from the midpoint of side ( A ) along the
perpendicular bisector of side ( A ).
Let ( M ) be the midpoint of side ( A ). The coordinates of ( M ) can be found as the average of the coordinates of the given points:
[ M = \left(\frac{7 + 8}{2}, \frac{5 + 2}{2}\right) = \left(\frac{15}{2}, \frac{7}{2}\right) ]
Now, we need to find a point ( P ) that is ( \frac{54}{\sqrt{10}} ) units away from ( M ) along the perpendicular bisector of ( A ).
Since the perpendicular bisector of a line segment passes through its midpoint, we only need to find a point on the perpendicular line that is ( \frac{54}{\sqrt{10}} ) units away from ( M ).
The equation of the line passing through ( M ) perpendicular to side ( A ) is given by: [ x - \frac{15}{2} = -\frac{3}{1} \times (y - \frac{7}{2}) ]
We solve this equation with the condition that the point is ( \frac{54}{\sqrt{10}} ) units away from ( M ) to find the possible coordinates of the third corner.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Differential Calculus
Differential calculus deals with the rate of change of one quantity with respect to another. Or you can consider it as a study of rates of change of quantities. For example, velocity is the rate of
change of distance with respect to time in a particular direction. If f(x) is a function, then f'(x) = dy/dx is the differential equation, where f’(x) is the derivative of the function, y is
dependent variable and x is an independent variable.
Calculus Definition
In mathematics, calculus is a branch that deals with finding the different properties of integrals and derivatives of functions. It is based on the summation of the infinitesimal differences.
Calculus is the study of continuous change of a function or a rate of change of a function. It has two major branches and those two fields are related to each other by the fundamental theorem of
calculus. The two different branches are:
• Differential calculus
• Integral Calculus
In this article, we are going to discuss the differential calculus basics, formulas, and differential calculus examples in detail.
Basics of Differential Calculus
In differential calculus basics, you may have learned about differential equations, derivatives, and applications of derivatives. For any given value, the derivative of the function is defined as the
rate of change of functions with respect to the given values. Differentiation is a process where we find the derivative of a function. Let us discuss the important terms involved in the differential
calculus basics.
A function is defined as a relation from a set of inputs to the set of outputs in which each input is exactly associated with one output. The function is represented by “f(x)”.
Dependent Variable
The dependent variable is a variable whose value always depends and determined by using the other variable called an independent variable. The dependent variable is also called the outcome variable.
The result is being evaluated from the mathematical expression using an independent variable is called a dependent variable.
Independent Variable
Independent variables are the inputs to the functions that define the quantity which is being manipulated in an experiment. Let us consider an example y= 3x. Here, x is known as the independent
variable and y is known as the dependent variable as the value of y is completely dependent on the value of x.
Domain and Range
The domain of a function is simply defined as the input values of a function and range is defined as the output value of a function. Take an example, if f(x) = 3x be a function, the domain values or
the input values are {1, 2, 3} then the range of a function is given as
f(1) = 3(1) = 3
f(2) = 3(2) = 6
f(3) = 3(3) = 9
Therefore, the range of the function will be {3, 6, 9}.
The limit is an important thing in calculus. Limits are used to define the continuity, integrals, and derivatives in the calculus. The limit of a function is defined as follows:
Let us take the function as “f” which is defined on some open interval that contains some numbers, say “a”, except possibly at “a” itself, then the limit of a function f(x) is written as:
\(\lim_{x\rightarrow a}f(x)= L\),
> 0, there exists
> 0 such that 0 < |x – a| <
implies that |f(x) – L| <
It means that the limit f(x) as “x” approaches “a” is “L”
An interval is defined as the range of numbers that are present between the two given numbers. intervals can be classified into two types namely:
• Open Interval – The open interval is defined as the set of all real numbers x such that a < x < b. It is represented as (a, b)
• Closed Interval – The closed interval is defined as the set of all real numbers x such that a ≤ x and x ≤ b, or more concisely, a ≤ x ≤ b, and it is represented by [a, b]
The fundamental tool of differential calculus is derivative. The derivative is used to show the rate of change. It helps to show the amount by which the function is changing for a given point. The
derivative is called a slope. It measures the steepness of the graph of a function. It defines the ratio of the change in the value of a function to the change in the independent variable. The
derivative of y with respect to x is expressed by dy/dx.
Graphically, we define a derivative as the slope of the tangent, that meets at a point on the curve or which gives derivative at the point where tangent meets the curve. Differentiation has many
applications in various fields. Checking the rate of change in temperature of the atmosphere or deriving physics equations based on measurement and units, etc, are the common examples.
1. f(x) = 6x^2-2 ⇒ f’(x) = 12x
2. f(x) = 2x ⇒ f’(x) = 2
3. f(x) = x^3 + 2x ⇒ f’(x) = 3x^2 + 2
Video Lesson
Differential calculus Questions
Differential Calculus Formulas
How do we study differential calculus? The differentiation is defined as the rate of change of quantities. Therefore, calculus formulas could be derived based on this fact. Here we have provided a
detailed explanation of differential calculus which helps users to understand better.
Suppose we have a function f(x), the rate of change of a function with respect to x at a certain point ‘o’ lying in its domain can be written as;
df(x)/dx at point o
Or df/dx at o
So, if y = f(x) is a quantity, then the rate of change of y with respect to x is such that, f'(x) is the derivative of the function f(x). Also, if x and y varies with respect to variable t, then by
the chain rule formula, we can write the derivative in the form of differential equations formula as;
In mathematics, differential calculus is used,
• To find the rate of change of a quantity with respect to other
• In case of finding a function is increasing or decreasing functions in a graph
• To find the maximum and minimum value of a curve
• To find the approximate value of small change in a quantity
Real-life applications of differential calculus are:
• Calculation of profit and loss with respect to business using graphs
• Calculation of the rate of change of the temperature
• Calculation of speed or distance covered such as miles per hour, kilometres per hour, etc.,
• To derive many Physics equations
Problems and Solutions
Go through the given differential calculus examples below:
Example 1: f(x) = 3x^2-2x+1
Solution: Given, f(x) = 3x^2-2x+1
Differentiating both sides, we get,
f’(x) = 6x – 2, where f’(x) is the derivative of f(x).
Example 2: f(x) = x^3
Solution: We know,
\(\frac{\mathrm{d} (x^n)}{\mathrm{d} x}\) = n x
Therefore, f’(x) = \(\frac{\mathrm{d} x^3}{\mathrm{d} x}\)
f’(x)= 3 x^3-1
f’(x)= 3 x^2
Video Lesson
Learn more Maths formulas and problems with us and download BYJU’S – The Learning App for interactive videos.
Frequently Asked Questions – FAQs
What is differential calculus?
Differential calculus is a method which deals with the rate of change of one quantity with respect to another. The rate of change of x with respect to y is expressed dx/dy. It is one of the major
calculus concepts apart from integrals.
Why do we use differential calculus?
To check the instantaneous rate of change such as velocity
To evaluate the approximate value of small change in a quantity
To know if a function is increasing or decreasing functions in a graph
What is the difference between differential calculus and integral calculus?
Differential calculus deals with the rate of change of quantity with respect to others. For example, velocity and slopes of tangent lines.
Integral calculus is a reverse method of finding the derivatives. We deal here with the total size such as area and volumes on a large scale. It is a process of finding antiderivatives.
What are derivatives?
The derivative is simply called a slope. It measures the steepness of the graph of a function. It defines the ratio of the change in the value of a function to the change in the independent variable.
The derivative is expressed by dy/dx.
What is the differential equation?
In Maths, when one or more functions and their derivatives are related with each other to form an equation, then it is said to be a differential equation. It includes derivatives of one variable
(dependent) with respect to other (independent).
For example, dy/dx=2, where y is the dependent variable and x is the independent variable. | {"url":"https://mathlake.com/Differential-Calculus","timestamp":"2024-11-06T03:06:55Z","content_type":"text/html","content_length":"20282","record_id":"<urn:uuid:5c331c80-639b-4d61-8703-0bad4001041b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00807.warc.gz"} |
The risk of playing Sic Bo is divided into 3 levels - pazisnimase.com
Low risk
Anyone who is new to playing, has never played or has played for a long time. Until he barely remembers anything. I think it’s best to start playing at this level of risk. Because it is very low
risk, the gameplay is not complicated. Because it focuses on 2 forms of betting namely high-low bets and Tod bets here. Online casinos will only have an advantage of 2.78% and our chances of losing
and winning are at the level of half and half. There are ways to play both formats.
High-low bets. How to play is not difficult. We just place bets in the Small position if we want to bet low. Or place a bet on the Big side if you want to bet high And because playing in this format.
It has a 50:50 chance of losing winning similar to Baccarat. So we can use the Baccarat money formula to be a 1324 or Martingal money formula to help manage money as well. It will give us the
opportunity to make more profit than usual.
Toad here, the odds will be 5 : 1 or bet 1 pays 5, excluding capital. The reason that this position should be played is because we have a chance to win 1 out of 6 times that bet. Which way of playing
we will use funds. Approximately 15 units bet 1 unit per round. If we win 3 times before using all the money. We can bounce. because it is considered profitable
It can be seen that playing Sic Bo at a low level of risk. Even if the profit is low. But it also cost a little money. They can also play for a long time. So it’s suitable for newbies to practice and
build confidence. Before going to play at a higher risk level than this.
This one will be a bit more difficult to play. along with having to prepare more money But it’s still safe. Only the style of play is quite fixed. Which is popularly played in 2 types:
Type 1 , bet a total of 9 points to 3 units, double bet on 1, 5 and 6, each 2 units, in total we will use the funds in this round 9 units. The reason for this bet is because
1. If the 1st and 2nd dice come out 1 and 1, even if the last dice comes out 6, they only get 8 points.
2. If it draws 5, 5, 1 its total is 11 or if it’s 5, 5, 6 it’s 16 and it’s not possible to sum 9 and 10 so we have to bet these 2 positions to cover it up.
Type 2 bets on a total of 12 points to 3 units and double bets on 1, 2 and 6, 2 units each, using the same amount of 9 units. For the reason that this type of stabbing is because
1. If exiting 1, 1, the chance that the total in that round will be 3 – 8.
2. If 2, 2 are issued then the chances that the total in that round will be 5 – 10.
3. If 6, 6 are drawn, the chance that the total in that round will be 13 – 18.
4. Out of all of this, there is no chance of getting a total of 11-12, so we bet on a total of 12 to cover as much as possible. But if you choose to bet on 11, it’s not wrong.
Which from both of these 2 there is a chance that we will win 1 in 4 or 25%, but that we have to bet on many positions even though there is no chance of stacking in one round It’s because we want to
bet to cover as much as possible, that’s all.
The rewards that will be received It can be divided according to the events as follows:
If the total is 9 or 12, profit is 3 x 6 = 18 units, subtracting 6 units of losing pairs, leaving 12 units from the total investment of 9 units, which means that we have a profit of 1.33 times, which
is considered effective. More rewards than playing high-low, but in accordance with the risks that we accept.
And if winning the double favorite, no matter what kind of play, we will have a profit of 2 × 10 = 20 units, minus 7 units of losses, remaining 13 units of profit.
Therefore, anyone who wants to win a profit at this level with an acceptable risk It is recommended that you look for a Sic Bo online table that gives a double bet of 10:1 or more, if less than this
is not worth it.
high risk
The ultimate bet of the gambler would be to throw large sums of money for big profits. which the bet will not focus on covering But focusing on throwing on either side of the end in the hope that it
will issue 2 positions at the same time from 3 positions, of course, that throwing together this size, this profit is definitely not normal. By betting, there will be 2 forms as follows.
Type 1 bets a total of 8 points to 3 units, bets on pairs 1, 2 and 3, 2 units each, followed by 2 units each, and 2 units each, totaling 11 units. The reason for this bet is because
1. If exiting 1, 1, 6, we will get money from betting a total of 8 points and another double favorite.
2. If 2, 2, 4 are drawn, it will give the same result, namely 2 places in front total and even favorite.
3. If 3, 3, 2 are issued, the result is not different from the first 2.
Now let’s see how much the profit will be.
• If you win a total of 8, you will get money 3×6 = 18 units.
• If you win the pair, you will get 2×10 = 20 units.
• If winning bets Tod 2, 3 will receive money 2 × 5 = 10 units
Type 2, we will bet on a total of 13 points to 3 units, bet on pairs 4, 5 and 6 of 2 units each, and then bet on Tod 4, 5 to finish off another 2 units each, which we predicted here. that
1. If out 4, 4, 5, we win a total of 13 points, coupled with double favorites.
2. If out 5, 5, 3 will win 2 positions as well.
3. or if output 6, 6, 1 gives the same result
From all events, if issued as follows, we will be profitable.
• If you win a total of 13 points, you will get a profit of 3×6 = 18 units.
• If you win the pair, you will get 2×10 = 20 units.
• If Tod wins 4, 5, you will get money 2×5 = 10 units.
At this point, you can decide whether we will play Sic Bo online with a good level of risk. But no matter what level of risk, do not forget that in the long run, we lose to Online casinos are good,
so if playing and making a profit, it is advisable to stop carrying the money back first. Come to play next time, it’s not too late. Most importantly, using the UFABET money walking formula together
will help you get more profit. | {"url":"https://pazisnimase.com/football/the-risk-of-playing-sic-bo-is-divided-into-3-levels/","timestamp":"2024-11-05T06:42:12Z","content_type":"text/html","content_length":"41619","record_id":"<urn:uuid:b79b4e2f-2b35-4c9b-8614-d8854f91314c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00647.warc.gz"} |
PHYS102: Introduction to Electromagnetism
The material in this unit is not directly related to electricity and magnetism, but it is the foundation of one of the most significant outcomes of Maxwell's theory – electromagnetic waves. In
PHYS101: Introduction to Mechanics, we learned how to describe the motion of particle-like masses using classical mechanics. In this unit, we begin the transition from mechanics to electromagnetism
by examining how objects of size – length, width, and depth – behave.
When we look at these types of extended objects, a mystery is hiding in plain sight: if you pull one end of a rope, how does the other end know? Your action somehow "propagates" from one end to the
other. The answer is related to the invisible hand of electromagnetism that can transmit information between different locations.
In this unit, we focus on vibrating systems and the propagation of mechanical waves through media; think of ripples traveling outward from a stone dropped into water. We also lay the basic foundation
for the development of a classical theory of mechanics for extended solids.
Completing this unit should take you approximately 7 hours. | {"url":"https://learn.saylor.org/course/view.php?id=18§ion=1","timestamp":"2024-11-04T21:40:53Z","content_type":"text/html","content_length":"714254","record_id":"<urn:uuid:63557ccc-8749-4ccd-a59a-5dd2441bb060>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00649.warc.gz"} |
Superstrings as Grand Unifier
10/27/2017 Science&
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by Himanshu Damle
The first step of deriving General Relativity and particle physics from a common fundamental source may lie within the quantization of the classical string action. At a given momentum,
quantized strings exist only at discrete energy levels, each level containing a finite number of string states, or particle types. There are huge energy gaps between each level, which
means that the directly observable particles belong to a small subset of string vibrations. In principle, a string has harmonic frequency modes ad infinitum. However, the masses of the
corresponding particles get larger, and decay to lighter particles all the quicker.
Most importantly, the ground energy state of the string contains a massless, spin-two particle. There are no higher spin particles, which is fortunate since their presence would ruin
the consistency of the theory. The presence of a massless spin-two particle is undesirable if string theory has the limited goal of explaining hadronic interactions. This had been the
initial intention. However, attempts at a quantum field theoretic description of gravity had shown that the force-carrier of gravity, known as the graviton, had to be a massless
spin-two particle. Thus, in string theory’s comeback as a potential “theory of everything,” a curse turns into a blessing.
Once again, as with the case of supersymmetry and supergravity, we have the astonishing result that quantum considerations require the existence of gravity! From this vantage point,
right from the start the quantum divergences of gravity are swept away by the extended string. Rather than being mutually exclusive, as it seems at first sight, quantum physics and
gravitation have a symbiotic relationship. This reinforces the idea that quantum gravity may be a mandatory step towards the unification of all forces.
Unfortunately, the ground state energy level also includes negative-mass particles, known as tachyons. Such particles have light speed as their limiting minimum speed, thus violating
causality. Tachyonic particles generally suggest an instability, or possibly even an inconsistency, in a theory. Since tachyons have negative mass, an interaction involving finite
input energy could result in particles of arbitrarily high energies together with arbitrarily many tachyons. There is no limit to the number of such processes, thus preventing a
perturbative understanding of the theory.
An additional problem is that the string states only include bosonic particles. However, it is known that nature certainly contains fermions, such as electrons and quarks. Since
supersymmetry is the invariance of a theory under the interchange of bosons and fermions, it may come as no surprise, post priori, that this is the key to resolving the second issue.
As it turns out, the bosonic sector of the theory corresponds to the spacetime coordinates of a string, from the point of view of the conformal field theory living on the string
worldvolume. This means that the additional fields are fermionic, so that the particle spectrum can potentially include all observable particles. In addition, the lowest energy level
of a supersymmetric string is naturally massless, which eliminates the unwanted tachyons from the theory.
The inclusion of supersymmetry has some additional bonuses. Firstly, supersymmetry enforces the cancellation of zero-point energies between the bosonic and fermionic sectors. Since
gravity couples to all energy, if these zero-point energies were not canceled, as in the case of non-supersymmetric particle physics, then they would have an enormous contribution to
the cosmological constant. This would disagree with the observed cosmological constant being very close to zero, on the positive side, relative to the energy scales of particle
Also, the weak, strong and electromagnetic couplings of the Standard Model differ by several orders of magnitude at low energies. However, at high energies, the couplings take on
almost the same value, almost but not quite. It turns out that a supersymmetric extension of the Standard Model appears to render the values of the couplings identical at approximately
1016 GeV. This may be the manifestation of the fundamental unity of forces. It would appear that the “bottom-up” approach to unification is winning. That is, gravitation arises from
the quantization of strings. To put it another way, supergravity is the low-energy limit of string theory, and has General Relativity as its own low-energy limit.
taken from:
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Mathematics in Dubai: 10 Best universities Ranked 2024
10 Best universities for Mathematics in Dubai
Below is a list of best universities in Dubai ranked based on their research performance in Mathematics. A graph of 50.5K citations received by 4.8K academic papers made by 10 universities in Dubai
was used to calculate publications' ratings, which then were adjusted for release dates and added to final scores.
We don't distinguish between undergraduate and graduate programs nor do we adjust for current majors offered. You can find information about granted degrees on a university page but always
double-check with the university website.
Acceptance Rate
Acceptance Rate
Acceptance Rate
Universities for Mathematics near Dubai
Mathematics subfields in Dubai | {"url":"https://edurank.org/math/dubai/","timestamp":"2024-11-13T15:07:17Z","content_type":"text/html","content_length":"71456","record_id":"<urn:uuid:966965a6-10e0-46c6-8cd1-2aade251235b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00538.warc.gz"} |
Lagrange Form (Interpolation)
The lagrange basis is a different basis for interpolating polynomials.
We will define the Lagrange basis functions, , to construct a polynomial as where are the coefficients (and are also our data values, .
One advantage of the Lagrange form is that the interpolating polynomial can be written down directly, without needing to solve a (Vandermonde) linear system. | {"url":"https://stevengong.co/notes/Lagrange-Form-(Interpolation)","timestamp":"2024-11-11T17:50:27Z","content_type":"text/html","content_length":"33799","record_id":"<urn:uuid:5680ad62-4eaf-4882-a68c-03dacdb34105>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00507.warc.gz"} |
Geometry Homework Help - Certified Online Math Helpers
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Data Analysis and Visualization
Geometry is used in data analysis and visualization in various fields. Geometric techniques such as data visualization, geometric modeling, and geometric algorithms are used to represent and analyze
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Can I take a Geometry class if I am not good at Drawing?
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8. Deadline Guaranteed | {"url":"https://homeworktypers.com/geometry-homework-help","timestamp":"2024-11-11T07:01:03Z","content_type":"text/html","content_length":"34972","record_id":"<urn:uuid:173da6ee-1bfe-4423-a957-d1ff9d0e0a02>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00736.warc.gz"} |
math nerds, advice needed on course selection. - AR15.COM
So here's the deal. I just dropped real analysis. Despite spending 15-20 hours a week on homework, losing my sanity and everything else associated with taking real analysis, I wasn't cutting it. My
grades were continuing to drop, and were in the low C range. Out of fear of getting a D and endangering my scholarship, I dropped. I am also in a difficult micro economics class, that incorporates
some real analysis, and lots of "WTF is this shit" material, and I'm currently doing very well in that. The rest of my classes are cake.
Currently, I have only taken Calc 1, 2, 3 (multivariable), and an mathematical structures class which is the prereq for the real analysis course that I just dropped. I have significantly less
background in math than my peers in that class. So, here's my plan.
Take 1 easy econ course.
Take 1 harder econ course, though much easier than what I'm doing now.
Take linear algebra, differential equations and probability theory.
This should get me on par with the average student in the real analysis class. Hopefully, while not directly helping me with the subject, it will broaden my math education to a point where the real
analysis might make a little more sense to me. Other things that contributed to my failure in the class were my participation in marching band and emotional issues from the arfcom curse. These won't
be issues next time around.
My main concern is I'm going to "overload" again. Some people have told me to only take two math classes. Others assured me that with the right teachers (which I'm confident I have selected), it
shouldn't be too bad.
So, arfcom mathematicians, does this sound reasonable, or am I setting myself up to fail? Until pure math entered my life, I was a straight A student and nothing could bring me down. But now I'm a
bit less confident in myself. | {"url":"https://www.ar15.com/forums/general/math_nerds__advice_needed_on_course_selection_/5-1106840/","timestamp":"2024-11-04T07:26:06Z","content_type":"text/html","content_length":"127469","record_id":"<urn:uuid:bc2a6136-f0b7-456d-8c43-5cdb204fb018>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00751.warc.gz"} |
HF-dataset from Edge Device
Published: 6 April 2021| Version 1 | DOI: 10.17632/w35nf2fw5m.1
The high-frequency (HF) data is retrieved from the Spinner U5-630 milling machine via an Edge Device. This data has a sampling rate of 2ms and hence, a frequency of 500Hz. The HF-data is from various
experiments performed. There are 2 experiments performed (parts 1 and 2). The experimented part 1 has 12 .json data files and part 2 has 11 .json files. In total, there are 23 files of HF-data from
23 experiments. The HF-data has vast potential for analysis as it contains all the information from the machine during the machining process. One part of the information was used in our case to
calculate the energy consumption of the machine. Similarly, the data can be used for retrieving information of torque, commanded and actual speed, NC code, current, etc. The spindle input power was
varied for every experiment to analyze the effects on machining of both parts with varying geometry.
Steps to reproduce
The HF-data was collected via Edge Device, Simatic IPC227E, which is a compact and flexible embedded industrial PC. This HF-data is from Spinner U5-630, 5-axis simultaneous milling CNC machine with a
Sinumerik 840d sl v4.8 NCU (Numeric Control Unit). The HF-dataset has a frequency of 500Hz. The channel utilized and analyzed were from the 'power' channel. However, several other channels can be
extracted and analyzed using this HF dataset. Channels such as Commanded and actual axis position, Torque, Load, Various encoder position, Commanded speed and actual speed, Current, and Power.
Moreover, this dataset has the relevant aforementioned channels along 7 axes; namely: X, Y, Z, B, C, Spindle, and tool change axis.
Technische Universitat Graz Fakultat fur Maschinenbau und Wirtschaftswissenschaften
Industrial Engineering, Machining, Energy Optimization Model, High-Frequency Data | {"url":"https://data.mendeley.com/datasets/w35nf2fw5m/1","timestamp":"2024-11-04T13:44:01Z","content_type":"text/html","content_length":"102780","record_id":"<urn:uuid:0b566094-67a7-433c-8d64-b726e3deb647>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00449.warc.gz"} |
Learn Statistics with Python: Quartiles, Quantiles, and Interquartile Range Cheatsheet | Codecademy
Quantiles are the set of values/points that divides the dataset into groups of equal size. For example, in the figure, there are nine values that splits the dataset. Those nine values are quantiles.
The three dividing points (or quantiles) that split data into four equally sized groups are called quartiles. For example, in the figure, the three dividing points Q1, Q2, Q3 are quartiles.
In Python, the numpy.quantile() function takes an array and a number say q between 0 and 1. It returns the value at the qth quantile. For example, numpy.quantile(data, 0.25) returns the value at the
first quartile of the dataset data.
If the number of quantiles is n, then the number of equally sized groups in a dataset is n+1.
The median is the divider between the upper and lower halves of a dataset. It is the 50%, 0.5 quantile, also known as the 2-quantile.
# The value 5 is both the median and the 2-quantile data = [1, 3, 5, 9, 20]Second_quantile = 5
The interquartile range is the difference between the first(Q1) and third quartiles(Q3). It can be mathematically represented as IQR = Q3 - Q1.
The interquartile range is considered to be a robust statistic because it is not distorted by outliers like the average (or mean).
# Eventhough d_2 has an outlier, the IQR is identical for the 2 datasets d_1 = [1,2,3,4,5,6,7,8,9]d_2 = [-100,2,3,4,5,6,7,8,9] | {"url":"https://www.codecademy.com/learn/learn-statistics-with-python/modules/quartiles-quantiles-and-interquartile-range/cheatsheet","timestamp":"2024-11-07T23:35:27Z","content_type":"text/html","content_length":"188298","record_id":"<urn:uuid:c9c66135-84ea-4ff9-9ebd-9cfe9b9939d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00721.warc.gz"} |
When Surfers Can’t Wait Their Turn
One really fun summer sport is surfing, where you stand on a long surfboard and try to ride it on waves. Now imagine sharing one surfboard with dozens of other people! 66 people in California
squeezed onto a 42-foot-long surfboard and rode it for a whole 15 seconds, beating the world record for the most surfers on the biggest surfboard. We hope once everyone fell off that somebody
remembered to grab the surfboard.
Wee ones: If you surf the 1st wave, then skip the 2nd, then ride the 3rd, then skip the 4th, what do you do on the 5th wave?
Little kids: The surfboard was 11 feet wide, maybe almost as wide as your room! Lie down head to toe with a grown-up, and guess whether the 2 of you could stretch across that board. If you lay across
it, by how many feet would you fall short of stretching across? Bonus: Once the 1st surfer fell off, how many of the 66 surfers were still hanging on?
Big kids: If the surfers weighed 10,100 pounds and the board weighed 1,300 pounds, how much did that weigh all together? (Hint if needed: Start with the board weighing just 1,000 pounds.) Bonus: If
your regular surfboard is only 1/6 as long as this monster 42-foot one, how long is your board?
Wee ones: You ride the 5th wave.
Little kids: See if your height in feet + a grown-up’s height add to 11. Bonus: 65 surfers.
Big kids: 11,400 pounds. Bonus: 7 feet long. | {"url":"https://bedtimemath.org/fun-math-surfing-world-record/","timestamp":"2024-11-11T07:25:36Z","content_type":"text/html","content_length":"87486","record_id":"<urn:uuid:028a2033-150f-45b3-9731-c63882fc62cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00434.warc.gz"} |
40+ Time and Money Worksheets for Children to Learn
Best Time and Money Worksheets for Kids! Download and print these free worksheets for kids, with the best printables for learning about time and money.
Time and Money, the two most significant ingredients for the recipe of a successful life.
A mathematical point of view on time and money starts with the basics and moves your way upwards to understand more complex and advanced concepts interwoven with other branches of maths.
Practice is what makes those concepts engraved into your minds. So, the more you practice, the higher hold you have on them. Here is a set of Time and Money worksheets for you to practice on.
If you want to understand more on the Time and Money Concept – go the last section of this article.
Time and Money Worksheets
Telling Time Worksheet – All Levels
These one page worksheets cover telling time. Students look at a clock and write the correct
Key concept: All Students need to practice their everyday time skills.
Creating Clocks – All Levels
These one page worksheets have students fill in the correct clock handles. Students look at the time
and draw in the clock.
Key concept: All Students need to practice their everyday time skills.
Answer, Find, and Shade Time Review Worksheet – Level 1
This art worksheet reviews time. Students read clocks, count minutes, and estimate hours.
Key concept: Students should be able to tell time, and determine the duration of intervals of time in minutes and hours.
Time Art Worksheet – Level 2
This one page art worksheet helps students practice time. Students answer the word problems, find the answers in the grid, and then shade the squares that match the answers.
Answer, Find, and Shade
Using and Reading a Schedule Worksheet
These one page art worksheets have students practice reading a schedule. The schedule and picture become more difficult as the levels get higher. For example: Information is left out of the schedule
and students must calculate what is missing.
Calendar Vocabulary Worksheet – Level 1
This one page worksheet reviews basic calendar vocabulary. Students fill in the blanks using the given words.
Key concept: Students should be able to understand time and know key vocabulary related to seconds, minutes, hours, days, weeks, months andyears.
Calendar Review Worksheet – Level 2
This one page worksheet reviews basic calendar knowledge. Students answer questions about dates and days of the week using calendars.
Key concept: Students should be able to solve time measurement problems and collect information from a calendar.
Counting Money Worksheet – Level 1
This one page worksheet has students add basic USA Currency. It only uses pennies, nickels, dimes, and quarters.
Key concept: Students should know and understand basic money and be able to add coins.
Answer, Find, and Shade Counting Money Worksheet – Level 2
This art worksheet reviews USA Currency. Students count dollars and change to calculate an amount.
Key concept: Students should be able to solve problems using different combinations of coins and bills.
Buying and Change with USA Money – Worksheet Level 1
This one page worksheet has students use their knowledge of currency to purchase items and then calculate how much money Alligator Jack will have left after the purchase. It uses pennies, nickels,
dimes, and quarters.
Key concept: Students should be able to count coins, and add and subtract money (currency).
Stained Glass Money Review Worksheet– Level 1
This art worksheet reviews money (USA Currency). It includes basic change, rounding to the nearest dollar, and adding and subtracting money. Each student chooses their own colors to create a unique
art project.
Key concept: Understand money amounts in decimal notation.
Money Word Problems Worksheet
These one page word problem worksheets review money, which are great for warm-ups or just some review. There are three different levels that have similar problems but the amounts are different.
Money with Change Worksheet – Level 1
These one page worksheets review purchasing items and calculating change. Students are given $10.00 to purchase three separate items, and calculate the total and the change they will receive back.
They do this process five different times.
Key concept: Students should practice everyday mathematical experiences.
Money with Change Worksheet – Level 2
These one page worksheets review purchasing items and calculating change. Students are told they’re going on a trip and need to purchase some new equipment. They need to purchase items, calculate the
total, and determine their change.
Key concept: Students should practice everyday mathematical experiences.
Who Done It? Word Problem Worksheet – All Levels
This one page worksheet has students use their basic math skills to solve a robbery. They read the information and look at the pictures to figure out who committed the crime.
Student misunderstanding: To solve the crime students need to use the information in the picture to confirm each person’s alibi.
Understanding Time & Money Math Concept
A few foundational and fundamental concepts of time and money will help you understand their practical use in your everyday life.
It is the measurement of a day or night or both to keep track of performing different activities like school or office hours, holidays, cooking time, when to go to sleep, and when to get up.
1. How to Draw a Clock?
A clock is of a simple circular shape with numbers 1-12 equally placed along its edges. These mark the 12 hours in a day and then again 12 hours of night, accounting for the total of 24 hours.
The 24-hour cycle begins from night 12 and is marked as mid-day when the clock strikes 12 again. The time between night 12 and day 12 falls under ‘am,’ and the time between day 12 and night 12 falls
under ‘pm.’
12:00 am is called ‘Midnight,’ and 12:00 pm is called ‘Afternoon.’
It also has two pointed needles pointing towards the digits. One is longer, and the other is shorter.
A third needle is also present, which is the same length as the longer one but thinner. These are of different shapes for you to easily differentiate between them.
You may think that how do I know about hour divisions? Well, to answer that question of yours, you will need to move to the next question.
2. What are Seconds, Minutes and Hours?
The time that passes between the two digits on the clock, for example, from 5 to 6, is called an hour. Let me elaborate.
60 seconds make up 1 minute, and 60 minutes make up 1 hour. The longer pointed needle or arm measures minutes, while the shorter arm measures hours. The third thinner arm measures seconds.
The thinner arm makes a complete circle the fastest as it calculates seconds, which occurs the fastest in time. The second-place taker in the speed is the minute’s arm, which circles with a medium
speed. The third-place taker and the slowest of all three is the hour’s arm as it accounts for one whole hour.
A more straightforward explanation is, for 1 hour to pass by, the seconds arm circles 3600 times and the minute’s arm circles 60 times.
Between two digits, there are five divisions. For seconds and minutes, 1 division equals 1 second and 1 minute, respectively.
3. How to Read Time on a Clock?
Nowadays, you have digital clocks that show the exact time without needing to calculate anything. But, if you were to come across an analog watch or clock, you will need to know how to read the time
on it.
Reading time on a clock is not hard. All you need is practice, and then it will be more than you can imagine. Let us take an example.
Note: You do not need to calculate the seconds too while reading time. Just the hour and minutes are enough.
If the minute hand is on the digit 6 and the hour hand is on 4, the time is 4:30 pm.
The number on which the hour arm is present defines the hour number. And if the hour arm is between two numbers, you should take the number before the arm.
For the minute calculation, instead of the 1-12 digits, you need to calculate the number of divisions beginning from 12. From 12 to 6, with 5 divisions between each, will add to 30 divisions. So, 30
is the number for minutes.
The last step: designating whether it’s day or night. Here pm is mentioned, and so it is daytime.
Hence, the time is 4:30 pm.
4. How does a Calendar show you Time?
A Calendar is an enlarged version of how time can be observed and recorded. Though it does have arms and numbers to be read, it works on the concept of not one 24-hour cycle like a clock, but many
such 24-hour cycles.
Seven 24-hour cycles make up 1 week, 30 or 31 such 24-hour cycles make up 1 month, 12 such months make up 1 year—the entire of which consists of 365 24-hour cycles.
To simplify, let us consider one 24-hour cycle as 1 day. So the 7 days in a week are Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday, and their respective abbreviations are Mon,
Tue, Wed, Thu, Fri, Sat, and Sun.
The 12 months in a year are January, February, March, April, May, June, July, August, September, October, November, and December. However, there is no name for years.
One exception to note is that February is not of 30 or 31 days, but 28 days. And once every four years, this month has 29 days, and the year is called a leap year.
Exchange of products was the ancient way of gaining the things you wish to acquire. Today, each country has its own currency and a symbol, not repeatable or copyable.
Further, Currency is divided into notes and coins for more accessible exchanges and to avoid mistakes. Notes are often made of rectangular-shaped paper, while coins are of the circular shape formed
of copper, zinc, or nickel (metals).
1. What are the Different Types of Coins?
USA’s currency is Dollar with the symbol ‘$’. So let us take Dollar as an example for understanding the concept better.
The basic types of dollar denominations (amount divisions) are cent, nickel, dime, quarter, half a dollar, and 1 dollar. There are also divisions above 1 dollar that are 2, 5, 10, 20, 50, and 100
dollars each.
All 1 and above 1 dollar are paper currency, whereas the below 1 dollar are all coins manufactured.
1 cent
5 cents = 1 nickel
2 nickels = 1 dime (10 cents)
2 dimes + 1 nickel = 1 quarter (25 cents)
2 quarters = 1 half dollar (50 cents)
2. How is Money Represented?
If a product has a price tag of $14.34, then what do you understand from this? Let’s break down.
The symbol of a currency is always written first, followed by the number of dollars. A tricky point is the decimal point.
Numbers written after the decimal point represent cents, i.e., less than 1 whole dollar, hence placed after denoting the whole dollars. So in the above-given example, the product costs 14 dollars and
34 cents.
Why spelled only cents when it can also be said in nickels, dimes, or quarters? Notation in cents is easier to know, display, and calculate, and so is used far more times than other measurements.
3. How to carry out different Arithmetic Operations with Money?
You will need money when you go to a grocery shop to buy chocolates, toffees, candies, lollipops, juice, bread, eggs, milk, etc. And, if you do not know how much the product costs and the remaining
balance money that should be returned to you after payment, how will you prevent being fooled and lose money?
On that account, you will need to learn different arithmetic operations like addition, subtraction, multiplication, and division. Under each sub-heading, the concept has been explained with examples
for better grasping.
This will involve simple to complex additions of money; all combinations:
note + note,
coin + coin, and
note + coin.
Example: You buy one chocolate for $0.34 and one candy for $0.66. What will be the total cost?
Method: Add the cents first and then the whole dollars. So, the addition of 34 and 66 will give 100, of which you will add the 1 to the whole dollar side.
Therefore, the total bill will be $1.00
This often happens when you have no change and only whole dollar bills. So, the grocery storekeeper will need to give you back the remaining amount.
Example: You have a $100 bill. The total bill of the list items bought by you comes to around $76.32. What is the difference the shopkeeper will have to give you back?
Method: Subtracting 76.32 from 100 will give you 23.68, which equals the cashback: 23 dollars and 68 cents.
Bulk buying is where multiplication comes into play. A wedding festivities list, a stationary inauguration, or a new auditorium opening, you will need multiple flowers, pencil boxes, and chairs,
Example: There are 40 classmates in your class. You are celebrating your birthday and are going to the convenience store to buy 40 chocolates. Each one cost $0.81. What is the total money you will
have to spend?
Method: Since you will need 40 0.81 dollars not to leave any of your classmates out, simply multiply with 0.81. The result is $32.40, which is the total money you will have to spend.
The division is an operation of rare occurrence. It is not usually required, but you should know how to go about it and not be clueless if it does come up.
Example: Suppose the bill format shows the total and not the individual division. Now how will you explain to your mother how much each of the 3 cupcakes cost you, given that the total is $14.70?
Method: All you have to do is perform a simple division of 14.70 by 3, and you will get the answer to be 4.9, which is the cost of $4.90 for each of the delicious vanilla strawberry and chocolate
flavored cupcakes.
4. What is Rounding Up to the nearest Amount?
Rounding up means, instead of writing or saying the numbers after the decimal for accurate calculation, you can simply tell the overall dollar number for easy understanding.
However, you should follow two rules for rounding up:
• If the number is less than half of the total, it should be rounded up to the previous number.
• If the number is half or more than half of the total, it should be rounded up to the following number.
Example 2: If the cost of a bread packet is $1.75, you can round up it to 2 dollars because 75 cents are more than the 50 cents or the halfway mark.
Example 1: If the cost of a pair of scissors is $1.49, you can round up it to 1 dollar because 49 cents are less than the 50 cents or the halfway mark. | {"url":"https://gosciencegirls.com/time-and-money-worksheets/","timestamp":"2024-11-14T08:38:16Z","content_type":"text/html","content_length":"182406","record_id":"<urn:uuid:55b44814-0ef7-4aa8-a058-44586bde35a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00820.warc.gz"} |
Data Science Archives - Page 25 of 33 - Analytics Yogi
Category Archives: Data Science
In this post, you will learn about concepts of neural networks with the help of mathematical models examples. In simple words, you will learn about how to represent the neural networks using
mathematical equations. As a data scientist / machine learning researcher, it would be good to get a sense of how the neural networks can be converted into a bunch of mathematical equations for
calculating different values. Having a good understanding of representing the activation function output of different computation units / nodes / neuron in different layers would help in
understanding back propagation algorithm in a better and easier manner. This will be dealt in one of the …
In this post, you will learn the concepts of Adaline (ADAptive LInear NEuron), a machine learning algorithm, along with Python example.As like Perceptron, it is important to understand the concepts
of Adaline as it forms the foundation of learning neural networks. The concept of Perceptron and Adaline could found to be useful in understanding how gradient descent can be used to learn the
weights which when combined with input signals is used to make predictions based on unit step function output. Here are the topics covered in this post in relation to Adaline algorithm and its Python
implementation: What’s Adaline? Adaline Python implementation Model trained using Adaline implementation What’s Adaptive …
This post highlights some great pages where python implementations for different machine learning models can be found. If you are a data scientist who wants to get a fair idea of whats working
underneath different machine learning algorithms, you may want to check out the Ml-from-scratch page. The top highlights of this repository are python implementations for the following: Supervised
learning algorithms (linear regression, logistic regression, decision tree, random forest, XGBoost, Naive bayes, neural network etc) Unsupervised learning algorithms (K-means, GAN, Gaussian mixture
models etc) Reinforcement learning algorithms (Deep Q Network) Dimensionality reduction techniques such as PCA Deep learning Examples that make use of above mentioned algorithms Here is an insight
into …
In this post, you will learn about how to use Python BeautifulSoup and NLTK to extract words from HTML pages and perform text analysis such as frequency distribution. The example in this post is
based on reading HTML pages directly from the website and performing text analysis. However, you could also download the web pages and then perform text analysis by loading pages from local storage.
Python Code for Extracting Text from HTML Pages Here is the Python code for extracting text from HTML pages and perform text analysis. Pay attention to some of the following in the code given below:
URLLib request is used to read the html page …
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In this post, you will learn about some of the top data science skills / concepts which may be required for product managers / business analyst to have, in order to create useful machine learning
based solutions. Here are some of the topics / concepts which need to be understood well by product managers / business analysts in order to tackle day-to-day challenges while working with data
science / machine learning teams. Knowing these concepts will help product managers / business analyst acquire enough skills in order to solve machine learning based problems. Understanding the
difference between AI, machine learning, data science, deep learning Which problems are machine learning problems? …
In this post, you will learn about the concepts of RANSAC regression algorithm along with Python Sklearn example for RANSAC regression implementation using RANSACRegressor. RANSAC regression
algorithm is useful for handling the outliers dataset. Instead of taking care of outliers using statistical and other techniques, one can use RANSAC regression algorithm which takes care of the
outlier data. In this post, the following topics are covered: Introduction to RANSAC regression RANSAC Regression Python code example Introduction to RANSAC Regression RANSAC (RANdom SAmple
Consensus) algorithm takes linear regression algorithm to the next level by excluding the outliers in the training dataset. The presence of outliers in the training dataset does impact …
In this post, you will learn about Beta probability distribution with the help of Python examples. As a data scientist, it is very important to understand beta distribution as it is used very
commonly as prior in Bayesian modeling. In this post, the following topics get covered: Beta distribution intuition and examples Introduction to beta distribution Beta distribution python examples
Beta Distribution Intuition & Examples Beta distribution is widely used to model the prior beliefs or probability distribution in real world applications. Here is a great article on understanding
beta distribution with an example of baseball game. You may want to pay attention to the fact that even if the baseball …
In this post, you will learn about the concepts of Bernoulli Distribution along with real-world examples and Python code samples. As a data scientist, it is very important to understand statistical
concepts around various different probability distributions to understand the data distribution in a better manner. In this post, the following topics will get covered: Introduction to Bernoulli
distribution Bernoulli distribution real-world examples Bernoulli distribution python code examples Introduction to Bernoulli Distribution Bernoulli distribution is a discrete probability
distribution representing the discrete probabilities of a random variable which can take only one of the two possible values such as 1 or 0, yes or no, true or false etc. The probability of …
In this post, you will get to learn deep learning through simple explanation (layman terms) and examples. Deep learning is part or subset of machine learning and not something which is different than
machine learning. Many of us when starting to learn machine learning try and look for the answers to the question “what is the difference between machine learning & deep learning?”. Well, both
machine learning and deep learning is about learning from past experience (data) and make predictions on future data. Deep learning can be termed as an approach to machine learning where learning
from past data happens based on artificial neural network (a mathematical model mimicking human brain). …
In this post, you will learn about Bayes’ Theorem with the help of examples. It is of utmost importance to get a good understanding of Bayes Theorem in order to create probabilistic models. Bayes’
theorem is alternatively called as Bayes’ rule or Bayes’ law. One of the many applications of Bayes’s theorem is Bayesian inference which is one of the approaches of statistical inference (other
being Frequentist inference), and fundamental to Bayesian statistics. In this post, you will learn about the following: Introduction to Bayes’ Theorem Bayes’ theorem real-world examples Introduction
to Bayes’ Theorem In simple words, Bayes Theorem is used to determine the probability of a hypothesis in the presence of more evidence or information. In other …
In this post, you will learn about joint and conditional probability differences and examples. When starting with Bayesian analytics, it is very important to have a good understanding around
probability concepts. And, the probability concepts such as joint and conditional probability is fundamental to probability and key to Bayesian modeling in machine learning. As a data scientist, you
must get a good understanding of probability related concepts. Joint & Conditional Probability Concepts In this section, you will learn about basic concepts in relation to Joint and conditional
probability. Probability of an event can be quantified as a function of uncertainty of whether that event will occur or not. Let’s say an event A is …
In this plot, you will quickly learn about how to find elbow point using SSE or Inertia plot with Python code and You may want to check out my blog on K-means clustering explained with Python
example. The following topics get covered in this post: What is Elbow Method? How to create SSE / Inertia plot? How to find Elbow point using SSE Plot What is Elbow Method? Elbow method is one of the
most popular method used to select the optimal number of clusters by fitting the model with a range of values for K in K-means algorithm. Elbow method requires drawing a line plot between SSE (Sum of
Squared errors) …
In this post, you will learn about boosting technique and adaboost algorithm with the help of Python example. You will also learn about the concept of boosting in general. Boosting classifiers are a
class of ensemble-based machine learning algorithms which helps in variance reduction. It is very important for you as data scientist to learn both bagging and boosting techniques for solving
classification problems. Check my post on bagging – Bagging Classifier explained with Python example for learning more about bagging technique. The following represents some of the topics covered in
this post: What is Boosting and Adaboost Algorithm? Adaboost algorithm Python example What is Boosting and Adaboost Algorithm? As …
In this post, you will learn about one of the popular and powerful ensemble classifier called as Voting Classifier using Python Sklearn example. Voting classifier comes with multiple voting options
such as hard and soft voting options. Hard vs Soft Voting classifier is illustrated with code examples. The following topic has been covered in this post: Voting classifier – Hard vs Soft voting
options Voting classifier Python example Voting Classifier – Hard vs Soft Voting Options Voting Classifier is an estimator that combines models representing different classification algorithms
associated with individual weights for confidence. The Voting classifier estimator built by combining different classification models turns out to be stronger meta-classifier that balances out the
individual …
In this post, you will learn about how to set up Keras and get started with Keras, one of the most popular deep learning frameworks in current times which is built on top of TensorFlow 2.0 and can
scale to large clusters of GPUs. You will also learn about getting started with hello world program with Keras code example. Here are some of the topics which will be covered in this post: Set up
Keras with Anaconda Keras Hello World Program Set up Keras with Anaconda In this section, you will learn about how to set up Keras with Anaconda. Here are the steps: Go to Environments page in
Anaconda App. …
In this post, you will learn about how to load and predict using pre-trained Resnet model using PyTorch library. Here is arxiv paper on Resnet. Before getting into the aspect of loading and
predicting using Resnet (Residual neural network) using PyTorch, you would want to learn about how to load different pretrained models such as AlexNet, ResNet, DenseNet, GoogLenet, VGG etc. The
PyTorch Torchvision projects allows you to load the models. Note that the torchvision package consists of popular datasets, model architectures, and common image transformations for computer vision.
Here is the command: The output of above will list down all the pre-trained models available for loading and prediction. You may …
I found it very helpful. However the differences are not too understandable for me
Very Nice Explaination. Thankyiu very much,
in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
Thank you....for your support. you given a good solution for me. | {"url":"https://vitalflux.com/category/data-science/page/25/","timestamp":"2024-11-13T14:13:43Z","content_type":"text/html","content_length":"140314","record_id":"<urn:uuid:9381f4d7-eca4-42c2-9117-d7181c1183ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00800.warc.gz"} |
Volume Variance in CVP in context of cost volume profit
01 Sep 2024
Volume Variance in Cost-Volume-Profit (CVP) Analysis: A Conceptual Exploration
Abstract: This article delves into the concept of volume variance in the context of cost-volume-profit (CVP) analysis, a fundamental tool for managerial decision-making. The volume variance is a
critical component of CVP that measures the difference between the actual and standard volumes of production or sales. This article provides an in-depth examination of the volume variance formula and
its implications for managerial accounting.
Introduction Cost-volume-profit (CVP) analysis is a widely used framework for evaluating the profitability of business operations. The three components of CVP are cost, volume, and profit. While
costs and profits are well-defined concepts, volume is often overlooked as a critical factor in determining profitability. This article focuses on the concept of volume variance, which measures the
difference between actual and standard volumes of production or sales.
Volume Variance Formula The volume variance (VV) can be calculated using the following formula:
VV = (Actual Volume - Standard Volume) × (Standard Cost per Unit)
In ASCII format: VV = (AV - SV) × SCPU
• AV = Actual Volume
• SV = Standard Volume
• SCPU = Standard Cost per Unit
Interpretation of Volume Variance A positive volume variance indicates that the actual volume is higher than the standard volume, resulting in increased costs and potentially lower profits.
Conversely, a negative volume variance suggests that the actual volume is lower than the standard volume, leading to decreased costs and potentially higher profits.
Implications for Managerial Accounting The volume variance has significant implications for managerial accounting. It highlights the importance of monitoring and controlling production or sales
volumes to ensure optimal profitability. A high volume variance can indicate inefficiencies in production or sales processes, which may require corrective action to improve profitability.
Conclusion In conclusion, the volume variance is a critical component of CVP analysis that measures the difference between actual and standard volumes of production or sales. Understanding the volume
variance formula and its implications for managerial accounting can help managers make informed decisions about production or sales levels, ultimately leading to improved profitability.
• Horngren, C. T., & Foster, W. (2015). Cost Accounting: A Managerial Emphasis. Pearson Education.
• Kaplan, R. S. (1984). The Role of Volume Variance in Cost-Volume-Profit Analysis. Journal of Accounting Research, 22(2), 241-254.
Note: This article does not provide numerical examples, but rather focuses on the conceptual framework and formula for volume variance in CVP analysis.
Related articles for ‘cost volume profit ‘ :
Calculators for ‘cost volume profit ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/468c0f28e2d8c4b9a9860e6ea5dee44b/JSON_TO_ARTCL_Volume_Variance_in_CVP_in_context_of_cost_volume_profit_.html","timestamp":"2024-11-05T07:10:31Z","content_type":"text/html","content_length":"17356","record_id":"<urn:uuid:f9dc56fb-2619-49dd-8fb2-44231e99b784>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00878.warc.gz"} |
$f(X_n)$ converges in probability to $f(X)$ implies $X_n$ converges in probability to $X$ ~ Mathematics ~ TransWikia.com
We will make use of the following simple observations:
Lemma 1. $$X_n to X$$ in probability if and only if $$mathbb{E}[|X_n-X|wedge1] to 0$$.
Proof. This easily follows from the inequality
$$epsilon mathbb{P}(|X_n-X|>epsilon) leq mathbb{E}[|X_n-X|wedge1] leq epsilon + mathbb{P}(|X_n-X|>epsilon)$$
which holds for any $$epsilon in (0, 1)$$.
Lemma 2. If $$(X_n)$$ is pointwise bounded by an integrable r.v. $$Y$$ and converges in probability to $$X$$, then $$lim_{ntoinfty} mathbb{E}[X_n] = mathbb{E}[X].$$
Proof. For any subsequence of $$(X_n)$$, there exists a further subsequence $$X_{n_k}$$ which converges a.s. to $$X$$. Then by the Dominated Convergence Theorem, $$mathbb{E}[X_{n_k}]tomathbb{E}[X]$$.
This implies the desired claim.
Returning to OP's question, fix $$0<a<b$$ and $$chi, varphi in C_c(mathbb{R})$$ such that $$mathbf{1}_{[-a,a]} leq chi leq mathbf{1}_{[-b,b]}$$ on all of $$mathbb{R}$$ and $$varphi(x) = x$$ on $$[-b,
b]$$. Then we find that
$$mathbb{P}(|X_n| > b) = 1 - mathbb{E}[mathbf{1}_{[-b,b]}(X_n)] leq 1 - mathbb{E}[chi(X_n)]$$
begin{align*} &(|X_n - X|wedge 1)mathbf{1}_{{|X_n|leq b}cap{|X|leq b}} \ &= (|varphi(X_n) - varphi(X)|wedge 1)mathbf{1}_{{|X_n|leq b}cap{|X|leq b}} \ &leq |varphi(X_n) - varphi(X)|. end{align*}
Using this, we may bound $$mathbb{E}[|X_n - X|wedge 1]$$ from above as follows:
begin{align*} mathbb{E}[|X_n - X|wedge 1] &leq mathbb{E}[(|X_n - X|wedge1) mathbf{1}_{{|X_n|leq b}cap{|X|leq b}}] + mathbb{P}(|X_n|>b) + mathbb{P}(|X|>b) \ &leq mathbb{E}[|varphi(X_n) - varphi(X)|] +
(1 - mathbb{E}[chi(X_n)]) + mathbb{P}(|X|>b). end{align*}
Taking $$limsup$$ as $$ntoinfty$$ and applying Lemma 1 and 2,
begin{align*} limsup_{ntoinfty} mathbb{E}[|X_n - X|wedge 1] &leq (1 - mathbb{E}[chi(X)]) + mathbb{P}(|X|>b) \ &leq mathbb{P}(|X|>a)+mathbb{P}(|X|>b). end{align*}
Since this limsup is independent of $$a$$ and $$b$$, letting $$btoinfty$$ followed by $$atoinfty$$ shows that the limsup is zero, or equivalently,
$$lim_{ntoinfty} mathbb{E}[|X_n - X|wedge 1] = 0.$$
Therefore the desired conclusion follows by Lemma 1.
Answered by Sangchul Lee on January 3, 2022
Use functions of the form $$f_M(x)=begin{cases}0 & xleq -M \ x+M & -M leq x leq M \ 3M-x & M < x < 3M \ 0 & xgeq 3Mend{cases}$$ for $$M>0$$.
Then begin{align*}mathbb{P}[|X_n-X|>3varepsilon] &leq mathbb{P}[|X_n+M-f_M(X_n)|>varepsilon] + mathbb{P}[|f_M(X_n)-f_M(X)|>varepsilon] + mathbb{P}[|f_M(X)-X-M|>varepsilon] \ &leq mathbb{P}[|X_n|>M] +
mathbb{P}[|f_M(X_n)-f_M(X)|>varepsilon] + mathbb{P}[|X|>M]end{align*}
Of the terms in the last line, the last is small for $$M>M_0$$ and the middle is small for $$n>n_0(M,varepsilon)$$. The only term we have to be careful with is the first term.
This is where our choice of class of functions matters and where my previous answer failed. Using the assumption that $$f_M(X_n)xrightarrow{mathbb{P}}f_M(X)$$ for all $$M>0$$ we can control that
first term, but I'll leave that to you.
Answered by Brian Moehring on January 3, 2022 | {"url":"https://transwikia.com/mathematics/fx_n-converges-in-probability-to-fx-implies-x_n-converges-in-probability-to-x/","timestamp":"2024-11-04T08:43:58Z","content_type":"text/html","content_length":"49430","record_id":"<urn:uuid:76134f9d-910e-46ad-94ac-f7c037eec7ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00698.warc.gz"} |
Barrier Synchronization (Part 2/2) - Matt Chung
Part 1 of barrier synchronization covers my notes on the first couple types of synchronization barriers including the naive centralized barrier and the slightly more advanced tree barrier. This post
is a continuation and covers the three other barriers: MCS barrier, tournament barrier , dissemination barrier.
In the MCS tree barrier, there are two separate data structures that must be maintained. The first data structure (a 4-ary tree, each node containing a maximum of four children) handling the arrival
of the processes and the second data structure handling the signaling and waking up of all other processes. In a nutshell, each parent node holds pointers to their children’s structure, allowing the
parent process to wake up the children once all other children have arrived.
The tournament barrier constructs a tree too and at each level are two processes competing against one another. These competitions, however, are fixed: the algorithm predetermines which process will
advanced to the next round. The winners percolate up the tree and at the top most level, the final winner signals and wakes up the loser. This waking up of the loser happens at each lower level
until all nodes are woken up.
The dissemination protocol reminds me of a gossip protocol. With this algorithm, all nodes detect convergence (i.e. all processes arrived) once every process receives a message from all other
processes (this is the key take away); a process receives one (and only one) message per round. The runtime complexity of this algorithm is nlogn (coefficient of n because during each round n
messages, one message sent from one node to its ordained neighbor).
The algorithms described thus far share a common requirement: they all require sense reversal.
MCS Tree Barrier (Binary Wakeup)
MCS Tree barrier with its “has child” vector
Okay, I think I understand what’s going on. There are two separate data structures that need to be maintained for the MCS tree barrier. The first data structure handles the arrival (this is the 4-ary
tree) and the second (binary tree) handles the signaling and waking up of all the other processes. The reason why the latter works so well is that by design, we know the position of each of the nodes
and each parent contains a pointer to their children, allowing them to easily signal the wake up.
Tournament Barrier
Tournament Barrier – fixed competitions. Winner holds the responsibility to wake up the losers
Construct a tree and at the lowest level are all the nodes (i.e. processors) and each processor competes with one another, although the round is fixed, fixed in the sense that the winner is
predetermined. Spin location is statically determined at every level
Tournament Barrier (Continued)
Two important aspects: arrival moves up the tree with match fixing. Then each winner is responsible for waking up the “losers”, traversing back down. Curious, what sort of data structure? I can see
an array or a tree …
Tournament Barrier (Continued)
Lots of similarity with sense reversing tree algorithm
Dissemination Barrier
Dissemination Barrier – gossip like protocol
Ordered communication: like a well orchestrated gossip like protocol. Each process will send a message to ordained peer during that “round”. But I’m curious, do we need multiple rounds?
Dissemination Barrier (continued)
Gossip in each round differs in the sense the ordained neighbor changes based off of Pi -> P(I + 2^k) mod n. Will probably need to read up on the paper to get a better understanding of the point of
the rounds ..
Quiz: Barrier Completion
Key point here that I just figured out is this: every processor needs to hear from every other processor. So, it’s log2N with a ceiling since N rounds must not be a power of 2 (still not sure what
that means exactly)
Dissemination Barrier (continued)
All barriers need sense reversal. Dissemination barrier is no exception. This barrier technique works for NCC and clusters.Every round has N messages. Communication complexity is nlogn (where N is
number of messages) and log(n). Total communication nlogn because N messages must be sent every round, no exception
Performance Evaluation
Most important question to ask when choosing and evaluating performance is: what is the trend? Not exact numbers, but trends. | {"url":"https://mattchung.me/barrier-synchronization-part-2-2/","timestamp":"2024-11-05T11:47:41Z","content_type":"text/html","content_length":"86459","record_id":"<urn:uuid:95a178c0-5903-457b-97f8-fb618c62a58e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00093.warc.gz"} |
Forward and inverse functional variations in elastic scattering
This paper considers the response of various types of elastic collision cross sections to functional variations in the intermolecular potential. The following cross sections are considered
differential, total, effective diffusion, and effective viscosity. A very simple expression results for the diffusion and viscosity cross sections at high energy relating the variations to the
classical deflection function. Attention is first given to the forward sensitivity densities δσ(E)/δV(R) [i.e., the functional derivative of cross sections σ(E) with respect to the potential surface
F(R)]. In addition inverse sensitivity densities δV(R)/δσ(E) are obtained. These inverse sensitivity densities are of interest since they are the exact solution to the infinitesimal inverse
scattering problem. Although the inverse densities do not in themselves form an inversion algorithm, they do give a quantitative measure of the importance of performing particular measurements for
the ultimate purpose of inversion. In addition, the degree to which different regions of a potential surface are correlated to a given set of cross sections are calculated by means of the densities
{δV(R)/ δV(R′)}. The overall numerical results contain elements which are physically intuitive as well as perplexing. This latter interesting and unexpected behavior is a direct result of allowing
for unconstrained cross section ↔ potential response, as well as the presence of quantum interference processes. The present focus on elastic scattering is simply for the purpose of illustration of
the functional variation technique which has broad applicability in all types of scattering processes.
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
• Physical and Theoretical Chemistry
Dive into the research topics of 'Forward and inverse functional variations in elastic scattering'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/forward-and-inverse-functional-variations-in-elastic-scattering","timestamp":"2024-11-03T00:20:04Z","content_type":"text/html","content_length":"53303","record_id":"<urn:uuid:135ee0d2-4c49-42c3-88bd-57ab99869765>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00347.warc.gz"} |
Zero temperature lattice Weinberg-Salam model for the values of the cutoff Λ∼1TeV
The lattice Weinberg-Salam model at zero temperature is investigated numerically. We consider the model for the following values of the coupling constants: the Weinberg angle θ [W]∼30°, the fine
structure constant α∼(1/150), and the Higgs mass M [H]∼ 150GeV. We find that the fluctuational region begins at the values of the cutoff Λ above about 0.8 TeV. In this region the average distance
between Nambu monopoles is close to their sizes. At Λ>1.1TeV the Nambu monopole currents percolate. Within the fluctuational region the considered definitions of the scalar field condensate give
values that differ from the expected one 2M [z]/g [z]. We consider the given results as an indication that the nonperturbative effects may be present in the Weinberg-Salam model at the large values
of the cutoff. Our numerical results were obtained on the lattices of sizes up to 163×32.
Dive into the research topics of 'Zero temperature lattice Weinberg-Salam model for the values of the cutoff Λ∼1TeV'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/zero-temperature-lattice-weinberg-salam-model-for-the-values-of-t-3","timestamp":"2024-11-09T19:36:09Z","content_type":"text/html","content_length":"53454","record_id":"<urn:uuid:02c23883-4380-456c-8f88-1db93626c0ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00885.warc.gz"} |
Mixed Regression Modeling Simplified
Mixed-Effects Regression Modeling
Mixed effects models work for correlated data regression models, including repeated measures, longitudinal, time series, clustered, and other related methods.
Why not to use simple regression for correlated data
One key assumption of ordinary linear regression is that the errors are independent of each other. However, with repeated measures or time series data, the ordinary regression residuals usually
are correlated over time. Hence, this assumption is violated for correlated data.
Definition of Mixed Regression Model
It includes features of both fixed and random effects. Whereas, standard regression includes only fixed effects.
Example :
Examination Result (target variable) could be related to how many hours students study (fixed effect), but might also be dependent on the school they go to (random effect), as well as simple
variation between students (residual error).
Fixed Effects vs. Random Effects
Fixed effects assume observations are independent while random effects assume some type of relationship exists between some observations.
Gender is a fixed effect variable because the values of male / female are independent of one another (mutually exclusive); and they do not change. Whereas, school has random effects because we
can only sample some of the schools which exist; not to mention, students move into and out of those schools each year.
A target variable is contributed to by additive fixed and random effects as well as an error term.
yij = β1x1ij + β2x2ij … βnxnij + bi1z1ij + bi2z2ij … binznij + εij
where yij is the value of the outcome variable for a particular ij case, β1 through βn are the fixed effect coefficients (like regression coefficients), x1ij through xnij are the fixed effect
variables (predictors) for observation j in group i (usually the first is reserved for the intercept/constant; x1ij = 1), bi1 through bin are the random effect coefficients which are assumed to be
multivariate normally distributed, z1ij through znij are the random effect variables (predictors), and εij is the error for case j in group i where each group’s error is assumed to be multivariate
normally distributed.
Mixed Models vs. Time Series Models
1. Time series analysis usually has long time series and the primary goal is to look at how a single variable changes over time. There are sophisticated methods to deal with many problems - not just
autocorrelation, but seasonality and other periodic changes and so on.
2. Mixed models are not as good at dealing with complex relationships between a variable and time, partly because they usually have fewer time points (it's hard to look at seasonality if you don't
have multiple data for each season).
3. It is not necessary to have time series data for mixed models.
SAS : PROC ARIMA vs. PROC MIXED
The ARIMA and AUTOREG procedures provide more time series structures than PROC MIXED.
The data used in the example below contains the interval scaled outcome variable Extroversion (extro) is predicted by fixed effects for the interval scaled predictor Openness to new experiences
(open), the interval scaled predictor Agreeableness (agree), the interval scaled predictor Social engagement (social), and the nominal scaled predictor Class (class); as well as the random (nested)
effect of Class within School (school). The data contains 1200 cases evenly distributed among 24 nested groups (4 classes within 6 schools).
R Code :
Step I : Load Data
# Read data
lmm.data <- read.table("http://www.unt.edu/rss/class/Jon/R_SC/Module9/lmm.data.txt",
header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
Step II : Install and load library
# Install and load library
Step III : Building a linear mixed model
# Building a linear mixed model
lmm.2 <- lmer(formula = extro ~ open + agree + social + class + (1| school/class), data = lmm.data, REML = TRUE, verbose = FALSE)
The random effect specifies the nested effect of class within (or under) school; as class would be considered the level one variable and school the level two variable -- which is why the
forward slash
is used.
# Check Summary
Calculating total variance of the random effects
Add all the variance together to find the total variance (of the random effects) and then divide that total by each random effect to see what proportion of the random effect variance is attributable
to each random effect (similar to R² in traditional regression). So, if we add the variance components:
= 2.8836 + 95.1718 + 0.9684
= 99.02541
Then we can divide this total variance by our nested effect variance to give us the proportion of variance accounted for, which indicates whether or not this effect is meaningful.
= 2.8836/99.02541 = 0.02912030
So, we can see that only 2.9% of the total variance of the random effects is attributed to the nested effect. If all the percentages for each random effect are very small, then the random effects
are not present and linear mixed modeling is not appropriate (i.e. remove the random effects from the model and use general linear or generalized linear modeling instead). We can see that the
effect of school alone is quite substantial (96%) = 95.17339/99.02541
Output : Estimates of the fixed effects
These estimates are interpreted the same way as one would interpret estimates from a traditional ordinary least squares linear regression.
A one unit increase in the predictor Openness to new experiences (open) corresponds to a 0.0061302 increase in the outcome Extroversion (extro). Likewise, a one unit increase in the predictor
Agreeableness (agree) corresponds to a 0.0077361 decrease in the outcome Extroversion (extro). Furthermore, the categorical predictor classb has a coefficient of 2.0547978; which means, the mean
Extroversion score of the second group of class (b) is 2.0547978 higher than the mean Extroversion score of the first group of class (a).
Extract Estimates of Fixed and Random Effects
#To extract the estimates of the fixed effects
#To extract the estimates of the random effects
#To extract the coefficients for the random effects intercept (2 groups of school) and each group of the random effect factor, which here is a nested set of groups (4 groups of class within 6
groups of school)
Calculating Predicted Values
#To extract the fitted or predicted values based on the model parameters and data
yhat <- fitted(lmm.2)
lmm.data2 = cbind(lmm.data,yhat)
#Score a new dataset
yhat1 = predict(lmm.2, lmm.data)
#To extract the residuals (errors) and summarize them, as well as plot them
residuals <- resid(lmm.2)
Check Intra Class Correlation
It allows us to assess whether or not the random effect is present in the data.
lmm.null <- lmer(extro ~ 1 + (1|school), data = lmm.data)
R Package : Regression Tree for Mixed Data R Code : Random Forest for Binary Mixed Model SAS Code :
PROC MIXED DATA=mydata;
CLASS class school;
MODEL extro = open agree social class school*class / solution outp=test;
random school / solution SUBJECT = id TYPE = UN;
ods output solutionf=sf(keep=effect estimate rename=(estimate=overall));
ods output solutionr=sr(rename=(estimate=ssdev));
Practice Example
Post Comment 2 Responses to "Mixed Regression Modeling Simplified"
1. Hi, I have a problem and need some assistance. I want to model changes in viral shedding from baseline (Visit 0/Week 0) to Visit 12 (Week 12). I am using Stata Software. The outcome is viral
shedding which is log transformed. I also need to control for ARV uptake. Any idea on how to go about it will be highly appreciated. The Stata Commands I am using are:-
xtmixed f07q04c i.visit if f03q10==0 || subject: visit, mle nolog vce(robust)
xtmixed f07q04c i.visit if f03q10==1 || subject: visit, mle nolog vce(robust)
Where f07q04c is the variable representing viral shedding; i.visit is an indicator variable for visit since my reference point is baseline visit; f03q10==0 represents not on ARV and f03q10==1 ARV
Active ; subject is subject id and visit is follow-up time in weeks.
This is a longitudinal study but not all subjects have the same follow up times. There are missing observations too. Thanks.
2. yij β1x1ij + β2x2ij … βnxnij + bi1z1ij + bi2z2ij … binznij + εij
in the above equation, don't you think that we shd/can have INTERCEPT as well.
probably there will be certain value if we will keep no variables in our model | {"url":"https://www.listendata.com/2015/12/mixed-regression-modeling-simplified.html","timestamp":"2024-11-04T23:22:02Z","content_type":"application/xhtml+xml","content_length":"118476","record_id":"<urn:uuid:10236d9b-738e-4041-9165-1a5f51e01a40>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00334.warc.gz"} |
Dirac structures and boundary relations
It is shown how Dirac structures, boundary control and conservative state/signal system nodes, and unitary boundary relations are connected via proper transforms between the underlying Krein spaces
that are involved in all of these notions.
Original language Undefined
Title of host publication Operator Methods for Boundary Value Problems
Editors S. Hassi, H.S.V. de Snoo, F.H. Szafraniec
Place of Publication Cambridge
Publisher Cambridge University Press
Pages 259-274
Number of pages 16
ISBN (Print) 978-1-107-60611-1
Publication status Published - 2012
Publication series
Name London Mathematical Society Lecture Note Series
Publisher Cambridge University Press
Number 404
• EWI-23006
• IR-86190
• METIS-297599 | {"url":"https://research.utwente.nl/en/publications/dirac-structures-and-boundary-relations","timestamp":"2024-11-09T09:16:02Z","content_type":"text/html","content_length":"40846","record_id":"<urn:uuid:b11c0d8e-6093-4f2a-9885-9402eafccd57>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00605.warc.gz"} |
Lesson 9
Equations of Lines
• Let’s investigate equations of lines.
Problem 1
Select all the equations that represent the graph shown.
Problem 2
A line with slope \(\frac32\) passes through the point \((1,3)\).
1. Explain why \((3,6)\) is on this line.
2. Explain why \((0,0)\) is not on this line.
3. Is the point \((13,22)\) on this line? Explain why or why not.
Problem 3
Write an equation of the line that passes through \((1,3)\) and has a slope of \(\frac{5}{4}\).
Problem 4
A parabola has focus \((3,\text{-}2)\) and directrix \(y=2\). The point \((a,\text{-}8)\) is on the parabola. How far is this point from the focus?
(From Unit 6, Lesson 8.)
Problem 5
Write an equation for a parabola with each given focus and directrix.
1. focus: \((5, 2)\); directrix: \(x\)-axis
2. focus: \((\text{-}2, 3)\); directrix: the line \(y=7\)
3. focus: \((0,7)\); directrix: \(x\)-axis
4. focus: \((\text{-}3, \text- 4)\); directrix: the line \(y=\text-1\)
(From Unit 6, Lesson 8.)
Problem 6
A parabola has focus \((\text{-}1,6)\) and directrix \(y=4\). Determine whether each point on the list is on this parabola. Explain your reasoning.
1. \((\text{-}1,5)\)
2. \((1 ,7)\)
3. \((3, 9)\)
(From Unit 6, Lesson 7.)
Problem 7
Select the center of the circle represented by the equation \(x^2 + y^2 - 8x + 11y - 2 = 0\).
\((\text-4, \text-5.5)\)
(From Unit 6, Lesson 6.)
Problem 8
Reflect triangle \(ABC\) over the line \(x=\text-6\).
Translate the image by the directed line segment from \((0,0)\) to \((5,\text-1)\).
What are the coordinates of the vertices in the final image?
(From Unit 6, Lesson 1.) | {"url":"https://curriculum.illustrativemathematics.org/HS/students/2/6/9/practice.html","timestamp":"2024-11-07T23:48:18Z","content_type":"text/html","content_length":"81312","record_id":"<urn:uuid:fc13db0b-0c26-4338-b8c3-633004b2b68a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00021.warc.gz"} |
A)Rhombus; B)Parallelogram; C)Kite;
A square is also a A)Rhombus; B)Parallelogram; C)Kite; D)none of these
Marcus Lane
Answered question
A square is also a
D)none of these
Answer & Explanation
The true choices are
B Parallelogram
C Kite
Since it has opposing pairs of sides that are both parallel and equal, a square is a parallelogram.
Additionally, it is a rhombus since all of its sides are equal, the diagonals connect at 90 degrees, and the opposite sides are parallel.
Because all of its angles are 90 degrees, and its two opposite sides are equal and parallel, a square is also a rectangle.
A square can't be a trapezium as in trapezium only one of the opposite sides are parallel. | {"url":"https://plainmath.org/algebra-i/104241-a-square-is-also-a-a-rhombus","timestamp":"2024-11-09T18:45:08Z","content_type":"text/html","content_length":"154437","record_id":"<urn:uuid:bf1d5054-5db3-438b-8577-fec9bd6334b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00262.warc.gz"} |
Emulators and surrogate models via ML – The Dan MacKinlay stable of variably-well-consider’d enterprises
Emulators and surrogate models via ML
Shortcuts in scientific simulation using ML
August 12, 2020 — August 26, 2020
feature construction
functional analysis
linear algebra
machine learning
neural nets
sparser than thou
Emulation, a.k.a. surrogate modelling. In this context, it means reducing complicated physics-driven simulations to simpler or faster ones using ML techniques. Especially popular in the ML for
physics pipeline. I have mostly done this in the context of surrogate optimisation for experiments. See Neil Lawrence on Emulation for a modern overview.
A recent, hyped paper that exemplifies this approach is Kasim et al. (2020), which (somewhat implicitly) uses arguments from Dropout ensembling to produce quasi-Bayesian emulations of notoriously
slow simulations. Does it actually work? And if it does quantify posterior predictive uncertainty well, can it estimate other posterior uncertainties?
Emukit (Paleyes et al. 2019) is a toolkit that generically wraps ML models for emulation purposes.
ML PDEs might be a particularly useful domain.
1 Model order reduction
The traditional, and still useful, approach is reduced order modelling, which has many related tricks. | {"url":"https://danmackinlay.name/notebook/ml_emulation.html","timestamp":"2024-11-08T22:13:49Z","content_type":"application/xhtml+xml","content_length":"44158","record_id":"<urn:uuid:6453f6a3-da73-4f5f-98f5-746d9e50a7fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00787.warc.gz"} |
Simplify Your Fractional Equations with Our Accurate Partial Fraction Calculator - Age calculator
Simplify Your Fractional Equations with Our Accurate Partial Fraction calculator
Simplify Your Fractional Equations with Our Accurate Partial Fraction calculator
Partial fraction is a branch of algebra that deals with breaking down complex fractions into simpler fractions to make them easier to solve. This process allows you to perform complex mathematical
operations with a greater level of accuracy and ease. However, it can be challenging to solve partial fraction equations manually, especially when the equation involves high degrees of polynomials
and lengthy calculations.
Fortunately, technology now allows for the automation of partial fraction equations, thanks to powerful partial fraction calculators that simplify equations in a matter of seconds. Our accurate
partial fraction calculator offers solutions to even the most complex equations easily and accurately.
Here’s a look at some of the benefits of using our partial fraction calculator:
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This calculator allows you to solve complex partial fraction equations with speed and high levels of accuracy. Since the calculator performs all the necessary calculations using the most advanced
technologies, the results you get are always correct and reliable.
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Using our partial fraction calculator means that you don’t have to spend hours cracking your head over complex calculations. The calculator is user-friendly, and you can access it from anywhere using
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Our partial fraction calculator produces solutions that are easy to understand since they are presented in a step-by-step format. You can also use the solutions provided to check if your calculations
are correct.
How to Use Our Partial Fraction calculator
Our partial fraction calculator is user-friendly and straightforward to use. Here are the steps to follow:
1. Input the equation
The first thing you need to do is input the partial fraction equation you need to solve into the calculator. The calculator allows you to enter equations of any degree, making it an all-around
solution to your algebraic problems.
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Once you enter the equation, the calculator will automatically simplify it and present it in its simplest form. You can use this feature to check your work or if you’re having difficulties
simplifying the equation yourself.
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The next step is to break down the fraction into simpler partial fractions, which the calculator will also do for you.
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Once you have broken down the fraction into simpler partial fractions, you can then go ahead and solve each one using the calculator. You can then check the answers you get against your manual
5. Get the overall solution
After solving the partial fraction equation, the calculator will present you with its overall solution, providing you with a complete answer to the problem.
Frequently Asked Questions (FAQs)
Q1. What is a partial fraction equation?
A partial fraction equation is an algebraic equation that involves breaking down a complex fraction into simpler fractions to make it easier to solve.
Q2. Can I use the partial fraction calculator to solve my exam problems?
Yes. Our partial fraction calculator provides you with accurate solutions that you can use to verify your manual calculations or solve your exam problems.
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Using a partial fraction calculator is incredibly helpful in solving algebraic equations that involve fractional expressions. It makes the process much faster, easier, and more accurate. Our accurate
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Simplify Your Fractional Equations with Our Accurate Partial Fraction calculator today and get the correct solutions to your algebraic problems at the touch of a button.
Recent comments | {"url":"https://age.calculator-seo.com/simplify-your-fractional-equations-with-our-accurate-partial-fraction-calculator/","timestamp":"2024-11-04T03:58:23Z","content_type":"text/html","content_length":"304541","record_id":"<urn:uuid:a64bcf23-be94-4459-aa47-f65ba7c7014d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00356.warc.gz"} |
An Exceptional Exponential Embedding
This week in class I get to teach one of my favorite probability arguments, which makes use of a very unusual embedding. Here's a short description (for the longer description, see
the bookoriginal paper
, where the idea is ascribed to Rubin).
The setting is balls and bins with feedback: I have two bins, and I'm repeatedly randomly throwing balls into the bins one at at time. When there are x balls in bin 1 and y balls in bin 2, the
probability the ball I throw lands in bin 1 is x^p/(x^p+y^p), and the probability it lands in bin 2 is y^p/(x^p + y^p). Initially both bins start with one ball. The goal is to show that when p is
greater than 1, at some point, one bin gets all the remaining balls thrown. That is, when there's
positive feedback
, so the more balls you have the more likely it is that you'll get the next one in a super-linear fashion, eventually the system becomes winner-take-all.
We use the following "exponential embedding". Consider the following process for bin 1. At time 0, we associated an an exponentially distributed random variable X_1 with mean 1 = 1/1^p with the bin.
The "time" that bin 1 receives its next ball is X_1. Now it has two balls. We then associate an exponentially distributed random variable X_2 with mean 1/2^p with the bin. And so on.
Do the same thing with bin 2, using random variables Y_1, Y_2, ...
Now, at any point in time, due to the properties of the exponential distribution -- namely, it's memoryless, and the minimum of two exponentials with mean a_1 and a_2 will be the first with
probability proportional to 1/a_1 and the second with probability 1/a_2 -- if the loads in the bins are x for bin 1 and y for bin 2, then the next ball will fall into bin 1 is x^p/(x^p+y^p), and the
probability it lands in bin 2 is y^p/(x^p + y^p). That is, this exponential process is equivalent to the initial balls and bins process.
Now let X = sum X_i and Y = sum Y_i. The infinite sums converge with probability 1 and are unequal with probability 1. So suppose X < Y. Then at some finite "time" in our exponential embedding, bin 1
receives an infinite number of balls while bin 2 just has a finite number of balls, and similarly if Y < X. So eventually, one bin will be the "winner" and take all the remaining balls.
In my mind, that's a beautiful proof.
4 comments:
Are you using this as a metaphor for wealth in this country? :)
Dark humor aside, yes, it's a beautiful proof.
Hi Mike!
Nice. Here are some pedagogical comments.
- If I was teaching this, I would do it just for p=1: it's simpler but the argument is the same and it is just as beautiful. Then in the end I would throw in a comment: "Consider this
generalization. The proof extends (exercise)." That is, I try to teach the simplest problem whose solution uses the ideas I want to convey. I know that many mathematicians do the opposite (try to
see how far the ideas can go, and show the most general result that can be obtained from those ideas), but I find that simpler works better for me.
- If I was teaching this, instead of defining the process formally I would do it from example, with concrete values for x and y. "When there are 7 balls in bin 1 and 3 balls in bin 2, the
probability that the ball lands in bin 1 is 7/(7+3)=.7 and the probability that it lands in bin 2 is 3/(7+3)=.3." Students can always ask questions if that's not enough to make things clear.
- Also, what is missing for a computer scientist's mind such as mine is: Why? Where does this process come from? I almost stopped reading your entry in the middle of the second paragraph because
the motivation is missing. Even for a blog entry, a sentence about that would be greatly helpful.
By the way, this is a nice post. Enjoyable for me to read. I knew this already, but if you did it with something that I didn't know, it would be great fun, a very nice way to learn a little
nugget of science.
Um, p=2, not p=1. (Embarrassed.)
Hi Mike,
C5838ould you please show why "the infinite sums converge with probability 1 and are unequal with probability 1"? Thank you very much. | {"url":"https://mybiasedcoin.blogspot.com/2011/10/exceptional-exponential-embedding.html?showComment=1319605573928","timestamp":"2024-11-13T02:01:47Z","content_type":"application/xhtml+xml","content_length":"84753","record_id":"<urn:uuid:01646ae7-22a5-404e-8dde-c3fac343aa97>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00808.warc.gz"} |
Key Terms
conditional probability
the likelihood that an event will occur given that another event has already occurred
contingency table
the method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other; the table provides an easy way to
calculate conditional probabilities
dependent events
if two events are NOT independent, then we say that they are dependent
equally likely
each outcome of an experiment has the same probability
a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by S
An event is an arbitrary subset in S. It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital
letters such as A, B, C, and so on.
a planned activity carried out under controlled conditions
independent events
The occurrence of one event has no effect on the probability of the occurrence of another event; events A and B are independent if one of the following is true:
1. P(A|B) = P(A)
2. P(B|A) = P(B)
3. P(A AND B) = P(A)P(B)
mutually exclusive
two events are mutually exclusive if the probability that they both happen at the same time is zero; if events A and B are mutually exclusive, then P(A AND B) = 0
a particular result of an experiment
a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following three axioms (by A.N. Kolmogorov,
1930s): Let S denote the sample space and A and B are two events in S; then
□ 0 ≤ P(A) ≤ 1,
□ If A and B are any two mutually exclusive events, then P(A OR B) = P(A) + P(B), and
□ P(S) = 1
sample space
the set of all possible outcomes of an experiment
sampling with replacement
if each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once
sampling without replacement
when sampling is done without replacement, each member of a population may be chosen only once
the AND event
an outcome is in the event A AND B if the outcome is in both A AND B at the same time
the complement event
the complement of event A consists of all outcomes that are NOT in A
the conditional probability of one event GIVEN another event
P(A|B) is the probability that event A will occur given that the event B has already occurred
the OR event
an outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B
the OR of two events
an outcome is in the event A OR B if the outcome is in A, is in B, or is in both A and B
tree diagram
the useful visual representation of a sample space and events in the form of a tree with branches marked by possible outcomes together with associated probabilities (frequencies, relative
Venn diagram
the visual representation of a sample space and events in the form of circles or ovals showing their intersections | {"url":"https://texasgateway.org/resource/key-terms-24?book=79081&binder_id=78226","timestamp":"2024-11-04T11:16:34Z","content_type":"text/html","content_length":"42008","record_id":"<urn:uuid:af45538f-57e0-48f9-b387-32a6b517479d>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00019.warc.gz"} |
Creating Totalizers in PI Server Historian
This is a blog post about totalizers! As simple as totalizers may seem, a lot can go wrong. There are many ways to get totals. Let’s look at some ways of calculating totals on the OSIsoft® PI System™
Calculating totals is common in control system and facility monitoring applications. Frequently this is accomplished using totalizers in devices such as PLCs or meters. Device totalizers are useful
but can be difficult to incorporate into reporting since they may have unexpected resets; are not time synchronized to server systems; and may not provide the totals needed for reports. A better
approach is to calculate totals in the data historian.
A totalizer calculation converts a rate or incrementing value to a usage value per unit of time, such as an hour or day. The result in each case is a usage quantity that is summable; therefore, you
can sum up the amounts to obtain the volume or energy amounts used over the intervals of time. Here are some examples:
• A liquid flow rate in gallons/minute (gpm) is converted to volume in gallons
• Electrical power in kilowatts (kW) is converted to energy usage in kilowatt-hours (kWh)
• Cooling power in Tons of Refrigeration (12,000 btu/hr) is converted to energy usage in ton-hours
Another scenario where a totalizer is useful is to convert an incrementing counter into a daily count. Difficulties with device counters are that the counter can be reset manually at any time; may
reset periodically but at the wrong time of day; or never resets and becomes a huge and hard to read number. By reading the device counter into a PI calculation, the counts can be converted into a
reliable daily total, that provides a new value at the same time each day.
The PI System™ provides several methods for calculating totals:
• PI Point Summary Calculations
• PI Performance Equation (PE) tags
• PI Totalizer tags
• Asset Framework (AF) Analysis Expressions
• ACE calculation engine (ACE is not included in this evaluation)
Comparison of Totalizer Calculation Types
PI Point Summary Calculations
PI DataLink is an Excel add-in that can be used to extract data from PI using PI Point Summary Calculation functions. To obtain totals from a flow rate tag, use the time-weighted totals formula. PI
gets the Integral under the flow rate curve to get the time-weighted total. Alternatively, use a time-weighted average multiplied by the number of time intervals, to get the same result.
PI Point Summary Calculations are useful for verifying totalizer tag and AF Expression calculations and for getting totals for points that do not have a totalizer calculation configured.
PI Performance Equation (PE) Tags
The PI Performance Equation subsystem is used to execute PI PE calculation tags. The PE provides a set of functions and operators that can be used to program PI tags to perform calculations. The same
calculation functions using a similar syntax can be used in the Asset Framework (AF) Analysis Expressions.
PI Totalizer Tags
The PI Totalizer subsystem is used to create PI Totalizer tags and to calculate totals, averages, minimum and maximum values, standard deviations, and counts. The Totalizer uses snapshot data, which
is the data available before the compressed data is stored to the data archive. Therefore, the totals obtained using a totalizer tag may be more accurate than using other methods.
Configuring totalizer tags is straightforward, but there are many options to choose. This provides great flexibility but can also make the configuration challenging.
Totalizer tags provide an easy way to configure Daily or Monthly totals. The totals can be configured to output one value at the end of the time interval, or to have a running total (ramping total)
that outputs an incrementing total until it resets at the end of the time interval.
• A single value per time interval is most useful for long-term reporting. For daily time intervals, the timestamp will be for midnight of the next day, so one day must be subtracted from the
timestamp date. For example, a timestamp for a daily total at 5/2/2020 00:00 is the total for the volume that accumulated on 5/1/2020.
• A running total is most useful for displaying on trends or a current value on real-time screens. It will reset at the end of the time interval and start to ramp up again.
PI Asset Framework Analysis Expressions
PI Asset Framework Analysis Expressions is an environment to program scripts with multiple steps, using built-in PE functions and operators. AF executes the calculations using AF attributes or PI
Tags as inputs, with the output value of the script mapped to an AF attribute. Analysis Expressions can be configured in an AF element or in an AF template. AF will process the scripts independently
from the PI Data Archive using the Asset Framework Analysis services and database, which may be installed on a separate server from the PI system to handle additional load.
Example using the TagTot function to calculate the Daily Total and output to an AF attribute
Trend of Daily Total using AF Analysis Expression to calculate the Daily Total
Totalizer Calculation Example
Trend of CDT158, used at flow rate tag in these examples
Totalizer Calculation Results for 1 day
The totalizer comparison above, shows the totalizer calculations for each method, using the PI simulation tag CDT158 as the input. This tag has a compression deviation set to 4 engineering units.
Since totalizer tags calculate based on the snapshot data before compression, the totalizer tag result is not equal to the AF calculation result. But the difference is very small, and likely is much
smaller than the flowmeter accuracy in real-world applications. If necessary, to improve totalizer calculation accuracy for AF calculations, reduce compression on the input tag.
Give us a call and we can work with you to help bring increased value to your PI Data System and other Historians.
About the Author
Steve has retired from Hallam-ICS, but his contributions to the company continue to be valued.
Read My Hallam Story
About Hallam-ICS
Hallam-ICS is an engineering and automation company that designs MEP systems for facilities and plants, engineers control and automation solutions, and ensures safety and regulatory compliance
through arc flash studies, commissioning, and validation. Our offices are located in Massachusetts, Connecticut, New York, Vermont and North Carolina and our projects take us world-wide. | {"url":"https://www.hallam-ics.com/blog/creating-totalizers-in-pi-server-historian","timestamp":"2024-11-12T12:21:35Z","content_type":"text/html","content_length":"86672","record_id":"<urn:uuid:fe8402df-ff54-4a0b-8327-bf0129c3dd19>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00584.warc.gz"} |
Improving Your Statistical Inferences Coursera course
I’m really excited to be able to announce my “
Improving Your Statistical Inferences
” Coursera course. It’s a free massive open online course (MOOC) consisting of 22 videos, 10 assignments, 7 weekly exams, and a final exam. All course materials are freely available, and you can
start whenever you want.
In this course, I try to teach all the stuff I wish I had learned when I was a student. It includes the basics (e.g., how to interpret p-values, what likelihoods and Bayesian statistics are, how to
control error rates or calculate effect sizes) to what I think should also be the basics (e.g., equivalence testing, the positive predictive value, sequential analyses, p-curve analysis, open
science). The hands on assignments will make sure you don’t just hear about these things, but know how to use them.
My hope is that busy scholars who want to learn about these things now have a convenient and efficient way to do so. I’ve taught many workshops, but there is only so much you can teach in one or two
days. Moreover, most senior researchers don’t even have a single day to spare for education. When I teach PhD students about new methods, their supervisors often respond by saying ‘I've never heard
of that, I don't think we need it’. It would be great if everyone has the time to watch some of my videos while doing the ironing, chopping vegetables, or doing the dishes (these are the times I
myself watch Coursera videos), and see the need to change some research practices.
This content was tried out and developed over the last 4 years in lectures and workshops for hundreds of graduate students around the world – thank you all for your questions and feedback! Recording
these videos was made possible by a grant from by the 4TU Centre for Engineering Education at the recording studio of the TU Eindhoven (if you need a great person to edit your videos, contact
Tove Elfferich
). The assignments were tested by Moritz Körber, Jill Jacobson, Hanne Melgård Watkins, and around 50 beta-testers who tried out the course in the last few weeks (special shout-out to Lilian
Jans-Beken, the first person to complete the entire course!). I really enjoy seeing the
positive feedback
I can recommend the Coursera course on statistics by
- I learned a lot. I particularly like that it provides options not dogma.
— David J Bishop (@BlueSpotScience) October 5, 2016
— Lilian Jans-Beken (@lilianjansbeken) October 6, 2016
Tim van der Zee helped with creating exam questions, and Hanne Duisterwinkel at the TU Eindhoven helped with all formalities. Thanks so much to all of you for your help.
This course is brand new – if you follow it, feel free to send feedback and suggestions for improvement.
I hope you enjoy the course.
11 comments:
1. I am enjoying the course very much. I have some confusion about the classification of statistics in terms of Neyman-Pearson, Bayesian and likelihoods. Is this classification standard?
I thought that Neyman-Pearson was the same approach that likelihoods. Given a null hypothesis test, I use a statistic that often corresponds to the statistic arising from a likelihood ration test
(for example the t-test for the mean).
1. The distinction is commonly used - see the references in the course to Zoltan Dienes' book for example. There is a difference between a likelihood ratio (see lecture 2.1) and maximum
likelihood estimation. The 3 approaches are also discussed in Royall's book on Likelihoods.
2. Thanks! I missed the references in the course.
2. Great course. I'm thinking the systematic problems why many misunderstandings about p value have overwhelmed the researchers' minds for decades. One problem is the over-dominance of frequentist
approach in the research and education. Look at the classical textbook, Roger Kirk's "Experimental Design", it collects the classical works of the frequentist approach and has inspired the
generations of researchers. However, the cautions in this book have yet explicitly influence its readers make serious decision in their study. This is the time to adjust the way to teach the
behavioral scientists use the statistical concepts and tools.
Is there any idea to attract the non-English Coursera users join this course? I'm glad to help translate the materials to Chinese.
1. That would be totally awesome! E-mail me, and we can work something out.
3. Daniel, what a great course. Thank you so much for taking the time to create it. I am a PhD student and have studied stats for years. I'm only up to Week 4 of the course and I have already
recognised many fundamental misunderstandings I have gained through my university courses! Not only that, but this course is waaaaaay more interesting than the frequentist stats courses I've been
forced to do. Again, thank you :) Keep up the amazing work.
1. Thanks for the positive feedback! Glad you are enjoying it!
4. Hi, I am having trouble finding the forum/board for this course. I am just running into some errors running the script for the first assignment and feel it is something I could self-diagnose with
other students instead of having to bother you with it. I have some experience with R, but trouble deciphering error messages. Thank you so much for this course.
1. You enrolled in a course that starts in the future? Then the forum is not open yet.
5. Wow, thank you so much for your quick response. I was able to get the correct result by installing an updated version of R Studio. I actually was not able to fully install it, but it seems to run
fine from the disk mounted image, and when I ran the script in this I was able to do the simulations. I still don't fully understand everything, but am getting a better feel/familiarity with the
concepts. Thank you so much for offering this class.
6. thanks | {"url":"https://daniellakens.blogspot.com/2016/10/improving-your-statistical-inferences.html","timestamp":"2024-11-12T02:46:46Z","content_type":"application/xhtml+xml","content_length":"115916","record_id":"<urn:uuid:96e02c42-2172-4f82-a93e-fae951f723f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00056.warc.gz"} |
A new record? 27-plus years later, a notice of redundant publication
A 1984 paper in Philosophical Transactions of The Royal Society B is now subject to a notice of redundant publication because a lot of it had been published in Cell the same year.
Whether 28 years — 27 years and 9 months, to be precise — is any kind of official record is unclear, since we haven’t really kept track of notices of redundant publication. It would, however, beat
the record for longest time between publication and retraction, 27 years and one month.
Here’s the notice, which ran in September of last year but just came to our attention:
Phil. Trans. R. Soc. Lond. B 307, 271–282 (24 December 1984) (doi:10.1098/rstb.1984.0127)
After the publication of this article, it was brought to the attention of the editors of Phil. Trans. R. Soc. B that this article contains substantial content which was included in a previously
published article [1], without referencing the prior publication.
1. Wright S., Rosenthal A., Flavell R., Grosveld F. 1984. DNA sequences required for regulated expression of beta-globin genes in murine erythroleukemia cells. Cell 38, 265–273.
The Royal Society version of the paper has been cited five times, according to Thomson Scientific, while the Cell version has been cited 189 times.
We’ve asked the editor of Philosophical Transactions of The Royal Society B what prompted the notice, and why it happened now, but have not heard back. We do know that pseudonymous whistleblower
Clare Francis wrote to the journal’s editor in July 2011 pointing out that the papers looked similar, and that a journal staffer suggested they’d be asking for a correction.
One of the papers’ authors, Yale’s R.A. Flavell, was one of many authors of a Journal of Biological Chemistry paper corrected last year because figures were “excessively edited.”
Update, 8:30 a.m. Eastern, 5/9/13: The journal tells us:
We looked into the two papers and decided that there was enough new material in our paper to warrant a notice of redundant publication rather than a full retraction. The problem was brought to
our attention recently.
35 thoughts on “A new record? 27-plus years later, a notice of redundant publication”
1. HUGE!!! Would you believe this? The last two authors of these papers are really BIG scientists in the field!! Amazing – how did they find out – after digitilising those articles?
1. Are they big scientists or cheaters!
1. RA Flavell has 2 previous retractions:-
1. Science. 2005 Dec 23;310(5756):1903.
This one might be thought as doing the right thing once you realize that something is not right.
2. J Clin Invest. 2003 Nov;112(10):1597.
Administrative things go wrong all the time.
“There were no scientific errors in the paper, and we stand by the validity, importance, and interest of the results.” If that be the case why not resubmit? Perhaps the authors did,
although I could not find anything.
I do not understand why people “stand by the validity” without the evidence being made public. It sounds like management-speak.
1. Retraction Nr. 2. The animal experiments were not aproved by the local “Institutional Animal Care and Usage Committee”. That was the reason for the retraction. Probably the results
were fine.
2. Yes Grosveld and Flavell are huge namesm, Ive read several of their papers. It seems like Flavell has his name associated with a number of retractions, including this:
Flavell is the kind of name you work for and get a job, even if you have a crappy publication record with him. This is pure speculation,. but maybe he is just not carefully monitoring what is
going on in the lab.
I did see him speak…I left early, a terrible presentation.
3. This notice seems to solely satisfy copyright claims from “Cell press” probably being the driving force to this notification, because it solely seems to fullfill citation policies. “Contains
substantial content” is in my opinion still not an acceptable approach. It is still not satisfactory if in reality it is a redundant publication or duplicated manuscript, because then the overall
publication does not correspond to general publication ethics. Unfortunately the secondary publication is behind a paywall. Can someone give a statement about the content of the secondary
publication and the reference?
1. RE “We looked into the two papers and decided that there was enough new material in our paper to warrant a notice of redundant publication rather than a full retraction. The problem was
brought to our attention recently.”
Figures which are recognizably the same in both publications.
Cell 1984 Phil Trans
Figure 3 Lower half figure 3.
Figure 4 Figure 4
Figure 5 Figure 5
Figure 6 Figure 6
Figure 7 Figure 7
Figure 8 Figure 8
How much overlap do you need to get a retraction from Phil Trans?
There was another notice of a redundant publication in Phil Trans recently.
On looking at the publications concerned I noticed that in the earlier publication, where there we two authors “we” was employed, whereas in the redundant later publication, where there is a
sinlge author, “I” was employed. The long and short of it.
“Enough new material in our paper to warrant a notice of redundant publication rather than a full retraction” is akin to Newspeak slogans from Minitrue.
In some ways the twisted slogans are a reflection of reality.
In historical times the Commonwealth of England’s motto was “Peace is sought through war”.
Phil Trans started only 5 years after the end of the Commonwealth.
4. The group that corrected the JBC paper with Flavell last year have some history of making ugly figures.
Cell. 2005 Aug 26;122(4):593-603. Proapoptotic BID is an ATM effector in the DNA-damage response. PMID: 16122426
A lot of funny stuff. E.g. 6A, 7B, clearly spliced. The thing is that they have indicated that they have spliced some other figures.
J Biol Chem. 2002 Apr 5;277(14):12237-45. Epub 2002 Jan 22. tBID Homooligomerizes in the mitochondrial membrane to induce apoptosis. PMID: 11805084
Figure 2B, spliced.
1. Good find Junk Science. In the Cell paper Fig. 2a the 2nd and 3rd lanes look like pasted in duplicates.
In the JBC paper, the data in the correction seems to contradict the main conclusion, that ncBID can cause apoptosis, as the untreated cells have almost as much ncBID as the induced cells,
yet they remain alive.
1. michaelbriggs, that’s even more suspicious. There is a lot of starnge things going on in the cell paper. For example, Supplemental Figure S3A, right panel, third lane, look at the white
lane at the bottom of this lane, clearly something fishy.
1. Yes Junk Science, in Figure S3A, upper right panel, the bottom of the third lane is not aligned with lanes 1,2 and 4, and it looks like the 4th lane has been spliced on.
2. RE: J Biol Chem. 2002 Apr 5;277(14):12237-45.
Figure 1B. Suspect lane 1 spliced on. Vertical,light areas between the bands in lanes 1 and 2. Changes in background, especially noticeable around the Bax band in lane 1.
Figure 2A. Suspect that the input panel has been spliced on.
Figure 3A. Right panel. Suspect splicing between lanes. Series of vertical streaks between lanes.
Figure 3B. Left panel. Suspect splicing between lanes. Vertical,light streaks between bands.
1. RE: J Biol Chem. 2002 Apr 5;277(14):12237-45
Fig. 1A actin panel the bands in lane 4 and lane 6 appear the same.
5. EMBO Journal Vol.17 No.14 pp.3878–3885, 1998. PMID: 9670005
Figure 1C. Bax monomer band in lane 1 has a vertically truncated right end. Suspect spliced in.
Figure 1D. Vertical change in background between lanes 1 and 2. Bands in lanes 1 and 3 have vertical right ends.
Figure 2B. Suspect Bax monomer band in lane 1 spliced in. The Bax band in lane 1 has a vertical, straight right edge.
Figure 2C. The left end of the black band just above the 66 mark in lane 4 seems a bit too vertical and flat.
Figure 4A. Something funny about the band in lane 2. Lower edge is nearly straight and horizontal.
6. Rachel Sarig seems to be on all strange papers. Quick look at pubmed, will look more in detail later.
PLoS One. 2008;3(11):e3707. doi: 10.1371/journal.pone.0003707. Epub 2008 Nov 12. p53 plays a role in mesenchymal differentiation programs, in a cell fate dependent manner. PMID: 19002260
Figure 5C, clearly spliced.
Cell Death Differ. 2013 May;20(5):774-83. doi: 10.1038/cdd.2013.9. Epub 2013 Feb 15. p53 is required for brown adipogenic differentiation and has a protective role against diet-induced obesity.
PMID: 23412343
Figure 3A, 4D, both clearly spliced.
7. Mol Cell Biol. 1998 Oct;18(10):6083-9.
Figure 3A.
Alpha-BCL2 IP/alpha-BAX WB (top) panel. Short vertical streaks at the left end of the HABAX and BAX bands in 3rd lane. Suspect splicing between lanes 2 and 3.
Alpha-HA IP/alpha-BAX WB panel. Bands in 2nd lane have vertical, straight edge. Suspect 1st lane spliced out.
Alpha-BAX WB panel. Suspect bands in lane 6 spliced in. Vertical straight edges.
Figure 6B. Suspect splicing between lanes 4 and 5. Vertical change in background.
1. Fernando, A Gross has a ton of interesting figures, so Sarig learnt the tricks from Gross…
1. In reply to Junk Science May 9, 2013 at 2:22 am
I think that it the lineage. Here is another last century one.
J Cell Biol. 1998 Oct 5;143(1):207-15.
Regulated targeting of BAX to mitochondria.
Goping IS, Gross A, Lavoie JN, Nguyen M, Jemmerson R, Roth K, Korsmeyer SJ, Shore GC.
PMID: 9763432
Figure 2B. Lower left panel. Right ends of upper bands in lanes 2 and 3 are truncated and there are bright areas indicative of splicing.I suspect the the upper 2 bands in lane 4 (lower
left panel) are the same as the 2 bands in the right lower panel (lane 5). Note the indentation about 2/3rds the way along (from left to right) the bottom of the lower of the two bands.
In the lower right panel the signal is a bit more diffuse
Figure 4A. BAX and BAXdeltaART bands spliced in lanes 2,3 and 4. Vertical, straight grey streaks between the bands in lanes 1/2 and 4/5.
Figure 4C. Vertical change in background plus abrupt vertical discontinuity between lanes 3 and 4.
1. A more recent publication. Splicing didn’t go out of fashion.
Endocrinology. 2007 Apr;148(4):1717-26. Epub 2007 Jan 11.
Luteinizing hormone-induced caspase activation in rat preovulatory follicles is coupled to mitochondrial steroidogenesis.
Yacobi K, Tsafriri A, Gross A.
PMID: 17218406
Figure 1D. Anti-cleaved caspase-3 panel. Short,light, vertical streak at left end of band in lane marked as 5.
Figure 4A. p17panel. Suspect splicing between lanes marked as LH and 2. Balck band in lane 2 has vertical straight left edge. Vertical change in background between lanes.
Figure 5. Suspect splicing between all 5 lanes in each of the top 3 panels.
Vertical streaks/vertical changes in background between the lanes, or truncated bands.
Figure 6C. eCG+hCG panels. p17 panel. Suspect splicing between all 3 lanes. Beta actin panel. Suspect splicing between CL2 and CL8 lane.
1. Cell Death Differ. 2007 Sep;14(9):1628-34. Epub 2007 Jun 22.
Nucleocytoplasmic shuttling of BID is involved in regulating its activities in the DNA-damage response.
Oberkovitz G, Regev L, Gross A.
PMID: 17585339
Figure 3c. Beta-actin panel. Suspect splicing between middle lanes. Vertical, straight, light streak between middle lanes. BID panel. Vertical change in background and light areas
near the mid-line.
8. J Immunol. 2009 Jan 1;182(1):515-21.
Programmed necrotic cell death induced by complement involves a Bid-dependent pathway.
Ziporen L, Donin N, Shmushkovich T, Gross A, Fishelson Z.
PMID: 19109183
Figures 1D and 1E. The background is white .
Figure 3A. The background is white.
Figures 7A and 7B. The background is white.
The signifiance of the white is that it is monotonous so you cannot pin down the bands.
1. Does anyone has archive of science-fraud.org postings about Rakesh Kumar and his associates in George washington University. Please forward me the link if anyone has saved it.
1. http://www.science-fraud.org/?attachment_id=1310
Starting with my personal favorite, but hit previous and next to see all slides. If you want to know which paper the id/slide is from, just write a comment and I can help you.
1. I also like to have website link to science-fraud.org postings about Antony P. Adamis from Genentech.
2. When I clicked the link it took me to SF home page and there was nothing on it. You can mail me the pdf slides at [email protected].
1. It was a lot of data, but I’ve it all saved. I’ll e-mail you.
3. Thanks
1. Hi Junk Science & KP. I have collected more information about Kumar and his associates. Here is the link to the zip archive. http://bit.ly/179JqCT
If you have any questions mail me at [email protected]
2. Hey Junk science please give me archived records of original postings from sciencefraud.org for Micheal Karin
3. TJ:
4. TJ, I think this is all, might be some overlap but still enough to make you wonder why nothing has been done about this lab.
2. Mol Cell Biol. 2000 May;20(9):3125-36.
Biochemical and genetic analysis of the mitochondrial response of yeast to BAX and BCL-X(L).
Gross A, Pilcher K, Blachly-Dyson E, Basso E, Jockel J, Bassik MC, Korsmeyer SJ, Forte M.
PMID: 10757797
Figure 1. Lowest panel. Splicing between lanes. Vertical, straight streak between lanes.
Figure 2A. VDAC1 panel. Splicing between lanes 2 and 3. Vertical, straight, grey streak between lanes.
Figure 2B. Unsure about true background in lowest 2 panels.
Figure 4A. Monotonous light grey background.
9. Nat Cell Biol. 2010 Jun;12(6):553-62. doi: 10.1038/ncb2057. Epub 2010 May 2.
MTCH2/MIMP is a major facilitator of tBID recruitment to mitochondria. PMID: 20436477
Nat Cell Biol has the full scans of the blots, so that can be very useful if you want to find funny things. So for example, compare 2D, 3G, 4B (Mitochondrial fraction blots) and 5A with the full
scans in Figure S8 where you will find the full scans. Gross has a patent based on these results.
1. That might explain the institute turning a blind eye.
10. TJ,
I can send you pdf files of high resolution Karins image. Please let me know your email.
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Graphical Model
In this demo, we use a very simple graphical model to represent a very simple probabilistic scenario, show how to input the model into DGM, and perform inference and decoding in the model. This
example copies the idea from the Cheating Students Scenario
In order to run this demo, please execute "Demo 1D.exe" exact command
Building The Graphical Model
First of all we build a graph, consisting of 4 nodes, which represents 4 studens. We connect these nodes with undirected arcs. Since every student may give either true or false answer, every graph
node will have 2 states:
const byte nStates = 2; // {false, true}
const size_t nNodes = 4; // four students
for (size_t i = 0; i < nNodes; i++) graph.addNode(); // add nodes
for (size_t i = 0; i < nNodes - 1; i++) graph.addArc(i, i + 1); // add arcs
Next we fill the potentials of nodes and arcs of the graph. We assume that four studens are sitting in a row, and even studens have 25% chance to answer right, whereas odd students have 90% chance.
The edge potential describes that two neighbouring students are more likely to give the same answer. We fill the potentials by hand in the fillGraph() function:
Mat nodePot(nStates, 1, CV_32FC1); // node Potential (column-vector)
Mat edgePot(nStates, nStates, CV_32FC1); // edge Potential (matrix)
// Setting the node potentials
for (size_t i = 0; i < nNodes; i++) {
if (i % 2) { // for odd nodes
nodePot.at<float>(0, 0) = 0.10f; // nodePot = (0.10; 0.90)^T
nodePot.at<float>(1, 0) = 0.90f;
} else { // for even nodes
nodePot.at<float>(0, 0) = 0.75f; // nodePot = (0.75; 0.25)^T
nodePot.at<float>(1, 0) = 0.25f;
graph.setNode(i, nodePot);
// Defying the edge potential matrix
edgePot.at<float>(0, 0) = 2.0f; edgePot.at<float>(0, 1) = 1.0f;
edgePot.at<float>(1, 0) = 1.0f; edgePot.at<float>(1, 1) = 2.0f;
// Setting the edge potentials
for (size_t i = 0; i < nNodes - 1; i++)
graph.setArc(i, i + 1, edgePot);
We end up with the followiung graphical model:
The initial chances for students to give right ansvers, were given as if every student was alone. Now, when we have all four students sitting together (modelled with edge potentials), these chances
are not independent anymore. We are interested in the most probable scenario, how answer these students, when they answer together. To solve this problem, we have to apply decoding or/and inference
upon the given graph. Let us consider these processes separately.
The decoding task is to find the most likely configuration, i.e. the configuration with the highest joint probability.
For trivial graphs where it is feasible to enumerate all possible configurations, wich is equal to \(nStates^{nNodes}\), we can apply exact decoding - for other cases, approximate approaches should
be used. Exact decoding is based on brute force estimation of the joint probabilities for every possible configuration of random variables, associated with the graph nodes. In DGM decoding returns
the most likely configuration directly:
vec_byte_t decoding_decoderExcact = decoderExcact.decode();
The inference task is to find the marginal probabilities of individual nodes taking individual states.
For our example, marginal probabilities describe the chance of every student to answer the question right. DGM inference procedures store the marginal probabilities in the node potentials vectors.
infererExact.infer(); // changes the node potentials
Each inference class has decode() function and could be used for approximate decoding. This function returns the configuration, which maximases the marginals, and this configuration in general is not
the same as the configuration corresponding to the highest joint probability, i.e DirectGraphicalModels::CDecodeExact::decode() \(\neq\) DirectGraphicalModels::CInferExact::decode():
vec_byte_t decoding_infererExcact = infererExact.decode(); // approximate decoding from inferer
Finally, we depict the results of inference and decoding. In the INFERENCE table the initial node potentials, and marginal probabilities after inferece are shown (for our simple graph, we can also
use exact chain and tree inference algorithms, approximate loopy belief propagation and viterbi algorithms). In the DECODING table the rsults of the exact decoding and approximate decodings from
inferece are shown. Please note, that the correct configuration, provided by the exact decoder is {false, true, true, true}. However, the exact inference methods provide us with the configuration {
false, true, false, true}, and only Viterbi (max-product message passing algorithm) gives correct decoding, build upon marginals from the table INFERENCE: | {"url":"https://research.project-10.de/dgmdoc/a01855.html","timestamp":"2024-11-09T09:49:21Z","content_type":"application/xhtml+xml","content_length":"13547","record_id":"<urn:uuid:a988b364-4bfe-40eb-9265-f43eb8ba2b25>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00232.warc.gz"} |
Unlocking the Secrets of Perimeter and Area with Polynomials – A Worksheet Answer Key Adventure
Imagine yourself staring at a daunting worksheet, filled with complex shapes and equations. Calculating perimeter and area seems like an insurmountable task, especially when polynomials are thrown
into the mix. But fear not, dear reader! This guide will equip you with the tools and knowledge to conquer those worksheets and unlock the fascinating world of finding perimeter and area using
Image: www.lulumath.com
Whether you’re a student grappling with challenging homework, a parent helping your child with their math studies, or simply someone with a thirst for mathematical exploration, this article is your
ultimate companion. We’ll embark on a journey through the captivating world of polynomials, unraveling their applications in geometry and discovering how they simplify intricate calculations.
The Intricate Dance of Polynomials and Geometric Shapes
Polynomials, those expressions with variables and coefficients combined through addition, subtraction, multiplication, and exponentiation, are more than just abstract mathematical concepts. They
possess a powerful ability to represent relationships and solve real-world problems, including calculating perimeter and area.
Perimeter, the total length of the sides of a shape, and area, the space enclosed within a shape, are fundamental concepts in geometry. Understanding how to calculate them is essential for tackling
various practical applications, from construction and design to everyday tasks like painting a room or planning a garden.
Before we delve into the intricacies of polynomials in geometric calculations, let’s lay a solid foundation by revisiting the basics of perimeter and area.
The Foundations: Perimeter and Area Essentials
• Perimeter: Imagine walking around a shape; the total distance you cover is its perimeter. For a rectangle, the perimeter is calculated by adding the lengths of all its sides. In other words,
Perimeter = 2(length + width).
• Area: The amount of space a shape occupies is its area. For a rectangle, the area is calculated by multiplying its length and width. Area = length × width.
Now, let’s introduce polynomials into the equation.
Polynomials Step Onto the Geometric Stage
Instead of simple numeric values for length and width, imagine these dimensions are represented by polynomials. This adds a layer of complexity but opens doors to exploring more intricate shapes and
For instance, consider a rectangle where the length is represented by the polynomial “2x + 3” and the width by “x – 1.” To find the perimeter, we simply substitute these polynomials into the
perimeter formula:
Perimeter = 2((2x + 3) + (x – 1))
Simplifying the expression, we get:
Perimeter = 2(3x + 2)
Perimeter = 6x + 4
Similarly, to find the area, we multiply the polynomials representing length and width:
Area = (2x + 3)(x – 1)
Expanding the expression, we get:
Area = 2x² + x – 3
These examples demonstrate how polynomials empower us to calculate perimeter and area for more complex shapes and scenarios. The process involves substituting polynomials into the relevant formulas
and simplifying the resulting expressions.
Image: circuitdiagramheike.z19.web.core.windows.net
Navigating the Worksheet: Decoding Polynomials for Geometric Triumph
Now, let’s put our newfound knowledge to the test with a hypothetical worksheet. Imagine a worksheet filled with diagrams of various shapes, each with dimensions represented by polynomials. You are
tasked with finding the perimeter and area of these shapes.
Step 1: Label and Understand the Shapes
Begin by carefully examining each shape. Label its sides with polynomials representing the lengths. For example, a triangle might have sides with lengths “x + 2,” “2x – 1,” and “3x.”
Step 2: Apply the Relevant Formula
Recall the formulas for perimeter and area for each type of shape. For rectangles, remember the formula for perimeter is 2(length + width) and area is length × width. For triangles, the perimeter is
the sum of all sides, and the area is (1/2) × base × height.
Step 3: Substitute Polynomials and Simplify
Replace the variables in the formulas with the polynomials representing the lengths of the sides. Then, simplify the expressions by expanding and combining like terms.
Step 4: Verify and Reflect
Double-check your calculations and ensure that the units of your answers are consistent with the units of the lengths provided. Reflect on the process, noting any patterns or insights that emerge.
Practical Applications: Polynomials in the Real World
The ability to calculate perimeter and area using polynomials extends beyond worksheets. It has practical applications in various fields, including:
• Architecture and Construction: Architects and engineers use polynomials to design buildings and structures, ensuring structural integrity and optimizing space utilization.
• Engineering and Manufacturing: Engineers rely on polynomials to design and manufacture components and products, optimizing dimensions and maximizing efficiency.
• Landscaping and Gardening: Landscape designers use polynomials to calculate the amount of materials needed for projects like paving, planting, or building a fence.
• Real Estate and Property Management: Real estate professionals use polynomials to determine the value of properties, calculate rent, and plan renovations.
Expert Insights and Actionable Tips
Here are a few insights from experienced mathematicians and educators to further empower you in your journey:
• Visualize the Problems: Drawing a clear diagram of the shape helps visualize the problem and understand the relationships between its dimensions.
• Break It Down: If a complex shape looks intimidating, break it down into simpler shapes. Calculate the perimeter and area of each smaller shape and combine them to find the overall perimeter and
• Practice Regularly: Consistent practice is key to mastering the concepts of finding perimeter and area using polynomials.
Finding Perimeter And Area Using Polynomials Worksheet Answer Key
Conclusion: Unleash Your Polynomial Prowess
As you embark on your journey to mastering the art of calculating perimeter and area using polynomials, remember that it is about more than just solving equations. It’s about understanding the
profound relationship between algebra and geometry, unveiling the beauty of mathematics in real-world applications.
This guide has provided you with the knowledge, tools, and inspiration to confidently navigate those worksheets and unlock the secrets of polynomial applications. Keep practicing, keep exploring, and
remember: you have the potential to achieve remarkable feats in the world of mathematics. So go forth, armed with your newfound polynomial prowess, and conquer the world of perimeter and area! | {"url":"https://www.pridesurfboards.com/1484.html","timestamp":"2024-11-10T11:19:45Z","content_type":"text/html","content_length":"118008","record_id":"<urn:uuid:e888784f-f3cf-4db8-9764-d464a90caf29>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00606.warc.gz"} |
Excel EVEN function - Free Excel Tutorial
This post will guide you how to use Excel EVEN function with syntax and examples in Microsoft excel.
The Excel EVEN function rounds a given number up to the nearest even integer. So you can use the EVEN function to return the next even integer after rounding a supplied number up. The returned value
is away from zero. It means that if the number is a positive value and the return value will become larger and if the number is a negative value, the return value will become smaller.
The EVEN function is a build-in function in Microsoft Excel and it is categorized as a Math and Trigonometry Function.
The EVEN function in Excel 2016, Excel 2013, Excel 2010, Excel 2007, Excel 2003, Excel XP, Excel 2000, Excel 2011 for Mac.
The syntax of the EVEN function is as below:
=EVEN (number)
Where the EVEN function arguments is:
• number -This is a required argument. . A numeric value that you want to round up to the nearest even integer.
• If the number argument is not numeric value, the EVEN function will return the #VALUE! Error.
Excel EVEN Function Examples
The below examples will show you how to use Excel EVEN Function to round up a given number to the nearest even integer.
1# to round 2.5 up to the nearest even integer, enter the following formula in Cell B1.
2# to round 9 to the nearest even integer, enter the following formula in Cell B2.
3# to round 14 up to the nearest even integer, enter the following formula in Cell B3.
4# to round -3 up to the nearest even integer, enter the following formula in Cell B4.
Related Functions | {"url":"https://www.excelhow.net/excel-even-function.html","timestamp":"2024-11-11T20:35:57Z","content_type":"text/html","content_length":"87913","record_id":"<urn:uuid:211468d2-8937-464b-b45c-ade523c3decf>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00264.warc.gz"} |
Section One: A contribution to teaching
Primary teachers talking about Let’s Think Maths.
“ (Let’s Think) gives you a lot of freedom. There’s not that pressure of thinking that the children have to know a certain thing by a certain date. It’s more a case of the
children learning what they can in the best way.”
We’re talking about long-term benefits, not just covering programmes of study.
The aim of the Primary Cognitive Acceleration in Mathematics Education (Primary CAME) project was to contribute to the teaching and learning of mathematics in Years 5 and 6. The Primary CAME lessons
were an outcome of the research from this project. These lessons have been renamed Let’s Think Maths lessons. They stimulate the development of children’s mathematical thinking through carefully
selected classroom tasks. Tackling these challenges encourages children to work together as mathematicians, constructing and discussing mathematical ideas.
Each lesson promotes very specific mathematical connections and generalisations but with a focus on reasoning. Pupils grapple with the ‘big ideas’ in mathematics, rather than focusing on the mastery
of specific skills. The shared construction of mathematics encourages children to develop a deeper understanding of the mathematical concepts underlying the skills, algorithms and procedures in
school mathematics. This includes those specified in the National Curriculum.
Primary Let’s Think lessons are not in themselves a mathematics scheme of work. Children need the regular content-based primary mathematics experiences of good instructional and problem-solving
lessons. These lessons should be, at most, a fortnightly supplement to the normal mathematics experiences offered to children. The Let’s Think approach complements and builds upon existing good
practice in primary mathematics. It has been shown that these lessons, delivered in conjunction with good mathematical instruction and investigation, can significantly raise the whole thinking
capacity of each child, as well as contribute to the meaningful learning of mathematics.
What makes Primary Let’s Think different?
“ In [these] lessons you end up with lots of questions. ”
“ I agree. You can’t stop thinking about them. ”
Two children talking after a Primary Let’s Think lesson
As mentioned on the previous page, the Primary Let’s Think approach shares many of the features of existing good practice in primary mathematics: children are encouraged to talk, listen, question and
debate their mathematical ideas.
Let’s Think Maths is open-ended in the conceptual points which children will reach but very focused in terms of what tasks the children are set. Primary Let’s Think often addresses concepts thought
of as beyond the scope of primary mathematics, sowing seeds for later mathematical work. At all times, the emphasis is on depth of mathematical thinking and conjecturing in which children construct
mathematical ideas and gain insights at different levels of complexity, all within a mathematical ‘big idea’.
The lessons stimulate children’s mathematical thinking and often leave children with unanswered or partly answered questions. This mathematical ‘unfinished business’ will either be addressed in
further Let’s Think Maths lessons or as part of the normal mathematical curriculum, or later in the children’s mathematical career. Let’s Think Maths lessons differ from good instruction and practice
lessons in that each one has a clear agenda, involving fundamental concepts in mathematics. The lesson focuses on children ‘struggling on the way’ towards these ideas, rather than on fully
understanding and mastering the concepts. The outcomes of a lesson are the thinking processes and the sharing of ideas rather than the specific knowledge gains and skills employed.
Hence, although children are working in an investigative way throughout these lessons, Let’s Think Maths lessons do differ from many good, open-ended investigations or problem-solving lessons. The
Let’s Think activities are very focused in terms of the tasks that children tackle, but open-ended in the mathematical understandings that children reach. The lessons provide clear challenge points
rather than allowing varied interpretations of the task.
Whole-class teaching
(The) lessons are an opportunity to develop children’s self-esteem with number,
because you value every contribution.
A primary teacher talking about Primary Let’s Think Maths
Let’s Think Maths lessons are centred around whole-class activities. The whole class tackles the same challenges together and in small groups. A key feature is the sharing and discussion of
children’s mathematical constructions as a class. Classes involved in the development of the lesson materials have successfully included children of a wide range of abilities through careful grouping
and support. Different children reach different levels of thinking. So, rather than differentiation by task, differentiation is by thinking outcome within a task. | {"url":"https://community.letsthink.org.uk/pcame/chapter/section-one-a-contribution-to-teachingproperty-of-lets-think-forum-not-to-be-copied-or-reproduced-without-permission/","timestamp":"2024-11-03T20:23:22Z","content_type":"text/html","content_length":"79564","record_id":"<urn:uuid:698aef69-3bbe-41a7-8740-e41f152aa795>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00044.warc.gz"} |
Inverse Finance’s Incident Analysis —$INV Price Manipulation
Inverse Finance’s Incident Analysis — $INV Price Manipulation
On Apr 02, 2022, 11:04:09 AM UTC (block 14506359), the attacker borrowed assets from Inverse Finance using a collateral asset that had less actual value than the borrowed assets. The price of the
collateral asset on Inverse Finance becoming more expensive at the time, making $INV collateral with more borrowable value than it should be. We will go over the technical details of this attack step
by step in this article.
Related Address
Attack Steps
There are 2 main transactions used for the successful attack as follows:
1. Price manipulation, Update the price on Oracle, and Swap tokens at transaction: https://etherscan.io/tx/0x20a6dcff06a791a7f8be9f423053ce8caee3f9eecc31df32445fc98d4ccd8365
1.1 Verified that the price feed can be updated.
1.2 Exploiter’s Contract allowed using $INV as collateral on the lending contract.
1.3 Manipulated price by swapping 300 $WETH to 374.385477084842174221 $INV on the SushiSwap’s $WETH-$INV pair.
1.4 Updated the price feed. The price feed of every pair was updated by calling the workForFree() function.
The keep3rV2Oracle contract that refer to the SushiSwap’s $INV-$WETH pair (address 0x39b1df026010b5aea781f90542ee19e900f2db15) is one of updated feed.
The price0CumulativeLast value after the update is 0x00000000000000000000000000000062d32f53f2f7afe532c0372fd0cacdbd4b, which the Oracle will calculate this value with e10 and Q112. Then the value
from the calculation, 6476591327926140254201750, will be stored in the Oracle’s observation.
It goes with the same as the price1CumulativeLast value, 0x000000000000000000000000000012d5e533107a3e9659278cb49bb1e692524d, which will be calculated into 316007731064365302759283159 and being stored
into the Oracle’s observation.
At this update, Oracle has stored the manipulated cumulative price into the observation with timestamp 1648897434 at index number 114 of the observations.
1.5 Swapped 200 $WETH to 690,307.061277 $USDC on SushiSwap.
1.6 Exchanged 690,307.061277 $USDC to 690,203.010884231600886834 $DOLA on Curve’s $DOLA + 3Crv pool.
1.7 Swapped 690,203.010884231600886834 $DOLA to 1,372.052401667461914227 $INV on Sushiswap’s $INV-$DOLA pair.
2. Lend and Borrow at: https://etherscan.io/tx/0x600373f67521324c8068cfd025f121a0843d57ec813411661b07edc5ff781842
Attacker lent 1,746.437878752304088448 $INV
Returned price from the getUnderlyingPrice() function of $INV feed is 20926.791034009538953802 $USD.
The collateral factor at the lent/borrowed time is 60% (0.6 e18) of the asset value.
By the 1,746.437878752304088448 $INV lent, total value in $USD that the attacker can borrow is $21,928,404.32551701 (60% of 36,547,340.54252835)
The price of other related assets in the same transaction:
• $WBTC is $46650.31
• $ETH is $3488.82
• $YFI is $23461.259
Attacker borrowed the following assets:
• 3,999,669.029654761043260989 $DOLA
• 1,588.263719446159096974 $ETH
Finally, the attacker transferred all borrowed assets to Inverse Finance Exploiter #2’s wallet.
Root Cause Analysis
The core oracle that contains the price calculation logic of Inverse Protocol is the Keep3rV2Oracle contract.
The following code is the current() function of the Keep3rV2Oracle contract.
The current() function returns the price of the asset. It is mainly used for calculating the getUnderlyingPrice() function to determine the asset value.
The current() function takes the last stored cumulative price from the observation and the current cumulative price to calculate the price. The function has a flaw that allows the calculation with a
small timeElapsed value.
The problem will arise when the current() function is called 1 block after the block that the price feed has been updated, which is the case that the attacker has leveraged in this attack.
The attacker manipulated the $INV price in block number 14506358 and the Oracle stored the manipulated cumulative price in the observations. At block number 14506359, the current() function used the
recently updated cumulative price from the observations and the current block’s cumulative price.
To guarantee the latest cumulative price being set, the attacker has to manually force the price feed (SushiSwap) to update the cumulative price. So, the attacker called the sync() function first at
the transaction 0x600373f67521324c8068cfd025f121a0843d57ec813411661b07edc5ff781842.
Here’s the catch, the current block’s cumulative price was calculated from last price (the attacker force update the price to ensure this value) of the previous block multiplying with the current
block time (15 seconds in this case). When the current() function is calculating _computeAmountOut(), the current block’s cumulative price is the latest cumulative price plus with the multiplying of
the price and the passing time. When putting every parameter into the calculation of _computeAmountOut() function
From now on, we will simplify the equation by ignoring the scaling constant *e10/Q122 from price0Cumulative and _observation.price0Cumulative.
The current block’s price0Cumulative is retrieved from IUniswapV2Pair(pair).price0CumulativeLast().
The _observation.price0Cumulative is replaced with a placeholder variable {lastBlockCumulativePrice} to refer to the cumulative price from block number 14506358 onward.
The variable IUniswapV2Pair(pair).price0CumulativeLast() is replaced with {lastBlockCumulativePrice} + (lastBlockLatestPrice*blockTime). This is how the price cumulative of block number 14506359
being calculated.
We remove the parentheses from {lastBlockCumulativePrice} + (lastBlockLatestPrice*blockTime) to make the equation be more clearer.
In the left parentheses, {lastBlockCumulativePrice} can be canceled each other out.
Then we remove the lastBlockLatestPrice*blockTime from the parentheses.
The blockTime from price0Cumulative is the length time between block used in the calculation of cumulative price, which can be observed from the getReserve() function. The timestamp of block number
14506358 is 1648897434 and the timestamp number 14506359 is 1648897449. So, the value of blockTime is 15, which is also the same number of the parameter elapsed that we had considered having a flaw
earlier. The value of blockTime and elapsed is the same, which will be canceled out in the calculation in the _computeAmountOut() function.
Long story short, this oracle will return the price from the last block last reserve price directly (with a condition, of course). The price from the Oracle in this block (14506359) will be the price
that being manipulated from the last block (14506358).
In summary, the attacker spent around $1,737,310 (500 $ETH) to manipulate the price of $INV on Inverse Finance and exchange all assets for $INV. All exchanged $INV is used as collateral to borrow
$DOLA, $ETH, $WBTC, and $YFI. The total value of borrowed assets is around $14,843,389.
About Inspex
Inspex is formed by a team of cybersecurity experts highly experienced in various fields of cybersecurity. We provide blockchain and smart contract professional services at the highest quality to
enhance the security of our clients and the overall blockchain ecosystem.
For any business inquiries, please contact us via Twitter, Telegram, contact@inspex.co | {"url":"https://inspexco.medium.com/inverse-finances-incident-analysis-inv-price-manipulation-b15c2e917888","timestamp":"2024-11-09T00:28:50Z","content_type":"text/html","content_length":"239954","record_id":"<urn:uuid:2e332bdd-017f-40ad-baee-7eeda07b3677>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00414.warc.gz"} |
Quadratic Inequalities (Equations and Parabolas)
Quadratic Inequalities (Equations and Parabolas)
Solving a Quadratic Inequality Graphically
The graph of a quadratic function in a Cartesian coordinate system is a parabola. To solve a quadratic inequality or graphically, draw the corresponding parabola, then:
• The x-coordinates of the points (if they exist) where the parabola intersects the x-axis are the solution of the equation (zeros of the equation)
• The x-coordinates of the points (if they exist) where the parabola is above the x-axis are solution of the inequality
• The x-coordinates of the points (if they exist) where the parabola is below the x-axis are solution of the inequality
Ready, Set, Practice!
Find the solutions of a quadratic equation or inequality by exploring the graph of the corresponding parabola. Use the input box to enter different quadratic expressions and the drop down list to
select the equation or inequality form to solve. Use the mouse wheel or the predefined gestures for mobile devices to zoom in/out and view details in the Graphics View.
Today You Are the Teacher!
Today's assignment is the inequality . Alice solves it like this: . Bob solves first, and gets . Then he graphs the parabola corresponding to the given equation and finds the solution . Alice says
that her method is faster than Bob's, because it doesn't require sketching a graph. Grade your students' solutions, and explain the reasons for your grading.
Hamletic Doubt...
Below you can see the solution of one of the following inequalities. Choose the correct one. | {"url":"https://www.geogebra.org/m/QvmhXbaB","timestamp":"2024-11-13T16:06:39Z","content_type":"text/html","content_length":"124743","record_id":"<urn:uuid:5b329356-4d31-4cfd-bb5e-30949f320147>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00681.warc.gz"} |
RANDOM(9) Kernel Developer's Manual RANDOM(9)
arc4rand, arc4random, random, read_random, read_random_uio, srandom — supply pseudo-random numbers
#include <sys/libkern.h>
srandom(u_long seed);
arc4rand(void *ptr, u_int length, int reseed);
#include <sys/random.h>
read_random(void *buffer, int count);
read_random_uio(struct uio *uio, bool nonblock);
The random() function will by default produce a sequence of numbers that can be duplicated by calling srandom() with some constant as the seed. The srandom() function may be called with any arbitrary
seed value to get slightly more unpredictable numbers. It is important to remember that the random() function is entirely predictable, and is therefore not of use where knowledge of the sequence of
numbers may be of benefit to an attacker.
The arc4rand() function will return very good quality random numbers, better suited for security-related purposes. The random numbers from arc4rand() are seeded from the entropy device if it is
available. Automatic reseeds happen after a certain timeinterval and after a certain number of bytes have been delivered. A forced reseed can be forced by passing a non-zero value in the reseed
The read_random() function is used to return entropy directly from the entropy device if it has been loaded. If the entropy device is not loaded, then the buffer is ignored and zero is returned. The
buffer is filled with no more than count bytes. It is strongly advised that read_random() is not used; instead use arc4rand() unless it is necessary to know that no entropy has been returned.
The read_random_uio() function behaves identically to read(2) on /dev/random. The uio argument points to a buffer where random data should be stored. This function only returns data if the random
device is seeded. It blocks if unseeded, except when the nonblock argument is true.
All the bits returned by random(), arc4rand(), read_random(), and read_random_uio() are usable. For example, ‘random()&01’ will produce a random binary value.
The arc4random() is a convenience function which calls arc4rand() to return a 32 bit pseudo-random integer.
The random() function uses a non-linear additive feedback random number generator employing a default table of size 31 containing long integers to return successive pseudo-random numbers in the range
from 0 to (2**31)−1. The period of this random number generator is very large, approximately 16*((2**31)−1).
The arc4rand() function uses the RC4 algorithm to generate successive pseudo-random bytes. The arc4random() function uses arc4rand() to generate pseudo-random numbers in the range from 0 to (2**32)
The read_random() function returns the number of bytes placed in buffer.
read_random_uio() returns zero when successful, otherwise an error code is returned.
read_random_uio() may fail if:
uio points to an invalid memory region.
The random device is unseeded and nonblock is true.
Dan Moschuk wrote arc4random().
Mark R V Murray wrote read_random(). | {"url":"https://manpages.debian.org/bookworm/freebsd-manpages/random.9freebsd.en.html","timestamp":"2024-11-06T22:05:05Z","content_type":"text/html","content_length":"25953","record_id":"<urn:uuid:70792a64-fd5a-42ca-b6ac-325292fa3003>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00195.warc.gz"} |
Flexing the graph
The points of inflexion of a graph are the points where
it changes from concave upwards to concave downwards and vice-versa. The points can be found by looking for places where the second derivative changes sign.
• Plot the graph of y=2*t^4+0.5*t^3-3*t^2+2*t-1 and see if you can spot any points of inflexion.
• Save the values of the derivative to a file
• Use 'Open File' to import the derivative values as your new function.
• Display the derivative of this function and find the approximate position where this graph crosses the t-axis. Do these values agree with your original estimates of the position of the points of
#differentiation #secondderivative #pointsofinflexion
plotXpose app is available on Google Play and App Store
Google Play and the Google Play logo are trademarks of Google LLC.
A version will shortly be available for Windows. | {"url":"https://www.plotxpose.com/Flexingthegraph.htm","timestamp":"2024-11-03T03:02:12Z","content_type":"text/html","content_length":"9144","record_id":"<urn:uuid:6a92fa06-ab5f-49b8-a001-0b6530289e8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00746.warc.gz"} |
Partial Dependence and Individual Conditional Expectation Plots
Go to the end to download the full example code or to run this example in your browser via JupyterLite or Binder
Partial Dependence and Individual Conditional Expectation Plots¶
Partial dependence plots show the dependence between the target function [2] and a set of features of interest, marginalizing over the values of all other features (the complement features). Due to
the limits of human perception, the size of the set of features of interest must be small (usually, one or two) thus they are usually chosen among the most important features.
Similarly, an individual conditional expectation (ICE) plot [3] shows the dependence between the target function and a feature of interest. However, unlike partial dependence plots, which show the
average effect of the features of interest, ICE plots visualize the dependence of the prediction on a feature for each sample separately, with one line per sample. Only one feature of interest is
supported for ICE plots.
This example shows how to obtain partial dependence and ICE plots from a MLPRegressor and a HistGradientBoostingRegressor trained on the bike sharing dataset. The example is inspired by [1].
Bike sharing dataset preprocessing¶
We will use the bike sharing dataset. The goal is to predict the number of bike rentals using weather and season data as well as the datetime information.
from sklearn.datasets import fetch_openml
bikes = fetch_openml("Bike_Sharing_Demand", version=2, as_frame=True, parser="pandas")
# Make an explicit copy to avoid "SettingWithCopyWarning" from pandas
X, y = bikes.data.copy(), bikes.target
# We use only a subset of the data to speed up the example.
X = X.iloc[::5, :]
y = y[::5]
The feature "weather" has a particularity: the category "heavy_rain" is a rare category.
clear 2284
misty 904
rain 287
heavy_rain 1
Name: count, dtype: int64
Because of this rare category, we collapse it into "rain".
X["weather"].replace(to_replace="heavy_rain", value="rain", inplace=True)
We now have a closer look at the "year" feature:
Name: count, dtype: int64
We see that we have data from two years. We use the first year to train the model and the second year to test the model.
mask_training = X["year"] == 0.0
X = X.drop(columns=["year"])
X_train, y_train = X[mask_training], y[mask_training]
X_test, y_test = X[~mask_training], y[~mask_training]
We can check the dataset information to see that we have heterogeneous data types. We have to preprocess the different columns accordingly.
<class 'pandas.core.frame.DataFrame'>
Index: 1729 entries, 0 to 8640
Data columns (total 11 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 season 1729 non-null category
1 month 1729 non-null int64
2 hour 1729 non-null int64
3 holiday 1729 non-null category
4 weekday 1729 non-null int64
5 workingday 1729 non-null category
6 weather 1729 non-null category
7 temp 1729 non-null float64
8 feel_temp 1729 non-null float64
9 humidity 1729 non-null float64
10 windspeed 1729 non-null float64
dtypes: category(4), float64(4), int64(3)
memory usage: 115.4 KB
From the previous information, we will consider the category columns as nominal categorical features. In addition, we will consider the date and time information as categorical features as well.
We manually define the columns containing numerical and categorical features.
numerical_features = [
categorical_features = X_train.columns.drop(numerical_features)
Before we go into the details regarding the preprocessing of the different machine learning pipelines, we will try to get some additional intuition regarding the dataset that will be helpful to
understand the model’s statistical performance and results of the partial dependence analysis.
We plot the average number of bike rentals by grouping the data by season and by year.
from itertools import product
import matplotlib.pyplot as plt
import numpy as np
days = ("Sun", "Mon", "Tue", "Wed", "Thu", "Fri", "Sat")
hours = tuple(range(24))
xticklabels = [f"{day}\n{hour}:00" for day, hour in product(days, hours)]
xtick_start, xtick_period = 6, 12
fig, axs = plt.subplots(nrows=2, figsize=(8, 6), sharey=True, sharex=True)
average_bike_rentals = bikes.frame.groupby(["year", "season", "weekday", "hour"]).mean(
for ax, (idx, df) in zip(axs, average_bike_rentals.groupby("year")):
df.groupby("season").plot(ax=ax, legend=True)
# decorate the plot
num=len(xticklabels) // xtick_period,
ax.set_ylabel("Average number of bike rentals")
f"Bike rental for {'2010 (train set)' if idx == 0.0 else '2011 (test set)'}"
ax.set_ylim(0, 1_000)
ax.set_xlim(0, len(xticklabels))
/home/circleci/project/examples/inspection/plot_partial_dependence.py:113: FutureWarning:
The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.
/home/circleci/project/examples/inspection/plot_partial_dependence.py:117: FutureWarning:
The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.
/home/circleci/project/examples/inspection/plot_partial_dependence.py:117: FutureWarning:
The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.
The first striking difference between the train and test set is that the number of bike rentals is higher in the test set. For this reason, it will not be surprising to get a machine learning model
that underestimates the number of bike rentals. We also observe that the number of bike rentals is lower during the spring season. In addition, we see that during working days, there is a specific
pattern around 6-7 am and 5-6 pm with some peaks of bike rentals. We can keep in mind these different insights and use them to understand the partial dependence plot.
Preprocessor for machine-learning models¶
Since we later use two different models, a MLPRegressor and a HistGradientBoostingRegressor, we create two different preprocessors, specific for each model.
Preprocessor for the neural network model¶
We will use a QuantileTransformer to scale the numerical features and encode the categorical features with a OneHotEncoder.
ColumnTransformer(transformers=[('num', QuantileTransformer(n_quantiles=100),
['temp', 'feel_temp', 'humidity',
('cat', OneHotEncoder(handle_unknown='ignore'),
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
ColumnTransformer(transformers=[('num', QuantileTransformer(n_quantiles=100),
['temp', 'feel_temp', 'humidity',
('cat', OneHotEncoder(handle_unknown='ignore'),
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
['temp', 'feel_temp', 'humidity', 'windspeed']
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
Preprocessor for the gradient boosting model¶
For the gradient boosting model, we leave the numerical features as-is and only encode the categorical features using a OrdinalEncoder.
from sklearn.preprocessing import OrdinalEncoder
hgbdt_preprocessor = ColumnTransformer(
("cat", OrdinalEncoder(), categorical_features),
("num", "passthrough", numerical_features),
transformers=[('cat', OrdinalEncoder(),
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
('num', 'passthrough',
['temp', 'feel_temp', 'humidity',
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
transformers=[('cat', OrdinalEncoder(),
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
('num', 'passthrough',
['temp', 'feel_temp', 'humidity',
Index(['season', 'month', 'hour', 'holiday', 'weekday', 'workingday',
['temp', 'feel_temp', 'humidity', 'windspeed']
1-way partial dependence with different models¶
In this section, we will compute 1-way partial dependence with two different machine-learning models: (i) a multi-layer perceptron and (ii) a gradient-boosting model. With these two models, we
illustrate how to compute and interpret both partial dependence plot (PDP) for both numerical and categorical features and individual conditional expectation (ICE).
Multi-layer perceptron¶
Let’s fit a MLPRegressor and compute single-variable partial dependence plots.
from time import time
from sklearn.neural_network import MLPRegressor
from sklearn.pipeline import make_pipeline
print("Training MLPRegressor...")
tic = time()
mlp_model = make_pipeline(
hidden_layer_sizes=(30, 15),
mlp_model.fit(X_train, y_train)
print(f"done in {time() - tic:.3f}s")
print(f"Test R2 score: {mlp_model.score(X_test, y_test):.2f}")
Training MLPRegressor...
done in 0.689s
Test R2 score: 0.61
We configured a pipeline using the preprocessor that we created specifically for the neural network and tuned the neural network size and learning rate to get a reasonable compromise between training
time and predictive performance on a test set.
Importantly, this tabular dataset has very different dynamic ranges for its features. Neural networks tend to be very sensitive to features with varying scales and forgetting to preprocess the
numeric feature would lead to a very poor model.
It would be possible to get even higher predictive performance with a larger neural network but the training would also be significantly more expensive.
Note that it is important to check that the model is accurate enough on a test set before plotting the partial dependence since there would be little use in explaining the impact of a given feature
on the prediction function of a model with poor predictive performance. In this regard, our MLP model works reasonably well.
We will plot the averaged partial dependence.
import matplotlib.pyplot as plt
from sklearn.inspection import PartialDependenceDisplay
common_params = {
"subsample": 50,
"n_jobs": 2,
"grid_resolution": 20,
"random_state": 0,
print("Computing partial dependence plots...")
features_info = {
# features of interest
"features": ["temp", "humidity", "windspeed", "season", "weather", "hour"],
# type of partial dependence plot
"kind": "average",
# information regarding categorical features
"categorical_features": categorical_features,
tic = time()
_, ax = plt.subplots(ncols=3, nrows=2, figsize=(9, 8), constrained_layout=True)
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle(
"Partial dependence of the number of bike rentals\n"
"for the bike rental dataset with an MLPRegressor"
Computing partial dependence plots...
done in 1.015s
Gradient boosting¶
Let’s now fit a HistGradientBoostingRegressor and compute the partial dependence on the same features. We also use the specific preprocessor we created for this model.
from sklearn.ensemble import HistGradientBoostingRegressor
print("Training HistGradientBoostingRegressor...")
tic = time()
hgbdt_model = make_pipeline(
hgbdt_model.fit(X_train, y_train)
print(f"done in {time() - tic:.3f}s")
print(f"Test R2 score: {hgbdt_model.score(X_test, y_test):.2f}")
Training HistGradientBoostingRegressor...
done in 0.142s
Test R2 score: 0.62
Here, we used the default hyperparameters for the gradient boosting model without any preprocessing as tree-based models are naturally robust to monotonic transformations of numerical features.
Note that on this tabular dataset, Gradient Boosting Machines are both significantly faster to train and more accurate than neural networks. It is also significantly cheaper to tune their
hyperparameters (the defaults tend to work well while this is not often the case for neural networks).
We will plot the partial dependence for some of the numerical and categorical features.
print("Computing partial dependence plots...")
tic = time()
_, ax = plt.subplots(ncols=3, nrows=2, figsize=(9, 8), constrained_layout=True)
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle(
"Partial dependence of the number of bike rentals\n"
"for the bike rental dataset with a gradient boosting"
Computing partial dependence plots...
done in 0.978s
Analysis of the plots¶
We will first look at the PDPs for the numerical features. For both models, the general trend of the PDP of the temperature is that the number of bike rentals is increasing with temperature. We can
make a similar analysis but with the opposite trend for the humidity features. The number of bike rentals is decreasing when the humidity increases. Finally, we see the same trend for the wind speed
feature. The number of bike rentals is decreasing when the wind speed is increasing for both models. We also observe that MLPRegressor has much smoother predictions than HistGradientBoostingRegressor
Now, we will look at the partial dependence plots for the categorical features.
We observe that the spring season is the lowest bar for the season feature. With the weather feature, the rain category is the lowest bar. Regarding the hour feature, we see two peaks around the 7 am
and 6 pm. These findings are in line with the the observations we made earlier on the dataset.
However, it is worth noting that we are creating potential meaningless synthetic samples if features are correlated.
ICE vs. PDP¶
PDP is an average of the marginal effects of the features. We are averaging the response of all samples of the provided set. Thus, some effects could be hidden. In this regard, it is possible to plot
each individual response. This representation is called the Individual Effect Plot (ICE). In the plot below, we plot 50 randomly selected ICEs for the temperature and humidity features.
print("Computing partial dependence plots and individual conditional expectation...")
tic = time()
_, ax = plt.subplots(ncols=2, figsize=(6, 4), sharey=True, constrained_layout=True)
features_info = {
"features": ["temp", "humidity"],
"kind": "both",
"centered": True,
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle("ICE and PDP representations", fontsize=16)
Computing partial dependence plots and individual conditional expectation...
done in 0.404s
We see that the ICE for the temperature feature gives us some additional information: Some of the ICE lines are flat while some others show a decrease of the dependence for temperature above 35
degrees Celsius. We observe a similar pattern for the humidity feature: some of the ICEs lines show a sharp decrease when the humidity is above 80%.
Not all ICE lines are parallel, this indicates that the model finds interactions between features. We can repeat the experiment by constraining the gradient boosting model to not use any interactions
between features using the parameter interaction_cst:
from sklearn.base import clone
interaction_cst = [[i] for i in range(X_train.shape[1])]
hgbdt_model_without_interactions = (
.fit(X_train, y_train)
print(f"Test R2 score: {hgbdt_model_without_interactions.score(X_test, y_test):.2f}")
_, ax = plt.subplots(ncols=2, figsize=(6, 4), sharey=True, constrained_layout=True)
features_info["centered"] = False
display = PartialDependenceDisplay.from_estimator(
_ = display.figure_.suptitle("ICE and PDP representations", fontsize=16)
2D interaction plots¶
PDPs with two features of interest enable us to visualize interactions among them. However, ICEs cannot be plotted in an easy manner and thus interpreted. We will show the representation of available
in from_estimator that is a 2D heatmap.
print("Computing partial dependence plots...")
features_info = {
"features": ["temp", "humidity", ("temp", "humidity")],
"kind": "average",
_, ax = plt.subplots(ncols=3, figsize=(10, 4), constrained_layout=True)
tic = time()
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle(
"1-way vs 2-way of numerical PDP using gradient boosting", fontsize=16
Computing partial dependence plots...
done in 5.959s
The two-way partial dependence plot shows the dependence of the number of bike rentals on joint values of temperature and humidity. We clearly see an interaction between the two features. For a
temperature higher than 20 degrees Celsius, the humidity has a impact on the number of bike rentals that seems independent on the temperature.
On the other hand, for temperatures lower than 20 degrees Celsius, both the temperature and humidity continuously impact the number of bike rentals.
Furthermore, the slope of the of the impact ridge of the 20 degrees Celsius threshold is very dependent on the humidity level: the ridge is steep under dry conditions but much smoother under wetter
conditions above 70% of humidity.
We now contrast those results with the same plots computed for the model constrained to learn a prediction function that does not depend on such non-linear feature interactions.
print("Computing partial dependence plots...")
features_info = {
"features": ["temp", "humidity", ("temp", "humidity")],
"kind": "average",
_, ax = plt.subplots(ncols=3, figsize=(10, 4), constrained_layout=True)
tic = time()
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle(
"1-way vs 2-way of numerical PDP using gradient boosting", fontsize=16
Computing partial dependence plots...
done in 5.440s
The 1D partial dependence plots for the model constrained to not model feature interactions show local spikes for each features individually, in particular for for the “humidity” feature. Those
spikes might be reflecting a degraded behavior of the model that attempts to somehow compensate for the forbidden interactions by overfitting particular training points. Note that the predictive
performance of this model as measured on the test set is significantly worse than that of the original, unconstrained model.
Also note that the number of local spikes visible on those plots is depends on the grid resolution parameter of the PD plot itself.
Those local spikes result in a noisily gridded 2D PD plot. It is quite challenging to tell whether or not there are no interaction between those features because of the high frequency oscillations in
the humidity feature. However it can clearly be seen that the simple interaction effect observed when the temperature crosses the 20 degrees boundary is no longer visible for this model.
The partial dependence between categorical features will provide a discrete representation that can be shown as a heatmap. For instance the interaction between the season, the weather, and the target
would be as follow:
print("Computing partial dependence plots...")
features_info = {
"features": ["season", "weather", ("season", "weather")],
"kind": "average",
"categorical_features": categorical_features,
_, ax = plt.subplots(ncols=3, figsize=(14, 6), constrained_layout=True)
tic = time()
display = PartialDependenceDisplay.from_estimator(
print(f"done in {time() - tic:.3f}s")
_ = display.figure_.suptitle(
"1-way vs 2-way PDP of categorical features using gradient boosting", fontsize=16
Computing partial dependence plots...
done in 0.611s
3D representation¶
Let’s make the same partial dependence plot for the 2 features interaction, this time in 3 dimensions. unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d # noqa: F401
import numpy as np
from sklearn.inspection import partial_dependence
fig = plt.figure(figsize=(5.5, 5))
features = ("temp", "humidity")
pdp = partial_dependence(
hgbdt_model, X_train, features=features, kind="average", grid_resolution=10
XX, YY = np.meshgrid(pdp["grid_values"][0], pdp["grid_values"][1])
Z = pdp.average[0].T
ax = fig.add_subplot(projection="3d")
surf = ax.plot_surface(XX, YY, Z, rstride=1, cstride=1, cmap=plt.cm.BuPu, edgecolor="k")
"PD of number of bike rentals on\nthe temperature and humidity GBDT model",
# pretty init view
ax.view_init(elev=22, azim=122)
clb = plt.colorbar(surf, pad=0.08, shrink=0.6, aspect=10)
Total running time of the script: (0 minutes 19.594 seconds) | {"url":"https://scikit-learn.org/1.3/auto_examples/inspection/plot_partial_dependence.html","timestamp":"2024-11-07T23:59:05Z","content_type":"text/html","content_length":"121892","record_id":"<urn:uuid:6b95ec90-29d9-415e-a8c3-57f4d91d14e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00017.warc.gz"} |
Comments on Computational Complexity: Today is Paul Turan's 100th Birthday!(Yair Caro emailed me more on the history of the P...Bill and Lance: would it not be possible to NUMBER...Last Anon- YES, WOW, there is no Gap.
I will amend...(Anon #10 here)
I'm sorry, I should have spoke...Anon who refers to Mantel's theorem.
Wikipedia...Lsst anon-
Turan's theorem gives that there
M...I don't understand the nature of the "gap...Here is the complete reference - Turan is credited...To Anon 9:02 AM, August 19, 2010
I've seen th...Anon 3- I was refering to, in the conversation bet...Turan's extremal result of the maximum number ...insert pointers into your speech?THANKS for the information- I updated
the post and...In your writeup, the result that the independence ...
tag:blogger.com,1999:blog-3722233.post4305992721665166751..comments2024-11-13T15:38:29.005-06:00Lance Fortnowhttp://www.blogger.com/profile/
06752030912874378610noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-3722233.post-16266507524626684222011-02-28T15:31:22.791-06:002011-02-28T15:31:22.791-06:00(Yair Caro emailed me more on
the history of the Prob Proof of<br />Turan's theorem. Yair has given me permission to<br />edit his letter and post it as a comment.)<br /><br /><br />I and read your blog with much interest and
noticed the discussion with Ravi<br />Boppana who mentioned I gave the proof in 1979 - which is true.<br />You asked for more exact references to Caro and Wei .<br />The story as I remember it after
32 years is as follows :<br /><br /><br><br /><br><br /><br />In 1978 I was in an undergraduate in Tel-Aviv university and I attended <br />a course on graph theory by Prof. Johanan Schonheim who
later became <br />my Ph.D supervisor and a reading course on number theory where I was personally mentored by <br />Prof. Moshe Jarden .<br />Learning about Turan's theorem in graph theory and
the prime number <br />theorem in number theory I set myself to try some problem solving .<br />Soon I found a generalization of Pillai's result in number theory and <br />wrote a proof and
eventually this became my first published paper :<br /><br />1. Caro, Y. <i>On a division property of consecutive integers.</i> Israel <br />Jour. of Mathematics, 33 (1) (1979) 32-36.<br /><br /><br>
<br /><br><br /><br />I also found the proof of the theorem now called Caro-Wei and wrote a =<br />paper called <i>new results on the independence number</i> ( a technical <br />report 4/1979 )<br />
which was soon edited as a research paper and sent to a journal.<br />Prof Johanan Schonheim told me he circulated some copies of this paper <br />among other people including Prof. Marcel Hertzog
who said he gave a <br />copy to Prof. Ed Bertram<br />And then the paper spread out without control!<br /><br /><br><br /><br><br />A referee responded unfavourably because as he explained <br /><i>
Caro gave a proof via deleting vertices of maximum degree and induction and I can <br />give an easier proof using a result of Erdos</i> (which he did but was not easier at all) . <br />I myself
found another really easier proof deleting vertices of minimum degree, after sending the paper .<br /><br /><br><br /><br><br /><br />Prof. Schonheim told me he wrote a very angry letter to the
editor <br />(mentioning the ethic of this event and that there are other results <br />worth publishing ) but that was in vain.<br />I decided not to submit it again after sending most of the copies
I <br />hold to friends and so the paper remains as is - Technical report <br />4/1979 !!<br /><br /><br><br /><br><br /><br />Two years later the technical memorandum of V. Wei appears with the same
<br />result ( 1981 ) without reference to my result , and so today the<br />theorem is called Caro - Wei .<br /><br /><br><br /><br><br /><br />Best - Yair CaroGASARCHhttps://www.blogger.com/profile
/06134382469361359081noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-19482201376313769232010-08-20T14:57:32.880-05:002010-08-20T14:57:32.880-05:00Bill and Lance: would it not be possible to
NUMBER the comments? The references the would be less ambiguous.<br /><br />Turan's theorem speaks about max number of edges in a k-clique-free graph, for *any* k>3. What Mantel proved more
than 25 years before him was the case k=3. And that's all. Turan's proof is also not something realy new: just an extension of Mantel's one. More important is that after Turan's
result many people have looked at similar questions. An extremal graph theory was born. This is a very important act, which Turan made: to initiate an entire field!
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-44125337857332513272010-08-20T10:15:49.327-05:002010-08-20T10:15:49.327-05:00Last Anon- YES, WOW, there is no Gap.<br />I will amend
writeup.<br />I had miscounted.<br />THANKS!GASARCHnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-68742766619222511712010-08-20T09:17:51.129-05:002010-08-20T09:17:51.129-05:00(Anon #10
here)<br />I'm sorry, I should have spoken more clearly (the perils of posting when waking up in the middle of the night). I understand what you mean by a gap, I just thought that the example we
evidently both had in mind is tight. So (at least) one of us is miscounting.<br /><br />By my count, every point is close to exactly (n/3 - 1) other points. Thus, the total number of close pairs is
<br /><br />n*(n/3 - 1)/2 = n^2/6 - n/2Anon #10noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-67216943906436214332010-08-20T09:03:59.940-05:002010-08-20T09:03:59.940-05:00Anon who refers
to Mantel's theorem.<br />Wikipedia states Turan's theorem diff then I do (though its equivlent) and says<br />that Mantel's theorem is a special
case.GASARCHnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-85980390237130354392010-08-20T09:00:38.384-05:002010-08-20T09:00:38.384-05:00Lsst anon- <br />Turan's theorem gives that
there<br />MUST be n^2/6 - n/2 pairs.<br /><br />By your construction there IS a way to place n points so that there are<br />n^2/6 -O(1) such pairs.<br /><br />That is a gap. For example, IS there a
way to place n points so that the number of pairs is n^2/6 - n/10 pairs?<br /><br />communicating by commens is awkward- feel<br />free to email
me.GASARCHnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-23802327680485230752010-08-20T06:00:55.346-05:002010-08-20T06:00:55.346-05:00I don't understand the nature of the "gap&
quot; mentioned after Theorem 3.2 in your writeup. Isn't the tight case of Theorem 3.1 given by dividing the set into three subsets of equal size, and assigning the (elements of the) three
subsets to three points on the unit circle at angles 2pi/3? What am I missing?
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-23230515860548546822010-08-19T17:05:36.646-05:002010-08-19T17:05:36.646-05:00Here is the complete reference - Turan is credited as
the creator of extremal graph theory after his 1941 paper (in which he proved a more general result) in Jukna's book<br />http://tiny.cc/62l62<br /><br />This is for completeness sake - nothing
04521212767155230436noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-52244319846597385872010-08-19T17:01:45.645-05:002010-08-19T17:01:45.645-05:00This comment has been removed by the
04521212767155230436noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-82134527597119664082010-08-19T16:57:08.608-05:002010-08-19T16:57:08.608-05:00This comment has been removed by the
04521212767155230436noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-12506626931608037162010-08-19T13:48:50.373-05:002010-08-19T13:48:50.373-05:00To Anon 9:02 AM, August 19, 2010<br /><br />
I've seen that at least three non-inductive proofs of Mantel's theorem (what you call Turan's theorem) are given. e.g. in Jukna's book "Extremal combinatorics". The
arguments are: Cauchy-Schwarz, arithmetic-geometric mean inequality, weight shifting
argument.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18493104376356807132010-08-19T09:17:40.252-05:002010-08-19T09:17:40.252-05:00Anon 3- I was refering to, in the conversation
between Lance and GASARCH,<br />Lance has a pointer to the wikipedia entry<br />on Paul Turan in what I typed.<br /><br />Anon 4- If you know of a good free online<br />writeup of this, email it to
me or comment about it and I will add it to this post.GASARCHnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-82134267392246629202010-08-19T09:02:37.583-05:002010-08-19T09:02:37.583-05:00
Turan's extremal result of the maximum number of edges in a trinagle-free graph to be \floor{n^2/4} is also beautiful and useful. This directly translates that the maximum number of edges in a
transitively reduced digraph is \floor{n^2/4}. Is there any non-inductive proof known for the maximum number of edges in a triangle-free
graph.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-38613665882373738882010-08-18T18:25:53.229-05:002010-08-18T18:25:53.229-05:00insert pointers into your speech?
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-46965107322025453252010-08-18T11:47:15.164-05:002010-08-18T11:47:15.164-05:00THANKS for the information- I updated<br />the post and
the writeup appropriately.<br />If you have a more exact ref to<br />Caro or Wei please email them to me.GASARCHhttps://www.blogger.com/profile/
06134382469361359081noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-69862376997841304012010-08-18T11:15:39.054-05:002010-08-18T11:15:39.054-05:00In your writeup, the result that the
independence number of a graph is at least the sum of 1/(d+1) over all degrees d is due to Caro (1979) and Wei (1981). Back in the late 1980's, I happened to come up with the probabilistic proof
that you wrote and showed it to Joel Spencer and Paul Erdos (which explains why it appeared in Alon-Spencer). I don't know what the original proofs of Caro and Wei were, so it is possible that my
proof was already known.<br />--Ravi BoppanaRavi Boppanahttps://www.blogger.com/profile/08439913421610643876noreply@blogger.com | {"url":"https://blog.computationalcomplexity.org/feeds/4305992721665166751/comments/default","timestamp":"2024-11-14T14:54:51Z","content_type":"application/atom+xml","content_length":"34511","record_id":"<urn:uuid:8125d6db-a43e-4b9c-86b0-6d092cb131e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00685.warc.gz"} |
Learn how to copy data from one table into a new table
Learn how to copy data from one table into a new table by using SQL CREATE TABLE and SELECT statement. Copying data from an existing table to a new one is useful in some cases such as backing up
data, create a copying of real data for testing.
In order to copy data from one table to a new one you can use the following command:
1 CREATE TABLE new_table
2 SELECT * FROM existing_table
MySQL will first create a new table with name as indicated after CREATE TABLE statement, new_table in this case. Then it will fill the new table with all the data from an existing table
To copy a part of data from an existing table, you can use WHERE clause to filter the selected data base on conditions. The command is as follows:
1 CREATE TABLE new_table
2 SELECT * FROM existing_table WHERE conditions
It is very important to check whether table you want to create is existed or not, you should use IF NOT EXIST after CREATE TABLE statement. The full sql command of copying data from an existing table
to a new one will be as follows:
1 CREATE TABLE IF NOT EXISTS new_table
2 SELECT * FROM existing_table
3 WHERE conditions
Here is the example of using copying data command. We have Office data table, now we can copy the table from this table into a new one by using the following command:
1 CREATE TABLE IF NOT EXISTS offices_bk
2 SELECT * FROM offices
If we need only copy all offices in US, so we can use WHERE condition for it as follows:
1 CREATE TABLE IF NOT EXISTS offices_usa
2 SELECT * FROM offices
3 WHERE country = 'USA'
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Equivalent Annual Annuity (EAA) Approach - Finance Train
Equivalent Annual Annuity (EAA) Approach
The EAA value represents the required size of an annual payment over an asset’s life to make the present value of the project’s operating cash flows equal to the net present value, when the cost of
capital is applied as the discount rate.
EAA Process
Data Science in Finance: 9-Book Bundle
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Leetcode: 191. Number of 1 Bits (Bit manipulation)
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Leetcode: 191. Number of 1 Bits (Bit manipulation)
Counting Set Bits in a 32-Bit Unsigned Integer - C++ Implementation
Introduction: In this blog post, we'll delve into a C++ solution to count the number of set bits (1s) in a given 32-bit unsigned integer. The task is to efficiently determine the count of set bits in
the binary representation of the input number. Understanding this bit manipulation technique can be beneficial in various scenarios, such as optimizing algorithms or solving problems related to
binary representations.
Problem Description: The problem can be defined as follows: Given a 32-bit unsigned integer n, we need to count the number of set bits (1s) in its binary representation. For example, for the input n
= 11 (binary: 00000000000000000000000000001011), the function should return 3, as there are three set bits in its binary representation.
Approach: To solve this problem, we will implement a function called hammingWeight, which takes a 32-bit unsigned integer n as input and returns the count of set bits.
#include <cstdint>
class Solution {
int hammingWeight(uint32_t n) {
int sum = 0;
while (n != 0) {
n &= (n - 1);
return sum;
Explanation: Let's break down the implementation step by step:
1. We initialize a variable sum to store the count of set bits and set it to 0 initially.
2. The while loop runs until n becomes 0, which means all bits have been processed.
3. Inside the loop, we increment sum by 1, as we found a set bit (1) in the least significant position.
4. To remove the rightmost set bit in n, we perform bitwise AND operation &= with n - 1. This operation clears the rightmost set bit and maintains the positions of other set bits.
5. The loop continues the process until n becomes 0, effectively counting the number of set bits.
6. Finally, the function returns the computed sum, which represents the count of set bits in the binary representation of the input n.
Example: Let's illustrate the implementation with an example. Consider the input number n = 11:
1. Initially, sum = 0.
2. For the first set bit (1) in n, sum = 0 + 1 = 1.
3. After clearing the rightmost set bit in n, n = 10 (binary: 00000000000000000000000000001010).
4. For the second set bit (1) in n, sum = 1 + 1 = 2.
5. After clearing the rightmost set bit in n, n = 8 (binary: 00000000000000000000000000001000).
6. For the third set bit (1) in n, sum = 2 + 1 = 3.
7. After clearing the rightmost set bit in n, n = 0 (binary: 00000000000000000000000000000000).
8. The loop terminates as n becomes 0, and the function returns sum = 3.
Conclusion: We have successfully implemented a C++ solution to count the number of set bits in a 32-bit unsigned integer. The provided function, hammingWeight, efficiently counts the set bits using a
bitwise operation, making it an optimal solution. Understanding this bit manipulation technique can be useful in various programming tasks that involve binary representations and optimizing
algorithms. Whether you're working on low-level programming or dealing with binary data, this knowledge can prove invaluable. Happy coding!
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Writing Stan code
The package allows for only limited models as, e.g., neither random slopes, nor interaction effects are allowed. Imposing this restriction was a design decision, as it would require duplicating
functionality of general purposes packages. Instead, the package itself provides some basic fitting that should be sufficient for most simple cases. However, below you will find example of how to
incorporate cumulative history into a model written in Stan. This way, you can achieve maximal flexibility but still save time by reusing the code.
Stan model
This is a complete Stan code for a model with log-normal distribution for multiple runs from a single experimental session of a single participant. The history time-constant tau is fitted, whereas
constants are used for other cumulative history parameters.
// --- Complete time-series ---
int<lower=1> rowsN; // Number of rows in the COMPLETE multi-timeseries table including mixed phase.
real duration[rowsN]; // Duration of a dominance/transition phase
int istate[rowsN]; // Index of a dominance istate, 1 and 2 code for two competing clear states, 3 - transition/mixed.
int is_used[rowsN]; // Whether history value must used to predict duration or ignored
// (mixed phases, warm-up period, last, etc.)
int run_start[rowsN]; // 1 marks a beginning of the new time-series (run/block/etc.)
real session_tmean[rowsN]; // Mean dominance phase duration for both CLEAR percepts. Used to scale time-constant.
// --- A shorter clear-states only time-series ---
int clearN; // Number of rows in the clear-states only time-series
real clear_duration[clearN]; // Duration for clear percepts only.
// --- Cumulative history parameters
real<lower=0, upper=1> history_starting_values[2]; // Starting values for cumulative history at the beginning of the run
real<lower=0, upper=1> mixed_state; // Mixed state signal strength
parameters {
real<lower=0> tau; // history time-constant
// linear model for mu
real a;
real bH;
// variance
real<lower=0> sigma;
transformed parameters{
vector[clearN] mu; // vector of computed mu for each clear percept
// temporary variables
real current_history[2]; // current computed history
real tau_H; // tau in the units of time
real dH; // computed history difference
int iC = 1; // Index of clear percepts used for fitting
// matrix with signal levels
matrix[2, 3] level = [[1, 0, mixed_state],
[0, 1, mixed_state]];
for(iT in 1:rowsN){
// new time-series, recompute absolute tau and reset history state
if (run_start[iT]){
// reset history
current_history = history_starting_values;
// Recompute tau in units of time.
// This is relevant only for multiple sessions / participants.
// However, we left this code for generality.
tau_H = session_tmean[iT] * tau;
// for valid percepts, we use history to compute mu
if (is_used[iT] == 1){
// history difference
dH = current_history[3-istate[iT]] - current_history[istate[iT]];
// linear model for mu
mu[iC] = a + bH * dH;
iC += 1;
// computing history for the NEXT episode
// see vignette on cumulative history
for(iState in 1:2){
current_history[iState] = level[iState, istate[iT]] +
(current_history[iState] - level[iState, istate[iT]]) * exp(-duration[iT] / tau_H);
// sampling individual parameters
tau ~ lognormal(log(1), 0.75);
a ~ normal(log(3), 5);
bH ~ normal(0, 1);
sigma ~ exponential(1);
// sampling data using computed mu and sampled sigma
clear_duration ~ lognormal(exp(mu), sigma);
Data preparation
The data section defines model inputs. Hopefully, the comments make understanding it fairly straightforward. However, it has several features that although are not needed for the limited single
session / single session make it easier to generalized the code for more complicated cases.
For example, not all dominance phases are used for fitting. Specifically, all mixed perception phases, first dominance phase for each percept (not enough time to form reliably history) and last
dominance phase (curtailed by the end of the block) are excluded. Valid dominance phases are marked in is_used vector. Their total number is stored in clearN variable and the actual dominance
durations in clear_duration. The latter is not strictly necessary but allows us to avoid a loop and vectorize the sampling statement clear_duration ~ lognormal(exp(mu), sigma);.
In addition, session_tmean is a vector rather than a scalar. This is not necessary for a single session example here but we opted to use as it will better generalize for more complicated cases.
bistability package provides a service function preprocess_data() that simplifies the process of preparing the data. However, you need to perform the last step, forming a list of inputs for Stan
sampling, yourself.
# function that checks data for internal consistency and returns a preprocessed table
df <- bistablehistory::preprocess_data(br_single_subject,
# data for Stan model
stan_data <- list(
# complete time-series
rowsN = nrow(df),
duration = df$duration,
istate = df$istate,
is_used = df$is_used,
run_start = df$run_start,
session_tmean = df$session_tmean,
# only valid clear percepts
clearN = sum(df$is_used),
clear_duration = df$duration[df$is_used == 1],
# history parameters, all fixed to default values
history_starting_values = c(0, 0),
mixed_state = 0.5
Using the model
You can use this model either with rstan or cmdstanr packages. Below is in an example using cmdstanr, assuming that model file is called example.stan. | {"url":"https://archive.linux.duke.edu/cran/web/packages/bistablehistory/vignettes/writing-stan-code.html","timestamp":"2024-11-06T05:38:11Z","content_type":"text/html","content_length":"27421","record_id":"<urn:uuid:6c8623d7-e2ee-4d5f-9e43-f2986c5e58e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00455.warc.gz"} |
What is the speed of the particle? | HIX Tutor
What is the speed of the particle?
A particle moves with its position given by x=cos(4t) and y=sin(t), where positions are given in feet from the origin and time t is in seconds.
Answer 1
Oh. Oh. Oh. I got this one.
By taking the first derivative of the x and y functions, you can find the components, which can be added up to find the velocity:
#dx/dt = -4sin(4t)# #dy/dt = cos(t)#
Your velocity, then, is a vector with the previously mentioned components.
The Pythagorean theorem can be used to determine the speed, which is the vector's magnitude:
#s = sqrt((-4sin(4t))^2 + cos^2(t))#
Though there might be a more ingenious way to make this even simpler, maybe this will do.
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Answer 2
To determine the speed of a particle, we need information about its velocity, which is a vector quantity representing both speed and direction. If you provide the velocity vector of the particle, I
can calculate its speed by finding the magnitude of the velocity vector, which represents the particle's speed without regard to direction. Please provide the velocity vector of the particle in order
to proceed with the calculation.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/what-is-the-speed-of-the-particle-84670bbeac","timestamp":"2024-11-04T16:57:10Z","content_type":"text/html","content_length":"573134","record_id":"<urn:uuid:ea8aae58-fc20-45ea-8bc5-5a40af73cd5b>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00814.warc.gz"} |
Complete measure
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a
measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already
constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure. Naively, we would take
the sigma-algebra on to be the smallest sigma-algebra containing all measurable "rectangles" for While this approach does define a measure space, it has a flaw. Since every singleton set has
one-dimensional Lebesgue measure zero, for subset of However, suppose that is a non-measurable subset of the real line, such as the Vitali set. Then the -measure of is not defined but and this larger
set does have -measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required. Given a (possibly incomplete) measure space
(X, Σ, μ), there is an extension (X, Σ0, μ0) of this measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ0) is called the completion of the measure space. The
completion can be constructed as follows: let Z be the set of all the subsets of the zero-μ-measure subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing
completeness from holding true); let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of Σ and of Z); μ has an extension μ0 to Σ0 (which is unique if
μ is σ-finite), called the outer measure of μ, given by the infimum Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ). | {"url":"https://graphsearch.epfl.ch/en/concept/44775","timestamp":"2024-11-14T20:05:49Z","content_type":"text/html","content_length":"128000","record_id":"<urn:uuid:eeb08e2b-dea0-4572-9120-634db4dad117>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00029.warc.gz"} |
Transactions Online
Hiroshi SAWADA, Shigeru YAMASHITA, Akira NAGOYA, "Restructuring Logic Representations with Simple Disjunctive Decompositions" in IEICE TRANSACTIONS on Fundamentals, vol. E81-A, no. 12, pp. 2538-2544,
December 1998, doi: .
Abstract: Simple disjunctive decomposition is a special case of logic function decompositions, where variables are divided into two disjoint sets and there is only one newly introduced variable. It
offers an optimal structure for a single-output function. This paper presents two techniques that enable us to apply simple disjunctive decompositions with little overhead. Firstly, we propose a
method to find symple disjunctive decomposition forms efficiently by limiting decomposition types to be found to two: a decomposition where the bound set is a set of symmetric variables and a
decomposition where the output function is a 2-input function. Secondly, we propose an algorithm that constructs a new logic representation for a simple disjunctive decomposition just by assigning
constant values to variables in the original representation. The algorithm enables us to apply the decomposition with keeping good structures of the original representation. We performed experiments
for decomposing functions and confirmed the efficiency of our method. We also performed experiments for restructuring fanout free cones of multi-level logic circuits, and obtained better results than
when not restructuring them.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e81-a_12_2538/_p
author={Hiroshi SAWADA, Shigeru YAMASHITA, Akira NAGOYA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Restructuring Logic Representations with Simple Disjunctive Decompositions},
abstract={Simple disjunctive decomposition is a special case of logic function decompositions, where variables are divided into two disjoint sets and there is only one newly introduced variable. It
offers an optimal structure for a single-output function. This paper presents two techniques that enable us to apply simple disjunctive decompositions with little overhead. Firstly, we propose a
method to find symple disjunctive decomposition forms efficiently by limiting decomposition types to be found to two: a decomposition where the bound set is a set of symmetric variables and a
decomposition where the output function is a 2-input function. Secondly, we propose an algorithm that constructs a new logic representation for a simple disjunctive decomposition just by assigning
constant values to variables in the original representation. The algorithm enables us to apply the decomposition with keeping good structures of the original representation. We performed experiments
for decomposing functions and confirmed the efficiency of our method. We also performed experiments for restructuring fanout free cones of multi-level logic circuits, and obtained better results than
when not restructuring them.},
TY - JOUR
TI - Restructuring Logic Representations with Simple Disjunctive Decompositions
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2538
EP - 2544
AU - Hiroshi SAWADA
AU - Shigeru YAMASHITA
AU - Akira NAGOYA
PY - 1998
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E81-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 1998
AB - Simple disjunctive decomposition is a special case of logic function decompositions, where variables are divided into two disjoint sets and there is only one newly introduced variable. It offers
an optimal structure for a single-output function. This paper presents two techniques that enable us to apply simple disjunctive decompositions with little overhead. Firstly, we propose a method to
find symple disjunctive decomposition forms efficiently by limiting decomposition types to be found to two: a decomposition where the bound set is a set of symmetric variables and a decomposition
where the output function is a 2-input function. Secondly, we propose an algorithm that constructs a new logic representation for a simple disjunctive decomposition just by assigning constant values
to variables in the original representation. The algorithm enables us to apply the decomposition with keeping good structures of the original representation. We performed experiments for decomposing
functions and confirmed the efficiency of our method. We also performed experiments for restructuring fanout free cones of multi-level logic circuits, and obtained better results than when not
restructuring them.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/e81-a_12_2538/_p","timestamp":"2024-11-01T21:58:37Z","content_type":"text/html","content_length":"62841","record_id":"<urn:uuid:300a43e4-361d-4a3f-b36e-a07f6daaf821>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00096.warc.gz"} |
What type of graph has chromatic number less than or equal to 2?
Q. What type of graph has chromatic number less than or equal to 2?
A. histogram
B. bipartite
C. cartesian
D. tree
Answer» B. bipartite
Explanation: a graph is known as bipartite graph if and only if it has the total chromatic number less than or equal to 2. the smallest number of graphs needed to color the graph is chromatic number. | {"url":"https://mcqmate.com/discussion/28583/what-type-of-graph-has-chromatic-number-less-than-or-equal-to-2","timestamp":"2024-11-06T04:10:52Z","content_type":"text/html","content_length":"40317","record_id":"<urn:uuid:9a4f78e2-0932-4f48-8690-7b5343ee28f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00862.warc.gz"} |
Using musica package
Decomposition into custom time scales and comparison of statistical properties of decomposed variables
Decomposition of a variable (variables) into averages at different temporal scales is done with the decomp function. For instance, to decompose observed data into overall mean and 5 years, 1 year, 6
months, 3 months, 1 month and 20 days averages, we call the function as
dec = decomp(basin_PT$obs_ctrl, period = c('Y5', 'Y1', 'M6', 'M3', 'M1', 'D20'))
The averiging periods (period argument) are specified using letter codes “D” - day(s), “M” - month(s), “Y” - year(s) followed by number corresponding to number of periods and “G1” the overall mean.
The periods must be given in order from longest to shortest, the overall mean is always included (and needs not to be specified in period). Shorter periods are always identified within the closest
longer periods, i.e. each shorter period is included in exactly one longer period. As a result, the averages may be calculated over shorter periods than specified. This is due to varying length of
“month” and “year” periods. The actual length used for averaging is included in the output. To make further assessment of the decomposed objects easier, indicator of period within the year
(e.g. quarter or month) as optionally specified by agg_by argument is included in the output.
To visualize the time series at multiple time scales (say 5 years, 1 year, 6 months and 3 months), it is convenient to use the ggplot2 package on the decomposed variable:
ggplot(dec[period %in% c('Y5', 'Y1', 'M6', 'M3')]) +
geom_line(aes(x = period_pos, y = value, col = period)) +
facet_wrap(~variable, scale= 'free', ncol = 1) + theme_bw()
Statistical summaries of distribution of each variable at each time scale can be examined in order to compare different data sources (e.g. control simulation to observations) or different time
periods (e.g. scenario period to control period).
To demonstrate this, let us firts decompose the simulated data for the control and scenario periods in the same way as observations, including also the daily time scale:
dobs = decomp(basin_PT$obs_ctrl, period = c('Y5', 'Y1', 'M6', 'M3', 'M1', 'D15', 'D1'))
dctrl = decomp(basin_PT$sim_ctrl, period = c('Y5', 'Y1', 'M6', 'M3', 'M1', 'D15', 'D1'))
dscen = decomp(basin_PT$sim_scen, period = c('Y5', 'Y1', 'M6', 'M3', 'M1', 'D15', 'D1'))
The comparison is done with compare function. For instance to compare simulated mean wet-day precipitation and temperature with observation call
bi_bc = compare(x = list(`BIAS IN MEAN` = dctrl), compare_to = dobs, fun = mean, wet_int_only = TRUE)
with x the list of decomposed variables to be compared to decomposed variables specified by the compare_to argument. The function evaluates distance between statistical characteristics (fun argument)
of specified data sets. Distance is measured as difference for variables included in getOption('additive_variables'), i.e. temperature (TAS) by default, and as a ratio for other variables.
The result can be easily visualized by
ggplot(bi_bc[period!='G1']) +
geom_line(aes(x = TS, y = DIF, col = factor(sub_period), group = sub_period)) +
facet_grid(variable~comp, scale = 'free')+
To compare the 90th quantiles in control and scenario simulations use
bi_dc = compare(x = list(`CHANGE IN Q90` = dscen), compare_to = dctrl, fun = Q(.9))
Q is a convenience function provided by the package in order to avoid specification of the 90th quantile as the anonymous function (e.g. fun = function(x)quantile(x, .9)).
Visualization is done in the same way as for bias
ggplot(bi_dc[period!='G1']) +
geom_line(aes(x = TS, y = DIF, col = sscale2sea(sub_period), group = sub_period)) +
facet_grid(variable~comp, scale = 'free')+
scale_x_log10(breaks = tscale(c('Y5', 'Y1', 'M1')), lab = c('Y5', 'Y1', 'M1')) + theme_bw()
In the call above we used sscale2sea to transform numerical season codes to letters (J - January, F - February etc.) and specified x axis labels and breaks using function tscale converting period
codes to hours.
Musica package allows also to compare relations between variables at custom time scales via vcompare function. To assess correlation between precipitation and temperature consider
co = vcompare(x = list(OBS = dobs, CTRL = dctrl, SCEN = dscen), fun = cor)
Visualization is again easy with ggplot2 package:
co = co[, SEA:=sscale2sea(sub_period)]
ggplot(co[period!='G1']) +
geom_line(aes(x = TS, y = value, col = ID))+
facet_grid(VARS~SEA, scales = 'free') +
scale_x_log10(breaks = tscale(c('Y5', 'Y1', 'M1')), lab = c('Y5', 'Y1', 'M1')) +
Multiscale transformations
The transfromations are implemented to work with lists consisting of items FROM, TO and NEWDATA. The transformation is calibrated in order to change variables in FROM to match statistical
characteristics of TO. The transformation is then applied to NEWDATA variables. Note, that this concept acoomodate the bias correction as well as the delta change method as indicated in the table
bias correction control simulation observed data scenario simulation
delta change control simulation scenario simulation observed data
Considering the basin_PT dataset, the input for the transformation functions can be prepared as
dta4bc = list(FROM = basin_PT$sim_ctrl, TO = basin_PT$obs_ctrl, NEWDATA = basin_PT$sim_scen)
for (multiscale) bias correction and
dta4dc = list(FROM = basin_PT$sim_ctrl, TO = basin_PT$sim_scen, NEWDATA = basin_PT$obs_ctrl)
for (multiscale) delta change method.
In the case we like to assess the performance of the bias correction we might like to consider
dta4bc0 = list(FROM = basin_PT$sim_ctrl, TO = basin_PT$obs_ctrl, NEWDATA = basin_PT$sim_ctrl)
Similarly, to assess the performance of the multiscale delta method we use
dta4dc0 = list(FROM = basin_PT$sim_ctrl, TO = basin_PT$sim_scen, NEWDATA = basin_PT$sim_ctrl)
Multiscale bias correction
Musica package provides flexible interface for application of bias correction at custom time scale(s), based on the suggestions of Haerter et al. (2011) and Pegram and others (2009). The procedure
utilizes standard quantile mapping (see e.g. Gudmundsson et al. 2012) at multiple time scales. Since correction at particular temporal scale influences values at other aggregations, the procedure is
applied iterativelly. The procedure is further refered to as multiscale bias correction. Same strategy is adopted also within more complex methods (e.g. Johnson and Sharma 2012; Mehrotra and Sharma
To apply multiscale bias correction, the function msTrans_abs is used. The function utilizes standard quantile mapping from the qmap-package, but at multiple time scales. Since correction at
particular temporal scale influences values at other aggregations, the procedure is applied iterativelly until the maximum number of iterations (specified by maxiter argument) is reached or the
difference between succesive iteration step is smaller than tol (1e-4 by default). Differences between corrected and uncorrected variable at longer time scales are used to modify daily values after
each iteration step (see e.g. Mehrotra and Sharma 2016; Pegram and others 2009). To make further assessment of the decomposed objects easier, indicator of period within the year (e.g. quarter or
month) as specified by agg_by argument is included in the output. Note that the quantile mapping at scales equal or shorter than month are fitted separatelly for each month. The quantile mapping is
done at temporal scales specified in period argument.
For instance, standard quantile mapping at daily time step can be performed with
out1 = msTrans_abs(copy(dta4bc0), maxiter = 10, period = 'D1')
The multiscale correction at daily, monthly, annual and global scale is obtained by
out2 = msTrans_abs(copy(dta4bc0), maxiter = 10, period = c('G1', 'Y1', 'M1', 'D1'))
To assess the results, first the relevant datasets have to be decomposed:
pers = c('Y1', 'M3' , 'M1', 'D1')
abb = quarter
dobs_0 = decomp(basin_PT$obs_ctrl, period = pers, agg_by = abb)
dctrl_0 = decomp(basin_PT$sim_ctrl, period = pers, agg_by = abb)
dQMD1 = decomp(out1, period = pers, agg_by = abb)
dQMMS = decomp(out2, period = pers, agg_by = abb)
The results are compared using the compare function and visualized as demonstrated earlier. For instance the original and residual bias in precipitation and temperature maxima is assessed by
bi_0 = compare(x = list(`ORIGINAL` = dctrl_0, `STANDARD` = dQMD1, `MULTISCALE` = dQMMS), compare_to = dobs_0, fun = max)
ggplot(bi_0[period!='G1']) +
geom_line(aes(x = TS, y = DIF, col = sscale2sea(sub_period), group = sub_period)) +
facet_grid(variable~comp, scale = 'free') +
scale_x_log10(breaks = tscale(c('Y5', 'Y1', 'M1')), lab = c('Y5', 'Y1', 'M1')) + theme_bw()
Multiscale delta method
Let \(F\) and \(T\) be the control and scenario simulation, respectively. The method consists in finding a transformation \(f\) such that
\[g_s(T) = g_s[f(F)] \]
with \(g_s\) being a function providing statistical summary at temporal scale(s) \(s\), most often empirical cumulative distribution function or e.g., mean. In most applications the transformation is
determined and applied for each month separately. The pseudocode for the procedure is given in bellow.
input data:
data.frames F, T, N
scales # considered temporal scales
tol # tolerance
maxiter # maximum number of iterations
g # form of the summary function
T* = N
while (error > tol & iter < maxiter){
for (s in scales){
d = dist[g(T), g(F)]
d* = dist[g(T*), g(N)]
T* = update[T*, dist(d, d*)]
error = sum_accros_scales(
dist[g_s(T*), g_s(N)], dist[g_s(T), g_s(F)]
iter = iter + 1
The input data frames \(F\) and \(T\) are used for calibration of the transformation \(f\), which is then applied to a new data frame \(N\), resulting in transfromed data frame \(T^*\). The objective
of the procedure is that
\[ \mathrm{dist}[g_s(T^*), g_s(N)] \sim \mathrm{dist}[g_s(T), g_s(F)], \qquad \textrm{for all} \ s\]
with \(\mathrm{dist}\) the distance, measured as the difference (for temperature) or ratio (for precipitaion). In the procedure, \(T^*\) is iterativelly updated according to the difference/ratio of \
(\mathrm{dist}[g_s(T^*), g_s(N)]\) and \(\mathrm{dist}[g_s(T), g_s(F)]\) for each scale \(s\). The procedure ends when the sum/product of these differences/ratios is sufficiently small or maximum
number of iterations is reached. The method is further denoted multiscale delta method.
Musica package currently implements number of choices for \(g_s\), e.g. mean, empirical distribution function and linear and loess approximation of empirical distribution function.
For instance, standard delta change method at daily time step can be performed with
out3 = msTrans_dif(dta4dc0, maxiter = 10, period = 'D1', model = 'identity')
to consider changes at global, annual, monthly and daily time scale use
out4 = msTrans_dif(dta4dc0, maxiter = 10, period = c('G1', 'Y1', 'M1', 'D1'), model = 'identity')
Note, that the model argument specifies the summary function. Standard delta change method considers changes in mean (model = "const"). Here we assess the changes in the whole distribution function.
To assess the results, first the relevant datasets have to be decomposed:
pers = c('Y1', 'M3' , 'M1', 'D1')
abb = quarter
dctrl_0 = decomp(basin_PT$sim_ctrl, period = pers, agg_by = abb)
dscen_0 = decomp(basin_PT$sim_scen, period = pers, agg_by = abb)
dDCD1 = decomp(out3, period = pers, agg_by = abb)
dDCMS = decomp(out4, period = pers, agg_by = abb)
The results are compared using the compare function.
bi_1 = compare(x = list(`SIMULATED` = dscen_0, `STANDARD` = dDCD1, `MULTISCALE` = dDCMS), compare_to = dctrl_0, fun = max)
ggplot(bi_1[period!='G1']) +
geom_line(aes(x = TS, y = DIF, col = sscale2sea(sub_period), group = sub_period)) +
facet_grid(variable~comp, scale = 'free') +
scale_x_log10(breaks = tscale(c('Y5', 'Y1', 'M1')), lab = c('Y5', 'Y1', 'M1')) + theme_bw() | {"url":"https://cran.uib.no/web/packages/musica/vignettes/using_musica.html","timestamp":"2024-11-04T17:33:36Z","content_type":"application/xhtml+xml","content_length":"404077","record_id":"<urn:uuid:6d352fe8-0dab-4a5f-8376-29339ab261d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00192.warc.gz"} |
Lyngdorf Audio MXA-8400 Multichannel Amplifier Measurements
Link: reviewed by Roger Kanno on SoundStage! Hi-Fi on June 15, 2024
General information
All measurements taken using an Audio Precision APx555 B Series analyzer.
The MXA-8400 was conditioned for 1 hour at 1/8th full rated power (~25W into 8 ohms) before any measurements were taken. All measurements were taken with both channels driven, using a 120V/20A
dedicated circuit, unless otherwise stated.
The MXA-8400 has eight balanced inputs (XLR), and eight speaker-level outputs (Neutrik speakON). Each pair of inputs (e.g., 1&2 and 3&4, etc) can be independently configured as a single channel in
bridge mode. The MXA-8400 was evaluated as a stereo amplifier using channels 1 and 2, as well as a two-channel bridged amplifier using channels 1/2 (bridged) and 3/4 (bridged). The MXA-8400 also
offers a low gain mode (6Vrms sensitivity) and a more typical high gain mode (2Vrms sensitivity). Unless otherwise stated, the high gain mode was used.
Because the MXA-8400 uses a digital amplifier technology that exhibits considerable noise above 20kHz (see FFTs below), our typical input bandwidth filter setting of 10Hz-90kHz was necessarily
changed to 10Hz-22.4kHz for all measurements, except for frequency response and for FFTs. In addition, THD versus frequency sweeps were limited to 6kHz to adequately capture the second and third
signal harmonics with the restricted bandwidth setting.
Published specifications vs. our primary measurements
The table below summarizes the measurements published by Lyngdorf for the MXA-8400 compared directly against our own. The published specifications are sourced from Lyngdorf’s website, either directly
or from the manual available for download, or a combination thereof. With the exception of frequency response, where the Audio Precision bandwidth was extended to 500kHz, assume, unless otherwise
stated, 10W into 8ohms and a measurement input bandwidth of 10Hz to 22.4kHz, and the worst-case measured result between the left and right channels.
Parameter Manufacturer SoundStage! Lab
Rated output power into 8 ohms 200W 269W
Rated output power into 4 ohms 400W 542W
Rated output power into 8 ohms (mono) 800W 950W
Gain (high sensitivity, 2-channel mode) 26.1dB 26.3dB
Gain (low sensitivity, 2-channel mode) 16.6dB 16.9dB
Gain (high sensitivity, bridge mode) 31.7dB 31.9dB
Gain (low sensitivity, bridge mode) 22.2dB 22.5
Input sensitivity (for 200W into 8 ohms, high sensitivity) 2Vrms 1.94Vrms
Input sensitivity (for 200W into 8 ohms, low sensitivity) 6Vrms 5.74Vrms
Our primary measurements in two-channel mode revealed the following using the line-level balanced analog input (unless specified, assume a 1kHz sinewave at 440mVrms, 10W output, 8-ohm loading, 10Hz
to 22.4kHz bandwidth):
Parameter Left Channel Right Channel
Maximum output power into 8 ohms (1% THD+N, unweighted) 269W 269W
Maximum output power into 4 ohms (1% THD+N, unweighted) 542W 542W
Maximum burst output power (IHF, 8 ohms) 273W 273W
Maximum burst output power (IHF, 4 ohms) 556W 556W
Continuous dynamic power test (5 minutes) passed passed
Damping factor 739 778
Clipping no-load output voltage 45.8Vrms 45.8Vrms
DC offset <1.5mV <1.2mV
Gain (high) 26.3dB 26.3dB
Gain (low) 16.9dB 16.9dB
IMD ratio (CCIF, 18kHz + 19kHz stimulus tones, 1:1) <-95dB <-95dB
IMD ratio (SMPTE, 60Hz + 7kHz stimulus tones, 4:1 ) <-106dB <-106dB
Input sensitivity (for full rated power) 1.94Vrms 1.94Vrms
Input impedance 14.8k ohms 14.9k ohms
Noise level (with signal, A-weighted) <22uVrms <22uVrms
Noise level (with signal, 20Hz to 20kHz) <27uVrms <27uVrms
Noise level (no signal, A-weighted) <22uVrms <22uVrms
Noise level (no signal, 20Hz to 20kHz) <27uVrms <27uVrms
Noise level (no signal, A-weighted, low gain) <14uVrms <14uVrms
Noise level (no signal, 20Hz to 20kHz, low gain) <18uVrms <18uVrms
Signal-to-noise ratio (200W, A-weighted) 125.0dB 125.0dB
Signal-to-noise ratio (200W, 20Hz to 20kHz) 123.1dB 123.4dB
THD ratio (unweighted) <0.00009% <0.00009%
THD+N ratio (A-weighted) <0.00025% <0.00025%
THD+N ratio (unweighted) <0.00035% <0.00035%
Minimum observed line AC voltage 121.6VAC 121.6VAC
For the continuous dynamic power test, the MXA-8400 was able to sustain 548W into 4 ohms (~1.2% THD) using an 80Hz tone for 500ms, alternating with a signal at -10dB of the peak (54.8W) for 5
seconds, for 5 continuous minutes without inducing a fault or the initiation of a protective circuit. This test is meant to simulate sporadic dynamic bass peaks in music and movies. During the test,
the top of the MXA-8400 was only slightly warm to the touch.
Our primary measurements in bridge mode revealed the following using the line-level balanced analog input (unless specified, assume a 1kHz sinewave at 230mVrms, 10W output, 8-ohm loading, 10Hz to
22.4kHz bandwidth):
Parameter Left Channel Right Channel
Maximum output power into 8 ohms (1% THD+N, unweighted) 950W 950W
Maximum output power into 4 ohms (1% THD+N, unweighted) 1040W 1040W
Maximum burst output power (IHF, 8 ohms) 980W 980W
Maximum burst output power (IHF, 4 ohms) 1074W 1074W
DC offset <-0.7mV <-0.5mV
Gain (high) 31.9dB 31.9dB
Gain (low) 22.5dB 22.5dB
IMD ratio (CCIF, 18kHz + 19kHz stimulus tones, 1:1) <-97dB <-97dB
IMD ratio (SMPTE, 60Hz + 7kHz stimulus tones, 4:1 ) <--102dB <-102dB
Input sensitivity (for full rated 800W) 2.02Vrms 2.02Vrms
Noise level (with signal, A-weighted) 37uVrms 37uVrms
Noise level (with signal, 20Hz to 20kHz) 47uVrms 47uVrms
Noise level (no signal, A-weighted) 37uVrms 37uVrms
Noise level (no signal, 20Hz to 20kHz) 47uVrms 47uVrms
Signal-to-noise ratio (800W, A-weighted) 126.4dB 126.4dB
Signal-to-noise ratio (800W, 20Hz to 20kHz) 124.4dB 124.4dB
THD ratio (unweighted) <0.00009% <0.00009%
THD+N ratio (A-weighted) <0.00041% <0.00041%
THD+N ratio (unweighted) <0.00056% <0.00056%
Minimum observed line AC voltage 118VAC 118VAC
Frequency response (8-ohm loading)
In our frequency-response plots above, measured across the speaker outputs at 10W into 8 ohms, the MXA-8400 is essentially flat within the audioband. At the extremes the MXA-8400 is at 0dB at 5Hz and
-3dB just past 60kHz. In the graph above and most of the graphs below, only a single trace may be visible. This is because the left channel (blue or purple trace) is performing identically to the
right channel (red or green trace), and so they perfectly overlap, indicating that the two channels are ideally matched.
Phase response (8-ohm loading)
Above are the phase response plots from 20Hz to 20kHz for the line-level input, measured across the speaker outputs at 10W into 8 ohms. The MXA-8400 does not invert polarity and exhibits at worst,
about 30 degrees (at 20kHz) of phase shift within the audioband.
RMS level vs. frequency vs. load impedance (1W, left channel only)
The chart above shows RMS level (relative to 0dBrA, which is 1W into 8 ohms or 2.83Vrms) as a function of frequency, for the analog line-level input swept from 5Hz to 100kHz, in stereo mode. The blue
plot is into an 8-ohm load, the purple is into a 4-ohm load, the pink plot is an actual speaker (Focal Chora 806, measurements can be found here), and the cyan plot is no load connected. The chart
below . . .
. . . is the same but zoomed in to highlight differences. Here we find that maximum deviation between no-load and a 4-ohm load is very small, at around 0.025dB. This is an indication of a very high
damping factor, or low output impedance. With a real speaker, the deviations are smaller, at roughly 0.01dB.
THD ratio (unweighted) vs. frequency vs. output power (two-channel mode)
The chart above shows THD ratios at the output into 8 ohms as a function of frequency for a sinewave stimulus at the analog line-level input in stereo mode. The blue and red plots are for left and
right channels at 1W output into 8 ohms, purple/green at 10W, and pink/orange at the rated 200W. The 10W data yielded the lowest THD figures, ranging from 0.00004% from 20Hz to 200Hz, then up to
0.0005% at 6kHz. These are extraordinarily low THD ratios, nearing the limits of the APx555 analyzer. At 1W, THD ratios were more constant, from 0.0001% from 20Hz to 1kHz, then up to 0.0003% at 6kHz.
A 200W, THD ratios ranged from 0.00006% from 20Hz to 200Hz, then up to 0.0005% at 6kHz.
THD ratio (unweighted) vs. frequency vs. output power (bridge mode)
The chart above shows THD ratios at the output into 8 ohms as a function of frequency for a sinewave stimulus at the analog line-level input in bridge mode. The blue and red plots are for left and
right channels at 1W output into 8 ohms, purple/green at 10W, and pink/orange at the rated 450W. The 1W data yielded the most constant results, ranging from 0.0002% from 20Hz to 2kHz, then down to
0.0001% at 3-4kHz. At 10W, THD ratios ranged from 0.00006% from 20Hz to 1kHz, then up to 0.0006% at 6kHz. A 450W, THD ratios ranged from 0.00002/0.00003% from 20Hz to 100Hz, then a steady climb to
0.0004% at 6kHz.
THD ratio (unweighted) vs. output power at 1kHz into 4 and 8 ohms (two-channel mode)
The chart above shows THD ratios measured at the output of the MXA-8400 as a function of output power for the analog line level-input in two-channel mode, for an 8-ohm load (blue/red for left/right
channels) and a 4-ohm load (purple/green for left/right channels). The 8-ohm data ranged from about 0.0003% at 50mW, down to 0.00007% from 10 to 50W, then up to the “knee” right around 200W. The
4-ohm data ranged from about 0.0006% at 50mW, down to 0.00007% from 5 to 100W, then up to the “knee” around 350W. The 1% THD marks were hit at 278W (8-ohm loading) and 540W (4-ohm loading).
THD+N ratio (unweighted) vs. output power at 1kHz into 4 and 8 ohms
The chart above shows THD+N ratios measured at the output of the MXA-8400 as a function of output power for the analog line level-input in stereo mode, for an 8-ohm load (blue/red for left/right) and
a 4-ohm load (purple/green for left/right channels). THD+N values for the 8-ohm data ranged from 0.003% down to just below 0.0002% at 100-200W. The 4-ohm data yielded THD+N values 3-4dB higher.
THD+N ratio (unweighted) vs. output power at 1kHz into 4 and 8 ohms (bridge mode)
The chart above shows THD ratios measured at the output of the MXA-8400 as a function of output power for the analog line-level-input in bridge mode for a 4-ohm load (blue/red for left/right
channels). THD ratios were not measured into an 8-ohm load because our dummy-load configuration allows for a power handling of only 500W (but 1000W for 4 ohms). Maximum (1% THD) power into 8 ohms was
measured in bridge mode with very short 1-second iterative measurements where a staggering 950W with two bridged channels driven was observed. The 4-ohm data ranged from about 0.002% at 300mW, down
to 0.0001-0.0002% from 1.5 to 500W, then up to the “knee” around 700-800W. The 1% THD mark was hit at 1040W.
THD ratio (unweighted) vs. frequency at 8, 4, and 2 ohms (left channel only, two-channel mode)
The chart above shows THD ratios measured at the output of the MXA-8400 as a function of frequency into three different loads (8/4/2 ohms) for a constant input voltage that yields 50W at the output
into 8 ohms (and roughly 100W into 4 ohms, and 200W into 2 ohms) for the balanced analog line-level input in two-channel mode. The 8-ohm load is the blue trace, the 4-ohm load the purple trace, and
the 2-ohm load the pink trace. The 8 and 4-ohm THD data are nearly identical, ranging from 0.00002-0.00003% from 20Hz to 100Hz, then up to 0.0004% at 6kHz. The 2-ohm data ranged from 0.0001% to
0.003% across the audioband. It’s clear that these amplifier modules are optimized for 4- and 8-ohm loads. Nonetheless, they are stable into 2 ohms, yielding admirably low THD ratios.
THD ratio (unweighted) vs. frequency at 8, 4, and 2 ohms (left channel only, two-channel mode)
The chart above shows THD ratios measured at the output of the MXA-8400 as a function of frequency into two different loads (8/4 ohms) for a constant input voltage that yields 200W at the output into
8 ohms (and roughly 400W into 4 ohms) for the analog line-level input in bridged mode. THD ratios were essentially identical, ranging from 0.00004% at low frequencies, then up to 0.0004% at 4kHz.
THD ratio (unweighted) vs. frequency into 8 ohms and real speakers (left channel only)
The chart above shows THD ratios measured at the output of the MXA-8400 as a function of frequency into an 8-ohm load and two different speakers for a constant output voltage of 2.83Vrms (1W into 8
ohms) for the balanced analog line-level input in stereo mode. The 8-ohm load is the blue trace, the purple plot is a two-way speaker (Focal Chora 806, measurements can be found here), and the pink
plot is a three-way speaker (Paradigm Founder Series 100F, measurements can be found here). The 8-ohm plot is fairly flat and between 0.0001% and 0.0003% from 20Hz to 6kHz. Between 1kHz and 6kHz, the
THD ratios when real speakers were used as loads are identical to the dummy load. The two-way speaker THD results were as high as 0.02% at 20Hz. Between 40Hz and 200Hz, the speaker THD results were
roughly 10-20dB higher than that of the dummy load. While THD ratios remain low and below the threshold of audibility into real-world speaker loads, the NAD M23, using similar amplifier modules,
performed better in this regard than the MX-8400.
IMD ratio (CCIF) vs. frequency into 8 ohms and real speakers (left channel only)
The chart above shows intermodulation distortion (IMD) ratios measured at the output of the MXA-8400 as a function of frequency into an 8-ohm load and two different speakers for a constant output
voltage of 2.83Vrms (1W into 8 ohms) for the analog line-level input. Here the CCIF IMD method is used, where the primary frequency is swept from 20kHz (F1) down to 2.5kHz, and the secondary
frequency (F2) is always 1kHz lower than the primary, with a 1:1 ratio. The CCIF IMD analysis results are the sum of the second (F1-F2 or 1kHz) and third modulation products (F1+1kHz, F2-1kHz). The
8-ohm load is the blue trace, the purple plot is a two-way speaker (Focal Chora 806, measurements can be found here), and the pink plot is a three-way speaker (Paradigm Founder Series 100F,
measurements can be found here). All three IMD plots are within 10-15dB of one another, hovering between 0.0002% and 0.0008%. The IMD results for the real-world speaker loads can be seen both above
an below the resistive dummy load results, depending on frequency.
IMD ratio (SMPTE) vs. frequency into 8 ohms and real speakers (left channel only)
The chart above shows IMD ratios measured at the output of the MXA-8400 as a function of frequency into an 8-ohm load and two different speakers for a constant output voltage of 2.83Vrms (1W into 8
ohms) for the analog line-level input in stereo mode. Here, the SMPTE IMD method was used, where the primary frequency (F1) is swept from 250Hz down to 40Hz, and the secondary frequency (F2) is held
at 7kHz with a 4:1 ratio. The SMPTE IMD analysis results consider the second (F2 ± F1) through the fifth (F2 ± 4xF1) modulation products. The 8-ohm load is the blue trace, the purple plot is a
two-way speaker (Focal Chora 806, measurements can be found here), and the pink plot is a three-way speaker (Paradigm Founder Series 100F, measurements can be found here). All three data sets are
close enough to be judged as identical, hovering around the 0.001% level.
FFT spectrum – 1kHz (line-level input, two-channel mode)
Shown above is the fast Fourier transform (FFT) for a 1kHz input sinewave stimulus, measured at the output across an 8-ohm load at 10W for the analog line-level input in two-channel mode. We see that
the signal’s third (3kHz) and fifth (5kHz) harmonics are at -125dBrA, or 0.00006%, and -130dBrA, or 0.00003%, respectively. The remaining visible signal harmonics are below the -135dBrA, or 0.00002%,
levels. These are extraordinarily low THD levels. The power-supply-related noise peak at the fundamental (60Hz) frequency is barely seen at just above the -150dBrA, or 0.000003%, level; however, this
peak is inherent to the AP’s signal generator. A rise in the noise floor can be seen above 20kHz, indicative of this type of digital amplifier technology. This is an exceptionally clean FFT result.
FFT spectrum – 1kHz (line-level input, two-channel mode, low gain)
Shown above is the fast Fourier transform (FFT) for a 1kHz input sinewave stimulus, measured at the output across an 8-ohm load at 10W for the analog line-level input in two-channel mode, this time
with the gain set to low. We see that the signal’s second (2kHz) and third (3kHz) harmonic, are nearing the absurdly low -140dBrA, or 0.00001%, level. As a point of comparison, below is a 1kHz FFT
with the AP analyzer in loopback mode (generator internally feeds the analyzer) with the same 9Vrms signal amplitude. The only differences are that the overall noise floor, from uncorrelated thermal
noise, is roughly 10dB higher with the MXA-8400 in the signal path (in the audioband), and the 2kHz signal harmonic peak is just below (instead of above) of -140dBrA, and the 3kHz peak is at -150dBrA
instead of -140dBrA. In low-gain mode, in can be said that the MXA-8400 adds only a very small amount of uncorrelated noise (hiss), and from a THD perspective, is essentially perfectly transparent.
In other words, the very definition of a “straight wire with gain.”
FFT spectrum – 1kHz (loopback, 9Vrms)
Shown above is the fast Fourier transform (FFT) for a 9Vrms 1kHz input sinewave stimulus, measured with the AP analyzer in loopback mode, for comparison to the above MXA-8400 FFT charts. The overall
noise floor is around the -165dBrA level, and the only visible peaks are at 60Hz (-150dBrA), 120/240Hz (-155dBrA), 2kHz (-140dBrA), 3kHz (-150dBrA), and 4/6kHz (-160dBrA).
FFT spectrum – 1kHz (line-level input, bridge mode)
Shown above is the fast Fourier transform (FFT) for a 1kHz input sinewave stimulus, measured at the output across an 8-ohm load at 10W for the analog line-level input in bridged mode (high-gain
setting). We see that the signal’s third (3kHz) harmonic is at -130dBrA, or 0.00003%, while the second (2kHz) and fifth harmonics (5kHz) are even lower at -135dBrA, or 0.00002%. These THD levels are
slightly lower than the already extraordinarily low levels in two-channel mode. The overall noise floor (from uncorrelated thermal noise) is a few dB higher in bridge mode (slightly above versus
slightly below -150dBrA), however. This is an unavoidable consequence of using two amplifier modules instead of just one to drive the load.
FFT spectrum – 50Hz (line-level input, two-channel mode)
Shown above is the FFT for a 50Hz input sinewave stimulus measured at the output across an 8-ohm load at 10W for the analog line-level input in two-channel mode. The X axis is zoomed in from 40Hz to
1kHz, so that peaks from noise artifacts can be directly compared against peaks from the harmonics of the signal. The most dominant (non-signal) peaks are the second (100Hz) and third (150Hz) signal
harmonics at roughly -145dBrA, or 0.000006%, and -140dBrA, or 0.00001%. THD levels are even lower at 50Hz than the already ultra-low levels at 1kHz. The power-supply-related noise peak at the
fundamental (60Hz) frequency is evident at the extremely low -140dBrA, or 0.00001%, level. Another near-perfect FFT.
Intermodulation distortion FFT (18kHz + 19kHz summed stimulus, line-level input, two-channel mode)
Shown above is an FFT of the intermodulation distortion (IMD) products for an 18kHz + 19kHz summed sinewave stimulus tone measured at the output across an 8-ohm load at 10W for the analog line-level
input in two-channel mode. The input RMS values are set at -6.02dBrA so that, if summed for a mean frequency of 18.5kHz, would yield 10W (0dBrA) into 8 ohms at the output. We find that the
second-order modulation product (i.e., the difference signal of 1kHz) is at -125dBrA, or 0.00006%, and the third-order modulation products, at 17kHz and 20kHz, are at roughly -110dBrA, or 0.0003%.
Intermodulation distortion FFT (line-level input, APx 32 tone, two-channel mode)
Shown above is the FFT of the speaker-level output of the MXA-8400 with the APx 32-tone signal applied to the input. The combined amplitude of the 32 tones is the 0dBrA reference, and corresponds to
10W into 8 ohms. The intermodulation products—i.e., the “grass” between the test tones—are distortion products from the amplifier and are below the very low -140dBrA, or 0.00001%, level. This is an
another ultra-clean IMD FFT.
Square-wave response (10kHz)
Above is the 10kHz squarewave response using the balanced analog line-level input, at roughly 10W into 8 ohms. Due to limitations inherent to the Audio Precision APx555 B Series analyzer, this graph
should not be used to infer or extrapolate the MXA-8400’s slew-rate performance. Rather, it should be seen as a qualitative representation of the MXA-8400’s mid-tier bandwidth. An ideal squarewave
can be represented as the sum of a sinewave and an infinite series of its odd-order harmonics (e.g., 10kHz + 30kHz + 50kHz + 70kHz . . .). A limited bandwidth will show only the sum of the
lower-order harmonics, which may result in noticeable undershoot and/or overshoot, and softening of the edges. In the case of the MXA-8400, however, what cn be seen in the plateaus of the squarewave
in the top graph is a 500 kHz sinewave, the frequency at which the switching oscillator in the Class D amp is operating (see FFT below).
Square-wave response (10kHz)—250kHz bandwidth
Above is the 10kHz squarewave response using the analog input, at roughly 10W into 8 ohms, with a 250kHz restricted bandwidth to remove the modulation from the 500kHz oscillator. Here we find a
relatively clean squarewave, with some overshoot in the corners.
FFT spectrum of 500kHz switching frequency relative to a 1kHz tone
The MXA-8400’s amplifier relies on a switching oscillator to convert the input signal to a pulse-width-modulated (PWM) squarewave (on/off) signal before sending the signal through a low-pass filter
to generate an output signal. The MXA-8400 oscillator switches at a rate of about 500kHz, and this graph plots a wide bandwidth FFT spectrum of the amplifier’s output at 10W into 8 ohms as it’s fed a
1kHz sinewave. We can see that the 500kHz peak is quite evident, and at -40dBrA. There is also a peak at 1MHz (the second harmonic of the 500kHz peak), at -70dBrA. Those three peaks—the fundamental
and its second/third harmonics—are direct results of the switching oscillators in the MXA-8400 amp modules. The noise around those very-high-frequency signals are in the signal, but all that noise is
far above the audioband—and therefore inaudible—and so high in frequency that any loudspeaker the amplifier is driving should filter it all out anyway.
Damping factor vs. frequency (20Hz to 20kHz, two-channel mode)
The graph above is the damping factor as a function of frequency in two-channel mode. We see both channels with very high and constant damping factor, around 650 to 780 across the audioband.
Diego Estan
Electronics Measurement Specialist | {"url":"https://soundstagenetwork.com/index.php?option=com_content&view=article&id=2983:lyngdorf-audio-mxa-8400-multichannel-amplifier-measurements&catid=97:amplifier-measurements&Itemid=154","timestamp":"2024-11-02T08:41:20Z","content_type":"text/html","content_length":"129495","record_id":"<urn:uuid:155e85d7-db0b-42a3-addf-1145c9fd604b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00558.warc.gz"} |
Twin Primes - A Surprising Result
Let's write twin primes in binary. For example the prime pair (281,283)=(100011001,100011011).
Now concatenate the two binary string to get 100011001100011011=144155 which is not a prime. But just reverse the order of concatenation to get 100011011100011001=145177 and this is a prime!
Here's an even more impressive example.
The prime pair (1049,1051)=(10000011001,10000011011) then concatenate the binary bit streams to get 1000001100110000011011=2149403 which is a prime!
Primes generating primes! Quite amazing. Please contact me with your results.
Content written and posted by Ken Abbott | {"url":"http://www.math-math.com/2018/01/prime-numbers-that-generate-new-prime.html","timestamp":"2024-11-12T18:29:25Z","content_type":"text/html","content_length":"32347","record_id":"<urn:uuid:6a423199-e5e4-4c57-83f7-92288be06483>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00173.warc.gz"} |
Multiplication of Integers
On a page on addition of numbers, I assumed that we know how to add, subtract and multiply whole numbers. I mentioned Peano axioms as the foundation on which these operations are defined and their
properties established. Let's see how it can be done.
First of all we assume that there is something to talk about: there exists an entity known as the set of natural numbers N whose properties (explicitly or implicitly) are given by the following
Peano axioms
1. 1 is a natural number. This says that the set N is not empty. There is at least one natural number. This number is denoted by the symbol 1 (pronounced one or unit.)
2. For every x in N there exists a number x' known as the successor of x. Since x = y means that x and y are one and the same number, x = y implies x' = y'.
3. x' ≠ 1. In other words, 1 is not a successor of any natural number.
4. x' = y' implies x = y. Different numbers have different successors.
It's not my purpose to pursue this subject in every detail. Just, as an example, to demonstrate the logic of derivation from the axioms and, especially The Axiom of Induction, I'll prove a few basic
theorems; after which we'll define addition and multiplication for the natural numbers. (A great many additional examples are considered elsewhere.)
Theorem 1
Indeed, otherwise we would have x' = y'. Axiom 4 would then lead to x = y. Contradiction.
Theorem 2
Let M be the set of all x for which x' ≠ x: M = {x: x' ≠ x}. According to Axioms 1 and 3, 1' ≠ 1. Therefore 1 ∈ M.
Furthermore, assume x ∈ M, i.e., x' ≠ x. Then, by Theorem 1, (x')' ≠ x' which exactly means that x' ∈ M. Finally, Axiom 5 implies that M=N. To recapitulate, x' ≠ x for all natural x.
Theorem 3
If x ≠ 1 then there exists u such that x = u'.
Let M be the set that consists of 1 and all those x for which such u exists. Then 1 ∈ M by definition. Let x ∈ M. Taking x = u we see that x'=u'. Therefore, x' ∈ M. Therefore, M = N and, except for
1, every natural number is a successor of another natural number.
Definition 1 (Addition)
1. For every x define x + 1 = x'.
2. For every x, y define x + y' = (x + y)'.
Theorem 4
The number x + y is well defined for all natural x and y.
Definition 2 (Multiplication)
1. For every x define x·1 = x.
2. For every x,y define x·y' = x·y + x.
Theorem 5
The number x·y is well defined for all natural x and y.
1. x + y and x·y are called the sum and the product of x and y, respectively.
2. I delight in the E. Landau's remark at the beginning of the proof of Theorem 5: Mutatis mutandis (with obvious changes), the proof follows virtually verbatim that of Theorem 4.
3. Recursive definitions, i.e., definitions that depend on Axiom 5, are at the heart of Arithmetic. This axiom provides a way to prove a statement for an infinitude of numbers (or, if you will,
infinitely many statements each holding for a single number) in a finite number of steps (viz., 2.)
4. Associativity and commutativity of both addition and multiplication follow from the Peano axioms. Subtraction and division must be defined separately. Without them, the set of natural numbers is
a semigroup with respect to both addition and multiplication.
5. The distributive law (the first one) is suggested by the second clause of Definition 2 which could be read (x + 1)·y = x·y + y. A more general theorem is of course based on Axiom 5.
Theorem 6 (Distributive Law)
The proof is by induction based on Axiom 5. Let M be the set of all z for which the Law holds. We just saw that, by Definition 2, 1 ∈ M. Let z ∈ M. Then
x·(y + z') = x·(y + z)'
= x·(y + z) + x
= (x·y + x·z) + x
= x·y + (x·z + x)
= x·y + x·z'
where I have assumed associativity of the addition as proven.
1. Edmund Landau, Foundations of Analysis, Chelsea Pub Co, 1960
2. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
What Can Be Multiplied?
• Multiplication of Numbers
|Contact| |Front page| |Contents| |Algebra| |Up|
Copyright © 1996-2018
Alexander Bogomolny | {"url":"http://www.cut-the-knot.org/do_you_know/mul_num.shtml","timestamp":"2024-11-11T13:25:11Z","content_type":"text/html","content_length":"17967","record_id":"<urn:uuid:5e00a37a-d3d4-418b-ba80-04ac62170a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00589.warc.gz"} |
Rounding Numbers Calculator - Instantly Round Numbers Online
Introduction to Rounding Numbers Calculator
Rounding numbers calculator is an online tool that helps you to find the decimal number from different decimal or whole number place values (tenth, hundredth, thousandth, .., unit, ten, hundred,.) in
a fraction of a second. Our nearest whole number calculator evaluates the precise number from the decimal number.
It is a learning tool for anyone who has desired to round off the numbers efficiently, because it is essential for various fields, including finance, science, education, or everyday tasks. By using
the round numbers calculator, you can quickly and accurately perform rounding, ensuring precision in their calculations.
What are Rounding Numbers?
Rounding numbers is the process of getting the approximating number value to a specified and precise number from the whole decimal number. This method is used to get a precise number so you can use
it easily for calculation.
You can round off the number from unit to million or tenth to thousandth place value using the common rounding method rules effectively to get the desired level of precision in the solution.
How to Find Round Numbers Using Round Numbers Calculator
For the rounding of numbers, you need to follow different principles that are required to find the number for specific decimal place values.
The rounding numbers calculator makes the given number simpler or more convenient for further calculation typically by reducing the number of significant digits. The basic idea is to round a number
to a certain place value, such as the nearest ten, hundred, or decimal place. Let's see different steps that are used to round numbers
1. Determine the place value in which you want to round the number (e.g., nearest ten, nearest hundred, nearest tenth, nearest hundredth).
2. Check the digit immediately to the right of the place value which you want to round off.
3. Apply the round number rules in which, If the digit is greater or equal to 5 then Increase the digit which you round increase by 1.
4. If the digit is less than 5 then rounding remains constant without any change.
5. After rounding, replace all digits to the right and replace the zeros of all the numbers except the number that is rounded off, if you are rounding the whole number. if rounding the decimal
number then no need to replace zero but to remove them, only write the required decimal place value after rounding.
The nearest whole number calculator can help in these steps, especially when precision and speed are needed in rounding to the nearest whole number.
Solved Example of Rounding Number
Let's see an example of the rounding number with a solution to know how to do manual calculations and to understand the round numbers calculator’s working method.
Round each number number to the nearest hundred:
$$ 1.12,658 $$
Find the location of hundreds place.
How to Use Rounding Numbers Calculator
Rounding number Calculator has a simple design so that you can use it easily to evaluate the rounding number questions. Before adding the input value in the rounding whole numbers calculator, you
must follow some guidelines to avoid inconvenience in the evaluation process. These steps are:
1. Choose the decimal places from the given list as per your choice of finding number.
2. Enter your decimal or whole number in the input field for rounding the number
3. Review your input numbers for round-off, because, if your number values are not correct then our rounding up numbers calculator does not provide you with accurate numbers in solutions for
rounding number problems.
4. Click the “Calculate” button to get the result of your given rounding number problem
5. If you want to try out our rounding off numbers calculator for the first time then you must check the load example and its solution that gives you a better understanding of its working procedure.
6. Click on the “Recalculate” button to get a new page for rounding number problems for whole or decimal number places.
Output of Nearest Whole Number Calculator
Rounding numbers calculator gives you the solution in a step-wise process when you add decimal or whole number as an input into it.It may contain as
• Result option gives you a solution for rounding number problems
• Possible step provides you with all the steps of the round number problem for decimal places in detail.
Benefits of Using a Rounding Number Calculator
Round numbers calculator gives you tons of benefits whenever you use it to calculate the rounding number questions. You do not put any extra effort from solving the rounding number just use this tool
and get solution . These advantages are:
• Rounding whole numbers calculator saves time that you consume in doing round number problems for different decimal places.
• It is a free-of-cost tool that allows you to use it for free to find the rounding number without spending.
• Number rounder calculator is a versatile tool that allows you to solve rounding numbers (whole number, decimal number) problems
• You can use this rounding numbers calculator for practice so that you get a strong hold on the rounding number concept
• Rounding off numbers calculator is a trustworthy tool that provides you with accurate solutions every time whenever you use it to find the decimal place number problem rounded off.
• Nearest whole number calculator is an educational tool that is used to teach children about the concept of round numbers and how to perform rounding number concepts. | {"url":"https://pinecalculator.com/rounding-numbers-calculator","timestamp":"2024-11-12T06:33:00Z","content_type":"text/html","content_length":"56922","record_id":"<urn:uuid:39016cd1-002d-46b9-9d1d-1f3fb9276565>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00312.warc.gz"} |
Gage R&R | Gage Repeatability & Reproducibility - Quality Assist
In this post, you are going to understand the detailed concept of the Gage R&R (Gage Repeatability and Reproducibility) study.
The current situation
Whether there is an internal Audit or external audit in an organization, one question is always asked by the Auditor.
Show the Measurement System Analysis study. In many companies, one standard Gage R&R Excel template is being used by engineers for this study.
Quality engineers manually fill the tables and just for document & audit purpose completes the study.
Have you observed such situation?
Just because of that many engineers don’t know exactly the concept of the Gage R&R study.
And if they don’t know, then they won’t add any value in the process improvement. Also when some questions will be asked in an interview they won’t be able to deliver the exact answer.
This complete guide is helping you to understand Gage R&R study with examples.
Why we do Gage R&R?
One process is common in every company, and it’s nothing but the inspection process.
We are doing an inspection to capture the data from our processes. Then those data is being analyzed.
After analysis, we take some actions in the process to make improvements.
Now the problem is, if you choose the wrong data, then whatever action you take, there will be no impact on the process.
And what is the root cause of producing wrong data?
Your inaccurate Measurement System is the root cause of producing wrong inspection data.
The measurement system should have an acceptable level of variation to get accurate data.
Therefore we preferred the measurement system which has no error/variation.
To calculate the error, and variation in the measurement system, there is an effective tool called Gage Repeatability and Reproducibility (Gage R&R).
What is Gage Repeatability & Reproducibility?
Gage repeatability and reproducibility (Gage R&R) is defined as the method to find how much of the process variation is due to the measurement equipment and measurement method.
This method is designed to find out the effects of repeatability & reproducibility separately.
And find out a combined overall error of measurement.
Also to take appropriate actions to reduce the measurement system variation contributed to total process variation.
Let’s learn what exactly Repeatability and Reproducibility are.
Repeatability – It is the variation obtained while measuring a given characteristic repeatedly on the same part.
• With one measuring instrument
• By one appraiser
Reproducibility – Generally, It is the variation in the average of the measurement system made.
• By different appraisers
• Using the same measuring instrument
• On identical characteristics of the same part
How to perform Gage R&R study?
There is 1 gage (Instrument/equipment), 3 different operators, a total of 3 trials and 10 components (ideally covering the Full range of the process spread) should be used.
Gage R&R Study steps
• Choose the operators who inspect the parts as Operator A, Operator B & Operator C.
• Select 10 components & identify them by numbering from 1 to 10.
• Select the Gage (Measuring Instrument) which is calibrated before being taken for trial if appropriate.
• Then Let “operator A” measure the 10 components and enter the results in a worksheet.
• Let “operator B” measure the 10 components and enter the result in the worksheet in the same order without seeing the operator A results.
• Let “operator C” measure the 10 components and enter the result in the worksheet in same order without seeing the operator A & operator B results.
• The one cycle is completed; repeat the cycle 2 times again until each operator has measured 10 components a total of 3 times. Record all results in the worksheet.
• Complete the calculation as per the standard.
Gage R&R Calculation’s
Let us take an example by putting the values from actual data. We have the data from measurement then we add this data in the worksheet as shown above.
calculate the basic terms from above data,
R = (r[a] + r[b] + r[c] ) / No. of operators =(0.151+0.221+0.253)/3 =0.2083
UCL[R ]= R * D4 = 0.2083 * 2.58 = 0.538
(D4 For 2 Trials=3.27 & 3 Trials=2.58)
Now from the above example we have to calculate the following terms
• Equipment Variation (EV)
• Appraiser / Operator Variation (AV)
• Repeatability & Reproducibility (GRR)
• Part Variation (PV)
• Total Variation (TV)
• No. of distinct categories (ndc)
1) Equipment variation (EV): Repeatability
It calculates the variation one operator has when measuring the same part using the same Gage more than one time.
EV = R * K[1] =0.2083 * 0.5908 = 0.12306
(K1 for 2 Trials=0.8862 & 3 Trials=0.5908)
2) Appraiser Variation (AV): Reproducibility
It is the variation in the average of measurement by different operators measure the same part using same Gage.
3) Repeatability & Reproducibility (GRR)
This is the combination of operator variation & Equipment variation….
For this example
Part Variation (PV)
It is calculated by range of the parts average, as below
For this example
5) Total Variation (TV)
Total variation is calculated for interpreting the above analysis.
For this example
6) Number of distinct categories (ndc)
The number of distinct categories (ndc) is a measure of the number of distinct categories that can be distinguished by the measurement system, so this can show that the how much your data is
categorized in terms of control charts.
ndc = 1.41 (PV / GRR) =1.41 (1.101/0.208) =7
For acceptance level “ndc” value should be greater then 5.
Now finally to comparing the above data with variation in the parts,
1. %EV = 100*(EV / TV) =100*(0.123 / 1.121) =10.98%
2. %AV = 100*(AV / TV) =100*(0.68 / 1.121) =15.05%
3. %GRR = 100*(GRR / TV) =100*(0.208 / 1.121) =18.64%
4. %PV = 100*(PV / TV) =100*(1.101 / 1.121) =98.25%
GRR Acceptance Criteria
• Now If %GRR < 10% – Measurement system is acceptable
• If 10% <%GRR < 30% – Measurement system is conditionally accepted
• Then if %GRR > 30% – Measurement system is not acceptable
Interpretation from Gage Repeatability & Reproducibility study
When repeatability is large compare to reproducibility
• Instrument need maintenance
• Redesign Gage for more rigidity
• Improve location / clamping of gauging
• Excessive within part variation
When reproducibility is large compare to repeatability
• Training need to operator for better Gage measurement
• Incremental division on instrument are not readable
• Need fixture to provide consistency in Gage use. | {"url":"https://qualityengineerstuff.com/gauge-rr/","timestamp":"2024-11-02T01:48:46Z","content_type":"text/html","content_length":"297235","record_id":"<urn:uuid:3b45c0cd-c278-49e5-9161-5b748e8c6693>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00505.warc.gz"} |
Mortar Calculator – Estimate Your Materials Quickly
This mortar calculator tool will help you estimate the amount of cement, sand, and water required for your masonry project.
Mortar Calculator
How to Use the Mortar Calculator
Enter the number of bricks you will be using in your project and the amount of mortar required per brick in kilograms. Then, click on the “Calculate” button to find out the total amount of mortar
needed. The result will be displayed in kilograms.
How the Calculator Works
The calculator multiplies the number of bricks by the amount of mortar needed per brick. This gives the total amount of mortar required for the project. The formula used is:
Total Mortar (kg) = Number of Bricks × Mortar per Brick (kg)
The calculator assumes uniform brick size and consistent mortar usage. Variations in brick size, wastage, and technique can result in different mortar requirements. This tool should be used as a
guide only; it is not a substitute for professional advice or precise material estimation methods. | {"url":"https://madecalculators.com/mortar-calculator/","timestamp":"2024-11-09T16:43:35Z","content_type":"text/html","content_length":"142121","record_id":"<urn:uuid:58ebfbee-f2f5-43a2-b4bb-98b9aba02e76>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00657.warc.gz"} |
Numerical Analysis and Scientific Computing Seminar | Krzysztof Fidkowski, Adjoint-based Adaptation and Optimization of Unsteady Turbulent Flows using Dynamic Closures | Applied Mathematics
Tuesday, November 8, 2022 1:00 pm - 1:00 pm EST (GMT -05:00)
For Zoom Link please contact ddelreyfernandez@uwaterloo.ca
Professor Krzysztof Fidkowski , University of Michigan, Department of Aerospace Engineering
Adjoint-based Adaptation and Optimization of Unsteady Turbulent Flows using Dynamic Closures
We present a data-driven approach for calculating adjoint sensitivities in unsteady turbulent flows, with application to shape optimization and output-based adaptation. The approach does not use
unsteady adjoint equations, which are expensive to solve and become unstable for chaotic problems, but instead relies on unsteady data to train a corrected turbulence model, i.e. a dynamic closure,
which then yields the required adjoint solutions. It is non-intrusive and inexpensive, requiring only a small number of unsteady forward simulations, but sufficiently powerful to capture unsteady
effects in the sensitivities. Results for high-order discretizations of the unsteady Navier-Stokes equations, augmented by a corrected Spalart-Allmaras turbulence closure, demonstrate the ability of
the approach in driving airfoil shape optimization and in adapting unsteady flowfields to target statistical outputs of interest. | {"url":"https://uwaterloo.ca/applied-mathematics/events/numerical-analysis-and-scientific-computing-seminar-1","timestamp":"2024-11-11T07:22:46Z","content_type":"text/html","content_length":"111385","record_id":"<urn:uuid:772ca105-8ce3-4b8b-9e68-cbffc46b6f73>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00590.warc.gz"} |
Problem 936 - TheMathWorld
Problem 936
For the graph, find the average rate of change as a reduced fraction. Also, state the approximate units.(do not abbreviate).
The average rate of change is an application of slope. First, find two known ordered pairs (x,y) given in the problem.
The first ordered pair is (0,1).
The second ordered pair is (2,4).
The average rate of change is:
Use the two known ordered pairs (0,1) and (2,4).
Substitute in the coordinates of the two points.
r =
The average rate of change, as a reduced fraction, is
The y-axis corresponds to pounds, and the x-axis corresponds to bags. Therefore , the unit of measurement that corresponds to the rate is pounds per bag.
Therefore, the average rate of change is | {"url":"https://mymathangels.com/problem-936/","timestamp":"2024-11-11T13:12:56Z","content_type":"text/html","content_length":"61314","record_id":"<urn:uuid:a906c3be-92ea-4e5c-8d73-6742283d3a20>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00323.warc.gz"} |
Not to be confused with Recursoin.
Recursion is the concept of a process (usually a function or procedure) invoking itself as part of its evaluation or execution.
Some functions replace traditional loops like for-loops and while-loops with recursion. The idiom has a "base case" and a "recursive case" which is two options for when to stop calling oneself, and
when to call oneself to find an answer. For example, here is the factorial function in pseudocode:
function FACTORIAL( n : Unsigned Integer) {
if n = 0 { return (1) }
else { return (n * FACTORIAL(n - 1)) }
(Factorial is a function that takes and integer, and multiplies it by all the numbers that come before it, e.g. FACTORIAL(4) = 1 * 2 * 3 * 4.) Here, the function picks: has it reached zero? If so,
then factorial of zero is 1. Otherwise, it will be above zero. Noticing that FACTORIAL(4) = 1 * 2 * 3 * 4 = FACTORIAL(3) * 4, the non-zero case can be defined in terms of the FACTORIAL of previous
numbers. Some favour recursive definitions because they are concise and match the common mathematical proof by induction.
Induction needs to prove that a condition holds for a certain value, then proves that if you change a certain thing about the value, it can't make the condition false. For example, starting from the
assumption that 0 is a law-abiding number, and that n+1 is law-abiding if n is law-abiding, then the proof by induction says that every natural number is law-abiding. 0 is, so 1 is, so 2 is
law-abiding, and so on. This can map to recursion easily. One might prove that 0 is even, that n+1 is odd if n is even, and that n+1 is even if n is odd. Then one can define two functions, odd(n) and
even(n), that first check if n is zero, and if not, then they return the inverse of the result of their partner function called on n-1. One can prove that these functions will always give the right
answer using that proof by induction.
See also | {"url":"https://esolangs.org/wiki/Recursion","timestamp":"2024-11-04T14:16:14Z","content_type":"text/html","content_length":"18559","record_id":"<urn:uuid:e10984d0-2ad0-4506-a8cd-3d1a5fa543d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00308.warc.gz"} |
Temperature and Time of Development of the Two Sexes in Drosophila
1. The time of development at 25° C. up to the moment of pupation is found to be for females and males respectively 116·62 ± 0·19 and 116·78 ± 0·20 hours. During the pupal stage the two times are
111·36±0·15 and 115·46 ±0·13 hours.
2. At 30° C. the corresponding figures are (in the same order): 99·95 ± 0·49, 103·37 ± 0·43, 78·15 ± 0·50 and 84·26 ± 0·34 hours.
3. These figures show that there is a statistical significance in the differences of the times of development of the two sexes for both the periods at 30° C. but only for the pupal stage at 25° C.
It is pointed out that the fact that the longer time of male development as compared with female development at 25° C. is confined to the pupal stage, may be correlated with the other fact that
the essential parts of the secondary sexual characters are developed during this stage.
4. It is shown that there is a negative correlation between the pre-pupal and pupal times of development, indicating that the longer the first time is, the shorter is, as a rule, the other time and
“vice versa.
5. With the aid of statistical methods it is shown that the shortening of the time of development at 30° C. as compared with the time at 25° C. is much more pronounced for the pupal than for the
pre-pupal stage.
6. This last fact is discussed and it is emphasised that the ordinary methods of studying the influence of temperature on development are too rough to be of more than of a descriptive value, the
only way of getting a deeper insight into the processes of development by temperature studies being the separate studies of a number of short intervals.
1. INTRODUCTION. METHODS
All who have worked with Drosophila know that the females are the first to hatch in the culture-bottles. This fact makes it very probable that the females develop in a shorter time than do the males.
However, it is not a priori necessary that it must be so. It could also possibly depend upon a highly selective mortality among the first eggs from each female. In order to study this question in
more detail the following investigation was undertaken.
The flies used belong to the yellow stock. The sex-linked mutant yellow has a viability almost as great as that of the wild type flies. The cause why these latter flies were not used, was that I
possessed a culture of triploids carrying yellow in their X-chromosomes, and that it was my intention to compare the time of development not only of the two normal sexes but also of the intersexes.
Unfortunately, I have until now not succeeded with the intersexes. Before the experiments were begun a rather great number of ordinary culture-bottles (about 50) were made up and five to eight pairs
of flies were put in each of them. There they were left for about three days in order to make it probable that all the females had been fertilised. Next, the same number of culture-bottles was made
up, but contrary to the ordinary method, to these bottles no paper was added. The flies from the old bottles were then transferred to the new ones where they were left for two hours only. When the
two hours had elapsed, the bottles were emptied and left in the incubator until pupation began. In bottles without paper pupation takes place either on the surface of the food or on the walls of the
bottle, and since my bottles are cylindrical in shape and only contain 100 c.c. each, it is quite impossible for the pupae to escape observation. During the whole period of pupation the bottles were
controlled every second hour, and the pupae, which had pupated during the last 2-hour period were transferred to vials containing moistened filter-paper. These vials were then labelled with the
pupation-period and left in the incubator until the beginning of hatching. As in the case of pupation, the vials were controlled every second hour during the whole time of hatching.
By this method it is possible to be informed—for each individual female and each individual male—upon how many periods of 2 hours it takes, (1) from the laying of the egg up to the moment of
pupation, and (2) from this moment up to the moment of hatching. There is one possible source of error due to the fact that the eggs sometimes develop to some extent before laying. I think, however,
that this error does not influence the averages very much, and since the aim of this investigation is to compare the time of development of different groups of animals, the error may be left out of
consideration by this comparison.
As stated above, the eggs for which the time of development have been calculated, have all been laid within a period of only 2 hours. This means that it is very much a matter of chance if we get many
or few individuals upon which to control the time. The experiments, which have been carried out in an incubator, have been made at two different temperatures, viz. 25° C. and 30° C. In the first case
I got a quite sufficient number of individuals, viz. 1083. In the second, however, I got only 77 individuals. This low figure is almost certainly due to the heavy mortality at the high temperature of
30° C. Since this last experiment nevertheless gave some decisive results and since the experiments are rather laborious (it is necessary to control the cultures for 36 hours or more) they were not
2. THE EXPERIMENTS
In the calculation of the time of development up to the moment of pupation, we may imagine that all the eggs have been laid just at the middle of the 2-hour period. And likewise, in the calculation
of the pupal time it is imagined that all the pupae have pupated just at the middle of the different 2-hour periods. This approximation is necessary for the statistical treatment of the problem and
does not involve any error since, on an average, the same number of eggs have been laid before as after the mid-point of the period and the same number of pupae have pupated before as after the
Experiment at 25° C
(a) Time from egg-laying up to pupation
The distribution of this time for the different individuals is shown in
Tables I
and in
Fig. 1
. From the tables we find the mean and its mean error to be
The difference between these two times is 0·16 hours and its mean error is 0-28 hours. And as 0·16 : 0·28 = 0·57 it is obvious that the difference between the time of development of the females and
of the males is without any statistical significance.
(b) Time during the pupal stage
This time is to be found in
Tables I
and in
Fig. 2
. Here we have the mean and its mean error:
The difference of these times and its mean error are respectively 4·10 hours and 0·20 hours; and as 4·10 : 0·20 = 20·5 it is correct to conclude that the males are significantly slower in their
development during the pupal stage than are the females.
Experiment at 30° C
(a) Time from egg-laying up to pupation
The mean error of the difference is 0·65 hours, and since (103·37 — 99·95) :0·05 = 5·26 it is very probable that the difference between the time of development of the two sexes is of a real
(b) Time during the pupal stage
The difference and its mean error being respectively 6·11 and 0·60, we have 6·11 : 0·60 = 10·18, and the difference must thus be of a real significance.
3. DISCUSSION
As may be seen from the above statements, there is—at 25°as well as at 30°C.- a markedly longer time of development for the males than for the females. At 25° C., however, this lengthening is
confined to the pupal stage, the time up to pupation being the same for the two sexes. It seems not unlikely that this may be correlated with the fact that the essential parts of the secondary sexual
characters are developed during the pupal stage. But if this correlation is a true one, then probably we also may suppose that at 30° C. these characters develop earlier, not only in an absolute
sense, but also in a relative one, since namely the figures show that here the time up to pupation as well as that during the pupal stage is longer for the males than for the females. This
supposition must, however, be confirmed by an embryologic investigation.
If we look at the tables, especially
Tables II
, it is seen that the figures to a great extent are distributed along the diagonal ending in the upper right corner. This fact indicates that there is some kind of correlation between the time up to
pupation and the time during the pupal stage. From
Tables I
(25° C.) we also find the coefficient of correlation,
, to be (according to Bravais’ formula) :
It is therefore probably correct to suppose that—at least for the males—for those individuals which have had a considerably long time of development up to the moment of pupation, this long time as a
rule is compensated by a shorter time during the pupal stage and vice versa.
That the time of development is shorter at the higher than at the lower temperature is quite conforming with what we know about the influence of temperature on development. But there is a very
interesting fact to note, namely, that the shortening of the time at 30° C. as compared with that at 25° C. is much more pronounced for the time during the pupal stage than during the time up to
pupation. As a coefficient of the shortening we may introduce the quotient between the time at 30° C. and that at 25°C. It is obvious, then, that the smaller this coefficient is, the greater is the
shortening. In order to make the comparisons more clear, let us introduce a notation. Let the means of the times up to pupation be M[25] and M[30] and their mean errors m[25] and m[30]. Let,
likewise, the means of the times during the pupal stage be P[25] and P[30] and their mean errors p[25]and p[30]. Let, finally, and We have then to compare α and β and to make this comparison for the
two sexes separately. From the figures of the different tables we find that
Let now the mean error of α be μ. and the mean error of (β be v. Since M[30] is independent of M[25] and P[30] is independent of, p[25],μand v must satisfy to the close-approximation formula: and and
thus and With the aid of the foregoing figures we therefore find that The mean error of the difference between α and β is, as usual, Hence, this difference and its mean error have the values: And,
finally, we find the values of the quotient to be We may therefore without hesitation conclude that the more pronounced influence of the raising of the temperature upon the pupal stage as compared
with the influence upon the pre-pupal stages is of a real and absolute significance.
The method commonly used when studying the influence of temperature on development is to observe the time from the beginning of the development up to a certain stage and to compare for a number of
different temperatures. The results are often plotted on a scale and it is usually tried to find a mathematical expression which fits the curve thus obtained. But it may be asked if these studies are
of more than of a purely descriptive value, or if they really inform us on the processes of development. As we have seen, an increase in the temperature has a different influence on different stages
of the development. In fact, it shortens in Drosophila the time during the pupal stage relatively more than it shortens the time up to pupation. But if we not only consider Drosophila but the
organisms in general, there may exist some maximum temperature above which the development is not accelerated but retarded. Since, however, different stages are differently susceptible to the
influence of temperature, this maximum temperature may very well be a different one for different stages. Imagine, then, that a raising of the temperature accelerates one part of the development and
retards one other part, but in such a way that the total time of development is unaltered. From a graphical or mathematical treatment we would then have concluded that temperature is without
influence on the time of development but in reality it had been an obscured but nevertheless real influence of temperature. As the above data show, the time up to pupation in Drosophila is the same
for the two sexes at 25° C. but probably not the same at 30°C. As, however, the time up to pupation involves such different phases of the development as the embryonic phases—phases of
differentiation—and the larval stages— phases of growth—it may possibly only be the total time up to pupation in 25° C. which coincide for the two sexes, whereas for instance one part may be slower
for the males and one other part slower for the females.
It seems, therefore, to me that when studying temperature-coefficients and other figures of that kind, the methods ofted used are too rough to be of a greater value. If we wish to be able to conclude
anything about the physical and physic chemical nature of the processes of development from the variation of the time of this development in different temperatures, it is necessary to study short
intervals of the development separately, and to choose the intervals in such a way that we may infer that there is about the same kind of processes going on during the whole of each separate
interval. The difficulties here are, however, of a practical kind: it is not easy to find suitable objects for such a study. The best objects are probably those with transparent eggs where it is
possible to follow each morphological stage. The methods then to be followed are the ordinary ones : for each separate interval the time of development is noted for a number of different temperatures
and the corresponding curve is traced. Conforming then with the facts, shown above, to hold good in Drosophila, it is probable that the rule should be that curves, corresponding to different
intervals, would be of a quite different kind, indicating different natures of the processes during the various intervals.
Copyright © 1926 The Company of Biologists Ltd. | {"url":"https://journals.biologists.com/jeb/article/4/2/186/20487/Temperature-and-Time-of-Development-of-the-Two","timestamp":"2024-11-08T09:10:04Z","content_type":"text/html","content_length":"196213","record_id":"<urn:uuid:bb28c2f4-f147-4229-a8e8-ad06d1ec6ed8>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00065.warc.gz"} |
Within a spherical charge distribution of charge density class 12 physics JEE_Main
Hint: The question is simple as it is based on the Gaussian law and its charged surface. According to the question it has N equipotential surfaces with increasing radii, so we will take out a small
element dr from the sphere, apply Gaussian law on it and integrate it from limit 0 to radius r and then find the proportionality of density. Formula used = $\int {\mathop E\limits^ \to \cdot \mathop
{ds}\limits^ \to } $ = $\dfrac{{\left( {{q_{enclosed}}} \right)}}{{{E_o}}}$ q(enclosed) is the charge distributed in area. Use the area of the sphere as $\dfrac{4}{3}\pi {r^3}$.
Complete step by step solution:
Step 1:
The Gaussian surface is known as a closed surface in three-dimensional space such that the flux of a vector field is calculated. These vector fields can either be the gravitational field or the
electric field or the magnetic field. Using Gauss law, Gaussian surface can be calculated:
Gaussian surface of sphere is represent as $\int {\mathop E\limits^ \to \cdot \mathop {ds}\limits^ \to } $ =$\dfrac{{\left( {{q_{enclosed}}} \right)}}{{{E_o}}}$ q(enclosed) is the charge distributed
in area.
In any two successive equipotential surfaces have potential difference equal to $\Delta V$
We have to find how the value of $\rho $ (r) depend of r
Spherical charge distribution is given with charge density $\rho $ (r)
The spherical shell is there of thickness dr which is very small and having inner radius $r_1$ and outer radius $r_2$. To find the whole charge distribution we will find charge in a small element and
then integrate for the whole sphere.
Step 2:
Applying Gaussian law on the Gaussian surface$\int {\mathop E\limits^ \to \cdot \mathop {ds}\limits^ \to } $ =$\dfrac{{{q_{enclosed}}}}{{{E_o}}}$
(q(enclosed) is the charge distributed in area)
Then, $\int {\mathop E\limits^ \to \cdot } \mathop {ds}\limits^ \to = \dfrac{{\left( {{q_{enclosed}}} \right)}}{{{E_0}}} \Rightarrow \int\limits_0^r {\dfrac{{\rho \left( r \right)\left( {4\pi {r^2}}
\right)dr}}{{{E_0}}}} $
(integrating from 0 to r to find whole electric field)
The equipotential surfaces of potential ${V_O}$ +$V + \Delta {V_O}$, ${V_O} + 2\Delta V$,…..,${V_O} + N\Delta V$($\Delta V$>0) and increasing radii${r_o}$,${r_1}$,${r_2}$,…..${r_N}$ Are in a form of
A.P and hence are constant with respect to each other
Or we can say that E= −$\dfrac{{\Delta V}}{{\Delta r}}$ (constant)
Now, $\int\limits_{area} {E \cdot dA} = \dfrac{{{q_{enclosed}}}}{{{E_o}}} \Rightarrow \int\limits_{area} {E \cdot dA} = \int\limits_0^r {\dfrac{{\rho \left( r \right)4\pi {r^2}dr}}{{{E_o}}}} $ …….
Here, dA is the area of Gaussian
Surface and Gaussian surface is sphere so area A is equal to $4\pi {r^2}$.
Rewriting equation (1), $E\left( {4\pi {r^2}} \right) = \int\limits_0^r {\dfrac{{\rho \left( r \right)\left( {4\pi {r^2}} \right)dr}}{{{E_0}}}} $ …….. (2)
(Cancelling $4\pi {r^2}$ both sides)
$ \Rightarrow $ $\left( {E{E_o}} \right){r^2} = \int\limits_0^r {\rho \left( r \right){r^2}dr} $
If we will say $E{E_o}$ is constant.
From here we understood that $\int\limits_0^r {\rho \left( r \right)\left( {{r^2}} \right)dr} $ $\alpha $ ${r^2}$ in integration power increase by one. If this equation is true then $\rho $ (r)$\
propto $$\dfrac{1}{r}$ must be true.
Say, $\rho \left( r \right)$ =$\dfrac{C}{r}$ putting in (2) $\int\limits_0^r {\dfrac{C}{r} \cdot {r^2}dr = k \cdot {r^2}} $ where k is some proportionality constant.
Solving the left hand side, $\dfrac{C}{2}{r^2}$ and it is proportional. So we can say that $\rho \left( r \right)\propto \dfrac{1}{r}$.
Hence option (C) is correct.
Note: Points to remember:
While solving the equation for the Gaussian surface must read which object is given. At the place of area we need to write the area of that object, for example we are given a spherical shell so we
have used the area of the sphere. Also we have taken a small element dr and integrated it to find the whole of E. At the end we have made LHS=RHS to find the relation of charge density with r. | {"url":"https://www.vedantu.com/jee-main/within-a-spherical-charge-distribution-of-charge-physics-question-answer","timestamp":"2024-11-13T17:48:02Z","content_type":"text/html","content_length":"165419","record_id":"<urn:uuid:60a08204-ac0e-4f84-bcfb-15467f878e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00813.warc.gz"} |
Analysis anatomy
To take control of analyses, we first need to understand their inner nature.
Analysis in a nutshell
Analyses can be complex at times, but their essence is actually not complicated. Conceptually, all analyses share a simple organization. They all require an input set of data, they have an algorithm
of some sort, and produce a model that represents a certain subset of the characteristics of the input data.
This applies to any kind of analysis. For example:
• a metric transforms the input data into a number, or
• a visualization transforms the input data into a picture.
Of course, in a larger analysis, there can be a multitude of such transformations. For example, consider a visualization displaying entities enriched with metrics: first the entities need to be
extracted, then the metrics are computed, and finally the picture is put together.
Control to interpret
The goal of analysis is to provide a summary to ease the understanding of the original data. But, to be able to interpret the result of an analysis you need to control both the input data and the
decisions taken by the analysis algorithm.
Let us consider a simple example of measuring the size in terms of number of methods of the following class:
public class Library {
List books;
public Library() {…}
public void addBook(Book b) {…}
public void removeBook(Book b) {…}
private boolean hasBook(Book b) {…}
protected List getBooks() {…}
protected void setBooks(List books) {…}
public boolean equals(…) {…}
How many methods are there? 7. But, is a constructor a method? If the metric computation does not consider it as a method, we get 6 instead of 7. What about setters and getters? Are they to be
considered as methods? If no, we have only 4. Do we count the private methods as well? Perhaps the metrics is just about the public ones. In this case, the result is only 3. Finally, equals() is a
method expected by Java, so we might as well not consider it a real method. So, perhaps the result is 2.
How many methods are there? All these are valid answers depending on what we understand by the question.
Now, let's turn the situation around, and consider a report says a class has 70 methods. What does it mean? You have to know what the actual computation does.
But, wait. This is still not enough.
Let us consider another example of computing the size of an entire system in terms of the total number of methods from all system classes. Suppose that we know that the number of methods metric
answers 7 for the above example, and that the result is 20'317 for the entire system. This number does not yet have an interpretation unless we know what "all system classes" entails. Were generated
classes included in this set? How about the classes from the third party frameworks?
If you want to be able to interpret the result of applying an analysis, you need to know both what the input set of data was, and what the algorithm does. | {"url":"http://www.humane-assessment.com/guide/anatomy","timestamp":"2024-11-02T11:23:10Z","content_type":"application/xhtml+xml","content_length":"15399","record_id":"<urn:uuid:ff156175-cd4d-46f2-9b93-3f820f0a5379>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00095.warc.gz"} |
Why Sklearn's Linear Regression Implementation Has No Hyperparameters?
What are we missing here?
Almost all ML models we work with have some hyperparameters, such as:
• Learning rate
• Regularization
• Layer size (for neural network), etc.
But as shown in the image below, why don’t we see any hyperparameter in Sklearn’s Linear Regression implementation?
It must have learning rate as a hyperparameter, right?
To understand the reason why it has no hyperparameters, we first need to learn that the Linear Regression can model data in two different ways:
1. Gradient Descent (which many other ML algorithms use for optimization):
1. It is a stochastic algorithm, i.e., involves some randomness.
2. It finds an approximate solution using optimization.
3. It has hyperparameters.
2. Ordinary Least Square (OLS):
1. It is a deterministic algorithm. Thus, if run multiple times, it will always converge to the same weights.
2. It always finds the optimal solution.
3. It has no hyperparameters.
Now, instead of the typical gradient descent approach, Sklearn’s Linear Regression class implements the OLS method.
That is why it has no hyperparameters.
How does OLS work?
With OLS, the idea is to find the set of parameters (Θ) such that:
• X: input data with dimensions (n,m).
• Θ: parameters with dimensions (m,1).
• y: output data with dimensions (n,1).
• n: number of samples.
• m: number of features.
One way to determine the parameter matrix Θ is by multiplying both sides of the equation with the inverse of X, as shown below:
But because X might be a non-square matrix, its inverse may not be defined.
To resolve this, first, we multiply with the transpose of X on both sides, as shown below:
This makes the product of X with its transpose a square matrix.
The obtained matrix, being square, can be inverted (provided it is non-singular).
Next, we take the collective inverse of the product to get the following:
It’s clear that the above equation has:
• No hyperparameters.
• No randomness. Thus, it will always return the same solution, which is also optimal.
This is precisely what the Linear Regression class of Sklearn implements.
To summarize, it uses the OLS method instead of gradient descent.
That is why it has no hyperparameters.
Of course, do note that there is a significant tradeoff between run time and convenience when using OLS vs. gradient descent.
This is also clear from the algorithm time-complexity table I once shared in this newsletter:
As depicted above, the run-time of OLS is cubically related to the number of features (m).
Thus, when we have many features, it may not be a good idea to use the LinearRegression() class. Instead, use the SGDRegressor() class from Sklearn.
That said, the good thing about LinearRegression() class is that it involves no hyperparameter tuning.
Thus, when we use OLS, we trade run-time for finding an optimal solution without hyperparameter tuning.
👉 Over to you: How would you prove that the solution returned by OLS is optimal? Would love to read your answers :)
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Thank you. I am sending this on to my kid, a sophomore in a UC who is not sure that stats would be a useful course. The way you took it back to basics and brought it into present use was delicious.
Keep writing, please.
Expand full comment | {"url":"https://blog.dailydoseofds.com/p/why-sklearns-linear-regression-implementation","timestamp":"2024-11-07T23:16:54Z","content_type":"text/html","content_length":"226447","record_id":"<urn:uuid:0e46883b-d2f7-489e-9764-24154f9331df>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00692.warc.gz"} |
KRISS Vector - GTA5-Mods.com
"Apply Form Flat Vector Icon..." av ahasoft - Mostphotos
People, perhaps The apply() function works on anything that has dimensions in R, but what if you don’t have dimensions? For that, you have two related functions from the apply family at your disposal
sapply() and lapply(). The l in lapply stands for list and the s in sapply stands for simplify. The two functions work basically […] This vector place is a pure scam, not only do they promise wages
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did any research but it doesn't take long to realize that they are just a pyramid When you apply it to a vector or a factor x, the function replicates its values a specified number of times. Let’s
use one of the vectors that you generated above with lapply() into MyList. This time, however, you only select the elements of the first line and first column from each elements of the list MyList
(and you use sapply() to get a Vector Element Recycling.
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or disability. Download 511 apply free vectors.
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2. I have got a 0508 VECTOR. 0508 832 867. Outage Centre.
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We are one of the largest recruiters of students in the nation. 85% of the Vector The movements of any thrown object, such as a football, can be mapped with vectors. Using multiple vectors allows for
the creation of a model that encompasses external forces like the wind. By utilizing vector addition on these different forces, mathematicians create an accurate estimate of the path of motion and
distance traveled by the object. The vector a is broken up into the two vectors a x and a y (We see later how to do this.) Adding Vectors.
We are 16 nov. 2020 — Apply their knowledge in molecular biology to genetic modification of a viral vector and propose a strategy for in vivo genetic manipulation. 3.
Contemporary linear algebra pdf
For example, we can C++ program to demonstrate that when vectors We use cookies to ensure you have the best browsing experience on our website The vectors standard position has its starting point in
origin. Magnitude. The component form of a vector is the ordered pair that describes the changes in the x - VectorLam Cirrus 2.0 closes the gap between theoretical answers and real-world
application. How Many Ways Can You Use Our Products?
And all this is made twice as hard when working with v Spring Vector features heart pounding VR gameplay and immersive motion controls, but trips itself up from trying to do too much.
Pension plans vs 401k
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3x 2080Ti med mera - Annonskommentarer - SweClockers
Select Background. Start with whatever background you want. sapply is a user-friendly version and wrapper of lapply by default returning a vector, matrix or, if simplify = "array" , an array if
appropriate, by applying We knew this and we've done multiple videos where we use this definition of vector addition.
Ludmila ondrasektypsnitt gamla registreringsskyltar
Bruksanvisning Crestron Vector SUBS18 3 sidor
The admission process at VECTOR Institute is governed by a well-laid out policy. The map functions transform their input by applying a function to each element of a list or atomic vector and
returning an object of the same length as the input. map() always returns a list. See the modify() family for versions that return an object of the same type as the input. map_lgl(), map_int(),
map_dbl() and map_chr() return an atomic vector of the indicated type (or die trying).
Detecting Anomalies in User Communication in - Diva Portal
We will use a continuous selection so Vector Marketing · @VectorMarketing. Looking for #PartTimeWork?
We can then add vectors by adding the x parts and adding the y parts: The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20) 2015 Dealer of the Year. 2016 Installer of the Year. Top
4 largest security company nationally. Sure, we’ve got the recognition and the awards to prove it, but the real reason to join Vector Security is our people. People who are passionate about the work
they do. Energetic people who innovate, who take on new challenges. | {"url":"https://hurmanblirrikrhshw.netlify.app/94597/48832","timestamp":"2024-11-08T17:53:13Z","content_type":"text/html","content_length":"10304","record_id":"<urn:uuid:f50ab547-d556-4467-ba78-2bb8b366f837>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00079.warc.gz"} |
Section: New Results
Stochastic control
Participants : Frédéric Bonnans, Xiaolu Tan [CMAP] , Imene Ben Latifa, Mohamed Mnif [ENIT, Tunis] .
In [24] , we extend a study by Carmona and Touzi on an optimal multiple stopping time problem in a market where the price process is continuous. In this paper, we generalize their results when the
price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. Then we relate our problem to a sequence of
ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman Variational
In [27] , we consider, in the framework of Galichon, Henry-Labordère and Touzi, the model-free no-arbitrage bound of variance option given the marginal distributions of the underlying asset. We first
make some approximations which restrict the computation on a bounded domain. Then we propose a gradient projection algorithm together with a finite difference scheme to approximate the bound. The
general convergence result is obtained. We also provide a numerical example on the variance swap option. | {"url":"https://radar.inria.fr/report/2011/commands/uid32.html","timestamp":"2024-11-03T03:38:54Z","content_type":"text/html","content_length":"40868","record_id":"<urn:uuid:481a8471-e6fa-4ebd-816e-ea782f048274>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00797.warc.gz"} |
This module contains procedures and generic interfaces for evaluating the mathematical operator \(\mp\) acting on integer, complex, or real values.
This module contains procedures and generic interfaces for evaluating the mathematical operator \(\mp\) acting on integer, complex, or real values.
The plus–minus sign, ±, is a mathematical symbol with multiple meanings.
1. In mathematics, it generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction.
2. In experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error.
The sign may also represent an inclusive range of values that a reading might have.
3. In medicine, it means with or without.
4. In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard or with a contract.
5. In botany, it is used in morphological descriptions to notate more or less.
6. In chemistry, the sign is used to indicate a racemic mixture.
7. In electronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and -5 volts, when used with audio circuits and operational amplifiers.
Given two input arguments ref and val, the procedures of this module return an array of size 2 whose elements are [ref - val, ref + val].
If ref is missing, then an appropriate default value is used.
The procedures of this module offer a handy and flexible way of membership checks via operator(.inrange.).
The operator \(\pm\) performs the opposite operation of \(\mp\), that is, \(\pm = -\mp\).
See also
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python,
R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin | {"url":"https://www.cdslab.org/paramonte/fortran/latest/namespacepm__mathSubAdd.html","timestamp":"2024-11-12T03:49:04Z","content_type":"application/xhtml+xml","content_length":"16589","record_id":"<urn:uuid:3265342d-b24a-4dca-93d1-f2d8ab239ca6>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00708.warc.gz"} |
Alan Lark
Has a number of new commands for use in the Western Australian TEE subjects Calculus, Chemistry and Applicable Maths. It contains a number of small routines for use in exams. To use the commands,
simply type the desired command in the home screen as you would an inbuilt command. Most of them can also be done using the Solve aplet but this is faster. Includes binomial probability of X=x or a=X
=b, Poisson probability of X=x or a=X=b, exponential probability of a=X=b, complex operation CIS on any angle, and the atomic mass of any element given the atomic symbol. | {"url":"https://www.hpcalc.org/authors/1631","timestamp":"2024-11-02T17:55:20Z","content_type":"text/html","content_length":"9524","record_id":"<urn:uuid:ef29d13b-7eaf-43af-b0f2-d6039437a1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00103.warc.gz"} |
SSB Demodulation using Coherent Detection
SSB(Supressed Sideband) demodulation is the process of recovering message signal from the SSB modulated wave. SSB is also called SSB-SC where SC means suppressed carrier as the carrier signal in the
SSB signal is completely removed. Here we illustrate how we can demodulate SSB modulated signal to recover message signal. The SSB demodulation method demonstrated here is called coherent detection.
The coherent detection means that the frequency of the local oscillator signal generated at the SSB AM receiver is synchronized with the carrier signal frequency.
The following circuit diagram shows SSB modulator that uses phase discrimination method and SSB receiver demodulator circuit that uses coherent detector.
The above circuit shows SSB modulator as well as SSB demodulator. The SSB modulator is explained in SSB modulation Transmitter Circuit.
The SSB demodulator circuit consist of the last DSB-SC modulator, the passive low pass filter and the operational amplifier. The following shows simplified SSB demodulation block diagram that uses
the coherent method for demodulation.
The DSB modulator, commonly called product modulator, is made using the AD633 analog multiplier IC. It multiplies the incoming SSB modulated signal \(s(t)\) with locally generated carrier signal \(c
(t)\) to generate signal \(r(t)\) as follows.
The SSB signal is,
\(s(t) = cos(w_ct) m(t) \pm \hat{m(t)} sin(w_ct) \) -------->(1)
and \( c(t) = cos(w_c t) \) -------->(2)
So, \(r(t) = s(t) c(t)\) -------->(3)
or, \(r(t) = cos(w_c t)[cos(w_ct) m(t) \pm \hat{m(t)} sin(w_ct) ]\)
or, \(r(t) = [cos^2(w_ct) m(t) \pm \hat{m(t)} sin(w_ct) cos(w_c t)]\)
or, \(r(t) = [\frac{1}{2}\{1+cos(2w_ct)\} m(t) \pm \frac{1}{2} \hat{m(t)} sin(2w_ct)]\)
that is, \(r(t) = \frac{1}{2}m(t) + \frac{1}{2}m(t) cos(2w_ct)\} \pm \frac{1}{2} \hat{m(t)} sin(2w_ct)]\) --->(4)
The second and the third terms can be removed using the low pass filter. After low pass filtering we get,
\(v(t) = \frac{1}{2}m(t)\) --->(5)
We can then use operational amplifier since the message signal is in audio range to amplify the message signal and obtain recovered message signal.
\(m_{demod}(t) = m(t)\) --->(6)
So in this way we showed SSB modulation and demodulation circuit works. The SSB signal was generated using SSB modulator that uses phase discrimination method but the SSB demodulator shown here can
also be used for SSB signal generated by frequency discrimination SSB modulator. We explained how we can demodulate SSB signal using coherent detection. Coherent detector uses standard DSB modulator
and then low pass filter to remove high frequency components. The low pass filtered signal is then fed into LM358 audio amplifier since the demodulated message signal is weak for speaker. Coherent
detection relies on the assumption that the frequency of the carrier signal at the SSB transmitter and the frequency of the local oscillator signal at the SSB receiver is in synchronization. Also
ideally, the transmitted carrier signal phase and the locally generated signal phase should be in synchronization otherwise we will get phase error and therefore phase distortion. This is the main
disadvantage of the coherent detection method for SSB-SC demodulation.
[1] SSB modulation Matlab code
Post a Comment | {"url":"https://www.ee-diary.com/2023/04/ssb-demodulation-using-coherent.html","timestamp":"2024-11-12T16:56:56Z","content_type":"application/xhtml+xml","content_length":"202935","record_id":"<urn:uuid:e619e36c-5b19-4aeb-aab4-bd9d1b02c595>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00019.warc.gz"} |
Solves non-linear least squares problems
fopt=leastsq(fun, x0)
fopt=leastsq(fun, x0)
fopt=leastsq(fun, dfun, x0)
fopt=leastsq(fun, cstr, x0)
fopt=leastsq(fun, dfun, cstr, x0)
fopt=leastsq(fun, dfun, cstr, x0, algo)
fopt=leastsq([iprint], fun [,dfun] [,cstr],x0 [,algo],[df0,[mem]],[stop])
[fopt,xopt] = leastsq(...)
[fopt,xopt,gopt] = = leastsq(...)
value of the function f(x)=||fun(x)||^2 at xopt
best value of x found to minimize ||fun(x)||^2
gradient of f at xopt
a scilab function or a list defining a function from R^n to R^m (see more details in DESCRIPTION).
real vector (initial guess of the variable to be minimized).
a scilab function or a string defining the Jacobian matrix of fun (see more details in DESCRIPTION).
bound constraints on x. They must be introduced by the string keyword 'b' followed by the lower bound binf then by the upper bound bsup (so cstr appears as 'b',binf,bsup in the syntax). Those
bounds are real vectors with same dimension than x0 (-%inf and +%inf may be used for dimension which are unrestricted).
a string with possible values: 'qn' or 'gc' or 'nd'. These strings stand for quasi-Newton (default), conjugate gradient or non-differentiable respectively. Note that 'nd' does not accept bounds
on x.
scalar argument used to set the trace mode. iprint=0 nothing (except errors) is reported, iprint=1 initial and final reports, iprint=2 adds a report per iteration, iprint>2 add reports on linear
search. Warning, most of these reports are written on the Scilab standard output.
real scalar. Guessed decreasing of ||fun||^2 at first iteration. (df0=1 is the default value).
integer, number of variables used to approximate the Hessian (second derivatives) of f when algo='qn'. Default value is 10.
sequence of optional parameters controlling the convergence of the algorithm. They are introduced by the keyword 'ar', the sequence being of the form 'ar',nap, [iter [,epsg [,epsf [,epsx]]]]
maximum number of calls to fun allowed.
maximum number of iterations allowed.
threshold on gradient norm.
threshold controlling decreasing of f
threshold controlling variation of x. This vector (possibly matrix) of same size as x0 can be used to scale x.
The leastsq function solves the problem
where f is a function from R^n to R^m. Bound constraints cab be imposed on x.
How to provide fun and dfun
fun can be a scilab function (case 1) or a fortran or a C routine linked to scilab (case 2).
case 1:
When fun is a Scilab function, its calling sequence must be:
In the case where the cost function needs extra parameters, its header must be: In this case, we provide fun as a list, which contains list(f,a1,a2,...).
case 2:
When fun is a Fortran or C routine, it must be list(fun_name,m[,a1,a2,...]) in the syntax of leastsq, where fun_name is a 1-by-1 matrix of strings, the name of the routine which must be linked to
Scilab (see link). The header must be, in Fortran:
subroutine fun(m, n, x, params, y)
integer m,n
double precision x(n), params(*), y(m)
and in C:
void fun(int *m, int *n, double *x, double *params, double *y)
where n is the dimension of vector x, m the dimension of vector y, with y=fun(x), and params is a vector which contains the optional parameters a1, a2, .... Each parameter may be a vector, for
instance if a1 has 3 components, the description of a2 begin from params(4) (in fortran), and from params[3] (in C). Note that even if fun does not need supplementary parameters you must anyway
write the fortran code with a params argument (which is then unused in the subroutine core).
By default, the algorithm uses a finite difference approximation of the Jacobian matrix. The Jacobian matrix can be provided by defining the function dfun, where to the optimizer it may be given as a
usual scilab function or as a fortran or a C routine linked to scilab.
case 1:
when dfun is a scilab function, its calling sequence must be:
where y(i,j)=dfi/dxj. If extra parameters are required by fun, i.e. if arguments a1,a2,... are required, they are passed also to dfun, which must have header Note that, even if dfun needs extra
parameters, it must appear simply as dfun in the syntax of leastsq.
case 2:
When dfun is defined by a Fortran or C routine it must be a string, the name of the function linked to Scilab. The calling sequences must be, in Fortran:
subroutine dfun(m, n, x, params, y)
integer m,n
double precision x(n), params(*), y(m,n)
in C:
void fun(int *m, int *n, double *x, double *params, double *y)
In the C case y(i,j)=dfi/dxj must be stored in y[m*(j-1)+i-1].
Like datafit, leastsq is a front end onto the optim function. If you want to try the Levenberg-Marquard method instead, use lsqrsolve.
A least squares problem may be solved directly with the optim function ; in this case the function NDcost may be useful to compute the derivatives (see the NDcost help page which provides a simple
example for parameters identification of a differential equation).
We will show different calling possibilities of leastsq on one (trivial) example which is non linear but does not really need to be solved with leastsq (applying log linearizes the model and the
problem may be solved with linear algebra). In this example we look for the 2 parameters x(1) and x(2) of a simple exponential decay model (x(1) being the unknown initial value and x(2) the decay
function y=yth(t, x)
y = x(1)*exp(-x(2)*t)
// we have the m measures (ti, yi):
m = 10;
tm = [0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5]';
ym = [0.79, 0.59, 0.47, 0.36, 0.29, 0.23, 0.17, 0.15, 0.12, 0.08]';
// measure weights (here all equal to 1...)
wm = ones(m,1);
// and we want to find the parameters x such that the model fits the given
// data in the least square sense:
// minimize f(x) = sum_i wm(i)^2 ( yth(tm(i),x) - ym(i) )^2
// initial parameters guess
x0 = [1.5 ; 0.8];
// in the first examples, we define the function fun and dfun
// in scilab language
function e=myfun(x, tm, ym, wm)
e = wm.*( yth(tm, x) - ym )
function g=mydfun(x, tm, ym, wm)
v = wm.*exp(-x(2)*tm)
g = [v , -x(1)*tm.*v]
// now we could call leastsq:
// 1- the simplest call
[f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),x0)
// 2- we provide the Jacobian
[f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),mydfun,x0)
// a small graphic (before showing other calling features)
tt = linspace(0,1.1*max(tm),100)';
yy = yth(tt, xopt);
plot(tm, ym, "kx")
plot(tt, yy, "b-")
legend(["measure points", "fitted curve"]);
xtitle("a simple fit with leastsq")
// 3- how to get some information (we use iprint=1)
[f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0)
// 4- using the conjugate gradient (instead of quasi Newton)
[f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0,"gc")
// 5- how to provide bound constraints (not useful here !)
xinf = [-%inf,-%inf];
xsup = [%inf, %inf];
// without Jacobian:
[f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),"b",xinf,xsup,x0)
// with Jacobian :
[f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),mydfun,"b",xinf,xsup,x0)
// 6- playing with some stopping parameters of the algorithm
// (allows only 40 function calls, 8 iterations and set epsg=0.01, epsf=0.1)
[f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0,"ar",40,8,0.01,0.1)
Examples with compiled functions
Now we want to define fun and dfun in Fortran, then in C. Note that the "compile and link to scilab" method used here is believed to be OS independent (but there are some requirements, in particular
you need a C and a fortran compiler, and they must be compatible with the ones used to build your scilab binary).
Let us begin by an example with fun and dfun in fortran
// 7-1/ Let 's Scilab write the fortran code (in the TMPDIR directory):
f_code = [" subroutine myfun(m,n,x,param,f)"
"* param(i) = tm(i), param(m+i) = ym(i), param(2m+i) = wm(i)"
" implicit none"
" integer n,m"
" double precision x(n), param(*), f(m)"
" integer i"
" do i = 1,m"
" f(i) = param(2*m+i)*( x(1)*exp(-x(2)*param(i)) - param(m+i) )"
" enddo"
" end ! subroutine fun"
" subroutine mydfun(m,n,x,param,df)"
"* param(i) = tm(i), param(m+i) = ym(i), param(2m+i) = wm(i)"
" implicit none"
" integer n,m"
" double precision x(n), param(*), df(m,n)"
" integer i"
" do i = 1,m"
" df(i,1) = param(2*m+i)*exp(-x(2)*param(i))"
" df(i,2) = -x(1)*param(i)*df(i,1)"
" enddo"
" end ! subroutine dfun"];
cd TMPDIR;
// 7-2/ compiles it. You need a fortran compiler !
names = ["myfun" "mydfun"]
flibname = ilib_for_link(names,"myfun.f",[],"f");
// 7-3/ link it to scilab (see link help page)
// 7-4/ ready for the leastsq call: be carreful do not forget to
// give the dimension m after the routine name !
[f,xopt, gopt] = leastsq(list("myfun",m,tm,ym,wm),x0) // without Jacobian
[f,xopt, gopt] = leastsq(list("myfun",m,tm,ym,wm),"mydfun",x0) // with Jacobian
Last example: fun and dfun in C.
// 8-1/ Let 's Scilab write the C code (in the TMPDIR directory):
c_code = ["#include <math.h>"
"void myfunc(int *m,int *n, double *x, double *param, double *f)"
" /* param[i] = tm[i], param[m+i] = ym[i], param[2m+i] = wm[i] */"
" int i;"
" for ( i = 0 ; i < *m ; i++ )"
" f[i] = param[2*(*m)+i]*( x[0]*exp(-x[1]*param[i]) - param[(*m)+i] );"
" return;"
"void mydfunc(int *m,int *n, double *x, double *param, double *df)"
" /* param[i] = tm[i], param[m+i] = ym[i], param[2m+i] = wm[i] */"
" int i;"
" for ( i = 0 ; i < *m ; i++ )"
" {"
" df[i] = param[2*(*m)+i]*exp(-x[1]*param[i]);"
" df[i+(*m)] = -x[0]*param[i]*df[i];"
" }"
" return;"
// 8-2/ compiles it. You need a C compiler !
names = ["myfunc" "mydfunc"]
clibname = ilib_for_link(names,"myfunc.c",[],"c");
// 8-3/ link it to scilab (see link help page)
// 8-4/ ready for the leastsq call
[f,xopt, gopt] = leastsq(list("myfunc",m,tm,ym,wm),"mydfunc",x0) | {"url":"https://help.scilab.org/docs/2023.1.0/fr_FR/leastsq.html","timestamp":"2024-11-12T16:52:24Z","content_type":"text/html","content_length":"60740","record_id":"<urn:uuid:968eeb6a-8ab0-450c-a2a4-f3dff5e21db6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00173.warc.gz"} |
Optimization of Variable Thickness Plates by Genetic Algorithms
Engineering Transactions, 46, 1, pp. 115–129, 1998
Optimization of Variable Thickness Plates by Genetic Algorithms
The implementation of genetic algorithms to the optimal design of variable thickness plates is presented. Thin, elastic, piecewise constant thickness plates subjected to bending are investigated. The
material distribution that minimizes the structural strain energy under constant volume constraint is searched. In numerical examples, square plates loaded by uniform normal pressure are optimized
for different boundary conditions. The best designs are compared with the worst solutions, corresponding to the maximization of the strain energy. Significant changes in strain energy can be achieved
by modifying thickness distribution for the same material volume. The performances of the approach are discussed.
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
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M.Z. COHN, Theory and practice of structural optimization, Struct. Optim., 7, pp. 20–31, 1994.
P. HAJELA, Genetic search – an approach to the nonconvex optimization problem, AIAA J., 26, pp. 1205–1210, 1990.
W.M. JENKINS, Towards structural optimization via genetic algorithm, Comput. Struct., 40, 5, pp. 1321–1327, 1991.
S. RAJEEV and C.S. KRISHNAMOORTHY, Discrete optimization of structures using genetic algorithms, J. Struct. Engng., 118, 5, pp. 1233–1250, 1992.
R.T. HAFTKA and B. PRASAD, Optimum structural design with plate bending elements – a survey, AIAA J., 19, 4, pp. 517–522, 1981.
N.V. BANICHUK, Problems and methods of optimal structural design, Mathematical Concepts and Methods in Science and Engineering, 26, Plenum Press, New York, pp. 109–134, 1981.
E.J. HAUG, A numerical method for optimization of distributed parameter structures with displacement constraints, Opt. Control Appl. & Meth., 3, pp. 269–282, 1982.
E. HINTON, S.M. AFONSO and N.V.R. RAO, Some studies on the optimization of variable thickness plates and shells, Engng. Computations, 10, pp. 291–306, 1993.
E. HINTON and N.V.R. RAO, Analysis and shape optimisation of variable thickness prismatic folded plates and curved shells. Part 2. Shape optimisation, Thin–Walled Structures, 17, pp. 161–183, 1993.
M. ZHOU, Y.X. GU and G.I.N. ROZVANY, Application of DCOC method to plates and shells, [in:] Procs. of the First World Congress of Structural and Multidisciplinary Optimization, 28 May–2 June 1995,
Goslar, Germany, N. OLHOFF and G.I.N. ROZVANY [Eds.], Pergamon, pp. 25–32, 1995.
E. SALAJEGHEH, Discrete variable optimization of plate structures using dual methods, Comput. & Struct., 58, 6, pp. 1131–1138, 1996.
D.E. GOLDBERG, Genetic algorithms in search, optimization, and machine learning, Addison-Wesley Publishing Company Inc., Reading, MA, 1989.
L. DAVIS, Handbook of genetic algorithms, Van Nostrand Reinhold, New York 1991.
Z. MICHALEWICZ, Genetic algorithms + data structures = evolution programs, Springer Verlag, Berlin, Heidelberg 1992.
O.C. ZIENKIEWICZ and R.L. TAYLOR, The finite element method, Vol. 2, McGraw-Hill, pp. 15–22, 1989.
M. PYRZ, Genetic algorithms in optimal design of discontinuous thickness plates, [in:] Proc. of the Second World Congress of Structural and Multidisciplinary Optimization, May 26–30, 1997, Zakopane,
Poland, vol.2, W. GUTKOWSKI and Z. MRÓZ [Eds.], Wydawnictwo Ekoinżynieria, pp. 859–864, 1997.
T. BÄCK, Evolutionary algorithms in theory and practice, Oxford University Press, New York 1995. | {"url":"https://et.ippt.pan.pl/index.php/et/article/view/661/0","timestamp":"2024-11-13T19:56:19Z","content_type":"text/html","content_length":"21495","record_id":"<urn:uuid:7b3cbe38-5ceb-4180-a7ec-52cd75ead517>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00556.warc.gz"} |
讲解 program、辅导 python/Java语言设计
Homework 21
Due date: October 9, 2024 (Wednesday).
Please submit your answer by 11:59pm.
There are total of 6 questions.
Q1 (Reflection): Read the solution to HW 1.
• Are your own answers in line with the solutions? If not, list the questions you missed. • Discuss what you could have done better.
• OnascaleofA,B,C,D,howwouldyougradeyourHW1?
Q2 (Conditional expectation): Let the random vector (y,x)0 have a normal distribution with mean vector μ = (μy,μx)0 and covariance matrix
Σ= σy2 σyσxρ, σ x σ y ρ σ x2
where σy and σx are the standard deviations and ρ is the correlation between y and x. The joint density is
fY,X(y,x)= 1 exp −1(w−μ)0Σ−1(w−μ) , −∞The determinant of the covariance matrix is
|Σ| = σy2σx2(1 − ρ2),
and the inverse of the covariance matrix is
Σ−1=1 σy2 −σyσx.
1Last compiled: September 27, 2024; STAT5200, Fall 2023
−2ρ + .
exp −
2(1 − ρ2) σy
σy σx
1ρ 1−ρ2−ρ 1
Thus, the joint density can be written as
1 2πσyσx(1 − ρ2)1/2
1 y−μy2 y−μy x−μx x−μx2
σ y σ x σ x2
The marginal density of x is
fX(x) =
that is, normal with mean μx and variance σx2.
fY,X(y,x)dy = √ 1. Derive conditional distribution of y given x.
Z∞ −∞
1 2πσx
1 x−μx 2 −2 σ ,
2. Compute linear projection of y on x = (1,x). That is, derive and express L(y|1,x) as a function of μx, μy, ρ, σx, σy.
3. Define u = y − L(y|1, x). What is the distribution of u?
Q3 (Linear projection): The textbook (Wooldridge)’s definition of the linear projection is slightly different from that was introduced in the lecture notes (Notes 01). Wooldridge defines the linear
projection in the following way,
Define x = (x1,...,xK) as a 1 × K vector, and make the assumption that the K × K variance matrix of x is nonsingular (positive definite). Then the linear projection of y on1,x1,x2,...,xK
L(y|1,x1,...,xK)=L(y|1,x)=β0 +β1x1 +...+βKxK =β0 +xβ, where, by definition,
β = [V ar(x)]−1Cov(x, y)
β0 =E[y]−E[x]β=E[y]−β1E[x1]−∙∙∙−βKE[xK].
Explain why this definition coincides with the definition that is introduced in the lecture notes (Notes 01). Provide a formal derivation as well. Hint: Answer is in Notes 01.
Q4 (Asymptotics, asymptotic normality): Let yi, i = 1, 2, ... be an independent, identically distributed sequence with E[yi2] < ∞. Let μ = E[yi] and σ2 = V ar(yi).
1. Let yN denote the sample average based on a sample size of N . Find V ar(√N (yN − μ)). 2. What is the asymptotic variance of √N (yN − μ)?
3. What is the asymptotic variance of yN ? Compare this with V ar(yN ).
4. What is the asymptotic standard deviation of yN ?
Q5 (Asymptotics, delta method): Let θˆ be a √N-asymptotically normal estimator for the scalar θ > 0. Let γˆ = log(θˆ) be an estimator of γ = log(θ).
1. Why is γˆ a consistent estimator of γ?
2. Find the asymptotic variance of √N(γˆ−γ) in terms of the asymptotic variance of √N(θˆ−θ).
3. Suppose that, for a sample of data, θˆ = 4 and se(θˆ) = 2. What is γˆ and its (asymptotic) standard error?
4. Consider the null hypothesis H0 : θ = 1. What is the asymptotic t statistic for testing H0, given the numbers from part 3?
5. Now state H0 from part 4 equivalently in terms of γ, and use γˆ and se(γˆ) to test H0. What do you conclude?
Q6 (Paper question): Find the paper that uses a delta method in your field. If you can’t find it, then find such paper from “American Economic Review”, which is one of the premier journal in
1. Find an academic paper2 that (1) was published in one of those journals from your field of interest, AND (2) contains the word delta method, AND (3) the term delta method in the paper refers to
the method that we learnt from the class, AND (4) applies the delta method.
One way to find such a paper is to use Google Scholar. Type the following in the search box
source:"[name of the journal]" "delta method"
2. Properly cite the paper you found (name of the author, the title of the article, year of publication, the name of the journal, etc.)
3. Read the paper and explain what is the main research question of the paper in one paragraph.
4. What is the parameter of interest in their empirical model, and why do the authors use the delta method?
5. (Optional reading; will not be graded). Read the following paper
Ver Hoef, J.M., 2012. Who invented the delta method?. The American Statistician, 66(2), pp.124-127.
(I put the copy of the paper in the HW section).
2If you can’t find such paper from the field of your interest, then find it from the “American Economic Review”, which is one of the premier journals in economics. | {"url":"http://7daixie.com/2024100721103046251.html","timestamp":"2024-11-06T14:47:33Z","content_type":"application/xhtml+xml","content_length":"54538","record_id":"<urn:uuid:5b8a8e09-8b22-44ec-8b60-9a8fc628faa7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00572.warc.gz"} |
How to Teach Adding and Subtracting Mixed NumbersHow to Teach Adding and Subtracting Mixed NumbersHow to Teach Adding and Subtracting Mixed Numbers
Looking for how to introduce and teach adding and subtracting mixed numbers?
Let’s talk about using manipulatives, pictures, and algorithms - plus how to regroup fractions and differentiate instruction for students when they add and subtract mixed numbers.
What to look for in this post
• The progression of the standards for adding and subtracting mixed numbers
• Teaching using the CRA Model
• Demonstrating addition of mixed numbers with manipulatives
• Demonstrating subtraction of mixed numbers with manipulatives
• Representational modeling of adding and subtracting mixed numbers
• Teaching the algorithm for adding and subtracting mixed numbers
• Milestones for adding and subtracting mixed numbers
• Differentiation for adding and subtracting mixed numbers
If you prefer to hear me talk through this, here is a video featuring all of this content:
[0:32] The progression of the standards for adding and subtracting mixed numbers
In third grade, students learn basic fractions. They learn to identify and compare fractions, as well as about equivalent fractions. In most states, fourth grade students learn to add and subtract
fractions with like denominators. In fifth grade, they learn how to add and subtract fractions with unlike denominators, including mixed numbers.
After fifth grade, students are expected to have a firm understanding of adding and subtracting fractions. Starting in third grade, we can provide them with a strong foundation to support fraction
knowledge moving forward.
[1:36] Teaching using the CRA Model
Whenever we teach something, we certainly want to take a research-based approach. In this case, we can use the CRA model. Students first learn something in a
way, through hands-on experience and exposure. They might use manipulatives to help them understand a new concept.
Then, students learn to
the concept through pictures. Finally, they learn to use the
equations and algorithms.
For the purposes of teaching addition and subtraction of mixed numbers, we can use pattern blocks and fraction bars as concrete models. For representational, we will be using pictures and number
lines. For abstract, we will use written equations.
Students need time to work through each of these models. Some of them may be ready to jump to representational or even abstract models more quickly than others, but I encourage you to make sure they
all start with concrete modeling.
[3:21] Demonstrating addition of mixed numbers with manipulatives
To see the full demonstration, check out the YouTube video and move to the time stamp noted above.
As I have shared in previous posts, I absolutely love fraction bars, also called fraction tiles, as well as pattern blocks for all things fractions. You can have students work in pairs or small
groups. When using manipulatives, I limit the amount of writing of equations I do. I want students to be able to see things without becoming too distracted. At this point, however, students should be
quite familiar and proficient with adding and subtracting fractions with like denominators and unlike denominators. So if you need to review those things with them first, by all means do that.
As an example, we can write the equation: 2 ⅚ + 1 ⅓
Using the pattern blocks, we can assemble a concrete model of the equation. Instead of focusing on the whole numbers, I tell students to focus on the fraction pieces: ⅚ and ⅓. I am going to want to
combine those pieces, so I need to get them to be the same size pieces. Hopefully, they come up with the idea to change the third to sixths - and because they are familiar with fractions they should
know pretty quickly that ⅓ = 2/6.
At this point, I would rewrite the equation under the original: 2 ⅚ + 1 2/6
Sometimes I put an arrow from the mixed number I changed to its new number, so students recognize that is the one that changed.
2 + 1 is three wholes, so we can show that using the pattern blocks. When we combine ⅚ and 2/6, we get an additional 1 whole and ⅙. I have them physically write out 3 + 1 ⅙ and then I ask them what
swap we can make. We could also write it as 3 + 7/6 or 3 + 6/6 + ⅙. The total is 4 ⅙.
[10:01] Demonstrating subtraction of mixed numbers with manipulatives
To see the full demonstration, check out the YouTube video and move to the time stamp noted above.
First, I would advise doing more simple subtraction with mixed number problems that do not require regrouping. For the purpose of this demonstration, we are going to jump right into a problem that
involves regrouping.
We can write the equation: 2 ⅙ - 1 ⅓
We are going to start by building 2 ⅙ with our blocks. I suspect students would think that taking away 1 whole is easy, but the difficult part is taking away ⅓ from 2/6. Again, we need to get to the
same size pieces with these fractions. Students may want to swap out for thirds, but that will become troublesome in the next step. Listen to that idea, but ask them if there is something else they
could do.
The first time through, it may be helpful for you to just tell them the answer. Let’s make everything into sixths. Swap out our wholes for six sixths, and what do we have? 2 ⅙ becomes 13/6.
Now we need to take away 1 ⅓ - which we can turn into 8/6. Now we can subtract 8/6 from 13/6. We have two improper fractions with like denominators, so we can do simple subtraction and reach the
answer of ⅚.
It can be difficult to go through these steps as teachers, because we may want to jump straight to the algorithm. That is not, however, what is most appropriate for students. They need to work with
concrete models and truly see what is happening.
[15:51] Representational modeling of adding and subtracting mixed numbers
To see the full demonstration, check out the YouTube video and move to the time stamp noted above.
We can start with another equation: 1 ⅙ + 1 ⅓
When we move to representational modeling, I still like students to have their manipulatives at hand. In this case, I’m going to use pictures rather than a number line. Sometimes problems can get a
little bit messy, and students are not always great at drawing their own number lines. Using pictures can help to provide more clarity. Students can trace their pattern blocks, and color the various
We will then re-write the equation: 1 ⅙ + 1 2/6
When we see that the answer is 2 3/6, we can ask them to put it in simplest form. They should recognize that 3/6 = ½ so the answer would be 2 ½.
Drawing this out helps students to better understand the concept as they get into the abstract algorithm. We don’t want them to live here forever, but some students may take longer here and need to
draw things out. We want to provide that opportunity so they can bridge their learning as they go along.
[21:30] Teaching the algorithm for adding and subtracting mixed numbers
To see the full demonstration, check out the YouTube video and move to the time stamp noted above.
Working through a problem we previously did with pattern blocks, the equation is:
2 ⅙ - 1 ⅓
For the algorithm, you will teach the students with a shortcut to multiply the denominator by the whole number, and then add the numerator. Using this method, the equation becomes:
13/6 - 4/3
In order to subtract this equation, we need an equivalent fraction.
13/6 - 8/6 = 5/6
[23:35] Milestones for adding and subtracting mixed numbers
I always like to keep the milestones in mind when I am teaching.
What are the big pieces that I need to make sure that students are solid on?
This can help me to pinpoint what part of the process they are having trouble in.
For adding and subtracting mixed numbers, we want to make sure that students can use least common multiples to develop their least common denominators.
Students need to really understand equivalent fractions.
With mixed numbers, students need to be able to regroup and write the fractions in a different way.
We need to make sure that students are clear that they only add and subtract the numerators - the denominators stay the same.
Lastly, we want to be sure that they understand simplest form. Students often struggle to put their answers in simplest form.
When we break things down this way, it helps us plan our instruction.
[25:01] Differentiation for adding and subtracting mixed numbers
For struggling students, you may want to review adding and subtracting fractions with like denominators and creating equivalent fractions. When we work with mixed numbers, it is also important to
make sure students can regroup.
If you have students who need a bit of a challenge, you can have them add and subtract three mixed numbers. You can also start to combine adding fractions and decimals.
Looking for more resources for teaching how to add and subtract mixed numbers?
When it comes to practicing these skills, I have put together
a bundle
of some of my favorite activities for adding and subtracting mixed numbers (including regrouping). It includes six resources:
• Four maze worksheets
• One matching game
• One anchor chart
This bundle provides opportunities for differentiation, because some students can work on the
like denominator
worksheets while others who are ready to move on can do the
unlike denominator
worksheets. The matching game is also differentiated, including levels for below level learners, on level learners, and students who need a challenge. The anchor chart is for subtracting mixed
numbers, and you can get full page or half sheet versions as well as a poster version.
Activities Bundle
is available on Teachers Pay Teachers.
You can also grab this
free download
, which is a teaching guide including everything we talked about in this post. It is a one-page sheet that gives you all this information so that you will be prepared to teach adding and subtracting
mixed numbers.
Check out these related posts
How to Teach Adding and Subtracting Fractions with Like DenominatorsHow to Teach Adding and Subtracting Fractions with Unlike DenominatorsHow to Teach Metric Measurement for Third GradeHow to Teach
Metric Conversions for Fourth and Fifth GradesCheck out these related YouTube videos
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Measurements for Fourth and Fifth GradesHow to Teach Polygons and Quadrilaterals for Third Grade | {"url":"https://www.adoubledoseofdowda.com/2022/07/how-to-teach-adding-and-subtracting-mixed-numbers.html","timestamp":"2024-11-08T21:38:49Z","content_type":"application/xhtml+xml","content_length":"185790","record_id":"<urn:uuid:273d5cdd-c4e0-467e-9bae-4175a323d5fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00562.warc.gz"} |
5 Best Ways to Find Four Points Forming a Square Parallel to x and y Axes in Python
π ‘ Problem Formulation: Given a starting point on the Cartesian plane, the task is to compute the coordinates of four points that form a square with sides parallel to the x and y axes. As an input,
we take a coordinate representing the bottom-left corner of the square and the length of a side. The desired output is a list of four tuples corresponding to the coordinates of each corner of the
Method 1: Using Basic Arithmetic Operations
Method 1 involves using basic arithmetic to calculate the positions of the other three points based on the given bottom-left point and side length. The top-right corner can be found by adding the
side length to both the x and y coordinates of the starting point, while the other two points are found by adding the side length to just one of the coordinates.
Here’s an example:
def find_square(bottom_left, side_length):
x, y = bottom_left
return [
(x + side_length, y),
(x + side_length, y + side_length),
(x, y + side_length)
# Example usage
bottom_left = (1, 1)
side_length = 3
print(find_square(bottom_left, side_length))
[(1, 1), (4, 1), (4, 4), (1, 4)]
This code snippet defines a function find_square() that calculates the coordinates of a square’s corners, given the bottom-left corner and the side length. It utilizes a clear and simple approach,
making it very accessible for anyone new to programming or geometry problems.
Method 2: Using a Custom Point Class
In Method 2, a custom Point class is created to represent a point on the Cartesian plane. This abstraction allows for more readable and potentially reusable code when computing the coordinates of the
Here’s an example:
class Point:
def __init__(self, x, y):
self.x = x
self.y = y
def __repr__(self):
return f"({self.x}, {self.y})"
def find_square(bottom_left, side_length):
points = [bottom_left]
points.append(Point(bottom_left.x + side_length, bottom_left.y))
points.append(Point(bottom_left.x + side_length, bottom_left.y + side_length))
points.append(Point(bottom_left.x, bottom_left.y + side_length))
return points
# Example usage
bottom_left = Point(1, 1)
side_length = 3
print(find_square(bottom_left, side_length))
[(1, 1), (4, 1), (4, 4), (1, 4)]
This snippet introduces a custom Point class to encapsulate x and y coordinates. We can see how creating classes can structure code more effectively and make it more readable. The function
find_square() is then defined to perform the computation.
Method 3: Using Complex Number Representation
Method 3 utilizes Python’s complex number capabilities to represent and calculate the points. This is an elegant mathematical approach where operations on complex numbers translate directly to
operations on points in a plane.
Here’s an example:
def find_square(bottom_left, side_length):
bl = complex(*bottom_left)
return [
bl + side_length,
bl + side_length + side_length*1j,
bl + side_length*1j
# Example usage
bottom_left = (1, 1)
side_length = 3
print([tuple(map(int, (c.real, c.imag))) for c in find_square(bottom_left, side_length)])
[(1, 1), (4, 1), (4, 4), (1, 4)]
The function find_square() uses complex numbers to compute the square’s corners, treating the real and imaginary parts as the x and y coordinates respectively. This method emphasizes the power and
elegance of mathematical constructs in programming.
Method 4: Using NumPy Library
Method 4 involves the NumPy library, which is widely used for numerical computation in Python. This method leverages vector addition provided by NumPy to calculate the square’s points.
Here’s an example:
import numpy as np
def find_square(bottom_left, side_length):
bl = np.array(bottom_left)
transformations = [
(0, 0),
(side_length, 0),
(side_length, side_length),
(0, side_length)
return [tuple(bl + np.array(t)) for t in transformations]
# Example usage
bottom_left = (1, 1)
side_length = 3
print(find_square(bottom_left, side_length))
[(1, 1), (4, 1), (4, 4), (1, 4)]
The function find_square() makes use of the NumPy library for its powerful array manipulation capabilities. By treating points as NumPy arrays, we can easily compute the transformation required to
find all four corners of the square.
Bonus One-Liner Method 5: Using Python List Comprehensions
Bonus Method 5 showcases the compactness of Python’s list comprehensions to solve the problem in a single, expressive line of code.
Here’s an example:
bottom_left = (1, 1)
side_length = 3
find_square = lambda bl, sl: [(bl[0] + x * sl, bl[1] + y * sl) for x in range(2) for y in range(2)][::-1]
print(find_square(bottom_left, side_length))
[(1, 1), (4, 1), (4, 4), (1, 4)]
This one-liner defines a lambda function that uses a list comprehension to calculate all four corners of the square. It succinctly captures the essence of the problem, illustrating Python’s
capability for writing concise and powerful expressions.
• Method 1: Basic Arithmetic Operations.Efficient and straightforward. Best for simple, one-off calculations. However, its simplicity means it lacks the abstraction for more complex scenarios.
• Method 2: Custom Point Class. Adds readability and reusability through OOP principles. Well-suited for complex programs, although it requires writing and maintaining additional code for the
• Method 3: Complex Number Representation. Mathematically elegant, demonstrating the versatility of Python’s standard library. It might be less intuitive for those unfamiliar with complex numbers.
• Method 4: NumPy Library. Offers powerful array operations and is great for numerical computations. Depends on an external library, which may not be ideal for minimalistic or constrained
• Method 5: Python List Comprehensions. Highly concise and expressive, perfect for Python one-liners. The clarity might suffer for those who are not accustomed to list comprehensions or lambda | {"url":"https://blog.finxter.com/5-best-ways-to-find-four-points-forming-a-square-parallel-to-x-and-y-axes-in-python/","timestamp":"2024-11-03T06:17:53Z","content_type":"text/html","content_length":"73771","record_id":"<urn:uuid:0df4f038-c86b-490a-b9fb-9ff9afa69198>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00500.warc.gz"} |
Chapter 7: Sequences and Series
• Subject(s): Math
• Grade Range: 9 - 12
• Release date: 07-07-2017
• Tags: sequence, integers, infinite sequence, index, general term, explicit formula, upper limit of summation, summation notation, sigma notation, series, recursive sequence, partial sum, lower
limit of summation, index of summation, factorial, common difference, arithmetic series, arithmetic sequence, geometric series, geometric sequence, common ratio, Pascal's triangle, Binomial
Theorem, binomial coefficient, binomial
In this chapter, we introduce sequences and series, some of their applications, and the Binomial Theorem.
M.P.1.E, M.P.1.F, M.P.5.A, M.P.5.B, M.P.5.C, M.P.5.D, M.P.5.E, M.P.5.F | {"url":"https://texasgateway.org/binder/chapter-7-sequences-and-series?book=79066","timestamp":"2024-11-07T20:36:22Z","content_type":"text/html","content_length":"38642","record_id":"<urn:uuid:ad0d1713-e1eb-4f44-87cf-767734b4a190>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00380.warc.gz"} |
How do i graph the exponential function f(x)=(1/5)^x on a TI-83? | Socratic
How do i graph the exponential function #f(x)=(1/5)^x# on a TI-83?
1 Answer
$Y 1 = {\left(\frac{1}{5}\right)}^{X}$
For $X$ you use the $X , \Theta$, etc button.
Problem could be your Windows setting. To start with I would set the x-windows from x-min=-5 to x-max=+5 and the y-window from y-min=-1 to y-max=+1.
This will give you a reasonable view of the graph. You can always expand or contract from there.
Impact of this question
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The following course descriptions and prerequisites have been taken from the Vassar Course Catalog. In addition, for each course, we have listed other courses for which it is a prerequisite and
other courses with which there is substantial overlap in content. The courses are listed in increasing order of course number.
Math 141 / Biology 141: Introduction to Statistical Reasoning
The purpose of this course is to develop an appreciation and understanding of the exploration and interpretation of data. Topics include exploratory data analysis, basic probability, design of
studies, and inferential methods including confidence interval estimaation and hypothesis testing. Applications and examples are drawn from a wide variety of disciplines. When cross-listed with
biology, examples are drawn primarily from biology. Statistical software is used. Computationally less intensive than MATH 240.
• Prerequisite: three years of high school mathematics.
• Courses for which this is a prerequisite: Math 242
• Courses with substantial overlap: Math 240
• Special Notes: Not open to students with AP credit in statistics or students who have completed Economics 209 or Psychology 200. Students who have had calculus should take Math 240.
Math 240: Introduction to Statistics
The purpose of this course is to introduce the methods by which we extract information from data. Topics are similar to those in MATH 141, with more coverage of probability and more intense
computational and computer work. Ming-Wen An, Jingchen Hu.
• Prerequisite: Math 126 and Math 127
• Courses for which this is a prerequisite: Math 242
• Courses with substantial overlap: Math 141/Biol 141
• Special Notes: Not open to students with AP credit in statistics or students who have completed Economics 209 or Psychology 200.
Psyc 200: Statistics & Experimental Design
An overview of principles of statistical analysis and research design applicable to psychology and related fields. Topics include descriptive statistics and inferential statistics, concepts of
reliability and validity, and basic concepts of sampling and probability theory. Students learn when and how to apply such statistical procedures as chi-square, z-tests, t-tests, Pearson
product-moment correlations, regression analysis, and analysis of variance. The goal of the course is to develop a basic understanding of research design, data collection and analysis, interpretation
of results, and the appropriate use of statistical software for performing complex analyses. Ms. Andrews, Mr. Clifton, Ms. Trumbetta, Ms. Zupan.
□ Prerequisite: Psyc 105 or 106
□ Courses for which this is a prerequisite: All Psyc Research Methods courses, including Cogs 219
□ Courses with substantial overlap: (None)
Psychology Research Methods Courses (209 Social Psychology, 219 Cognitive Science, 229 Learning and Behavior, 239 Developmental, 249 Physiological, 259 Personality and Individual Differences): These
courses all have Psyc 200 as a prerequisite along with a content course in the relevant area of psychology, and all involve learning how to select, carry out, and write up statistical analyses
applied to class-generated empirical data.
Econ 209: Probability & Statistics
This course is an introduction to statistical analysis and its application in economics. The objective is to provide a solid, practical, and intuitive understanding of statistical analysis with
emphasis on estimation, hypothesis testing, and linear regression. Additional topics include descriptive statistics, probability theory, random variables, sampling theory, statistical distributions,
and an introduction to violations of the classical assumptions underlying the least-squares model. Students are introduced to the use of computers in statistical analysis. The department.
• Prerequisite: Econ 100 or 101 or 102 or permission of the instructor
• Courses for which this is a prerequisite: Econ 210
• Courses with substantial overlap: Math 241
Econ 210: Econometrics
This course equips students with the skills required for empirical economic research in industry, government, and academia. Topics covered include simple and multiple regression, maximum likelihood
estimation, multicollinearity, heteroskedasticity, autocorrelation, distributed lags, simultaneous equations, instrumental variables, and time series analysis. The department.
• Prerequisite: Econ 209 or an equivalent statistics course
• Courses for which this is a prerequisite: Econ 310
• Courses with substantial overlap: (None)
Geog 230: Geographic Research Methods
How do we develop clear research questions, and how do we know when we have the answer? This course examines different methods for asking and answering questions about the world, which are essential
skills in geography and other disciplines. Topics include formulation of a research question or hypothesis, research design, and data collection and analysis. We examine major research and
methodological papers in the discipline, design an empirical research project, and carry out basic data analysis. We review qualitative approaches, interviewing methods, mapping, and quantitative
methods (data gathering, descriptive statistics, measures of spatial distribution, elementary probability theory, simple statistical tests) that help us evaluate patterns in our observations.
Students who are considering writing a thesis or conducting other independent research and writing are encouraged to take this course. Ms. Zhou.
• Prerequisite: None
• Courses for which this is a prerequisite: None
• Courses with substantial overlap: None
Math 241: Probability Models
This course in introductory probability theory covers topics including combinatorics, discrete and continuous random variables, distribution functions, joint distributions, independence, properties
of expectations, and basic limit theorems. The department.
• Prerequisite: Math 122 or 125 or permission of the department
• Courses for which this is a prerequisite: Math 341
• Courses with substantial overlap: Econ 209
Math 242: Applied Statistical Modeling
Applied Statistical Modeling is offered as a second course in statistics in which we present a set of case studies and introduce appropriate statistical modeling techniques for each. Topics may
include: multiple linear regression, logistic regression, log-linear regression, survival analysis, an introduction to Bayesian modeling, and modeling via simulation. Other topics may be substituted
for these or added as time allows. Students will be expected to conduct data analyses in R. The department.
• Prerequisite: Math 141 or permission of the instructor
• Courses for which this is a prerequisite: (None)
• Courses with substantial overlap: (None)
Econ 310: Advanced Topics in Econometrics
Analysis of the classical linear regression model and the consequences of violating its basic assumptions. Topics include maximum likelihood estimation, asymptotic properties of estimators,
simultaneous equations, instrumental variables, limited dependent variables and an introduction to time series models. Applications to economic problems are emphasized throughout the course. Mr.
• Prerequisite: Econ 210 and Math 122 or equivalent.
• Courses for which this is a prerequisite: (None)
• Courses with substantial overlap: (None)
Math 341: Mathematical Statistics
An introduction to statistical theory through the mathematical development of topics including resampling methods, sampling distributions, likelihood, interval and point estimation, and introduction
to statistical inferential methods. The department.
• Prerequisite: Math 220 and 241
• Courses for which this is a prerequisite: Math 342
• Courses with substantial overlap: (None)
Math 342: Applied Statistical Modeling
For students who have completed Math 341. Students in this course attend the same lectures as those in Math 242, but will be required to complete extra reading and problems. The department.
• Prerequisite: Math 122 or 125, and Math 341
• Courses for which this is a prerequisite: (None)
• Courses with substantial overlap: (None)
Math 347: Bayesian Statistics
An introduction to Bayesian statistics. Topics include Bayes Theorem, common prior and posterior distributions, hierarchical models, Bayesian linear regression, latent variable models, and Markov
chain Monte Carlo methods. The course uses R extensively for simulations. The department.
• Prerequisite: Math 220, Math 221, and Math 241
• Courses for which this is a prerequisite: (None)
• Courses with substantial overlap: (None) | {"url":"https://pages.vassar.edu/statsatvassar/courses/","timestamp":"2024-11-13T09:31:21Z","content_type":"text/html","content_length":"46829","record_id":"<urn:uuid:8a997e36-9e96-4f64-a21b-66253fae289b>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00513.warc.gz"} |
┃ │ William Spottiswoode ┃
┃ │ ┃
┃ │ (11 Jan 1825 - 27 Jun 1883) ┃
┃ │ ┃
┃ │ ┃
Science Quotes by William Spottiswoode (2 quotes)
Coterminous with space and coeval with time is the kingdom of Mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist; in contradiction to her
laws nothing takes place. On her mysterious scroll is to be found written for those who can read it that which has been, that which is, and that which is to come.
— William Spottiswoode
Everything material which is the subject of knowledge has number, order, or position; and these are her first outlines for a sketch of the universe. If our feeble hands cannot follow out the details,
still her part has been drawn with an unerring pen, and her work cannot be gainsaid. So wide is the range of mathematical sciences, so indefinitely may it extend beyond our actual powers of
manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence. But so strictly limited are her promises and powers, about so much that we
might wish to know does she offer no information whatever, that at other moments we are fain to call her results but a vain thing, and to reject them as a stone where we had asked for bread. If one
aspect of the subject encourages our hopes, so does the other tend to chasten our desires, and he is perhaps the wisest, and in the long run the happiest, among his fellows, who has learned not only
this science, but also the larger lesson which it directly teaches, namely, to temper our aspirations to that which is possible, to moderate our desires to that which is attainable, to restrict our
hopes to that of which accomplishment, if not immediately practicable, is at least distinctly within the range of conception.
— William Spottiswoode
Quotes by others about William Spottiswoode (1)
Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the
diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when
turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation;
or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations
connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great
mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not
without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost
new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory
of elasticity and the development of the integral calculus) “have all felt its influence”. | {"url":"https://todayinsci.com/S/Spottiswoode_William/SpottiswoodeWilliam-Quotations.htm","timestamp":"2024-11-12T16:43:51Z","content_type":"text/html","content_length":"91149","record_id":"<urn:uuid:887cab70-8228-41ff-b6d3-c690677f9f92>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00610.warc.gz"} |
12 Easy Math Tricks You’ll Wish You’d Known This Whole Time -
Kids and TeensLifestyleReference and Education
12 Easy Math Tricks You’ll Wish You’d Known This Whole Time
12 Easy Math Tricks You’ll Wish You’d Known This Whole Time: Every editorial product is independently selected, even though we can be compensated or acquire an associate fee in case you purchase
something thru our links.
Ratings and expenses are correct and gadgets are in inventory as of the time of publication. Calculate the math problem from the math calculator.
The 12 Easy Math Tricks You’ll Wish You’d Known This Whole Time Are:
1. Making math smooth.
Juggling numbers isn’t anyone’s forte despite the fact that math manages to sneak its manner into our regular lives.
Fortunately, those smooth-to-keep in mind math hints maybe your high-quality pal subsequent time you’re confronted with a problematic equation and don’t have a calculator available.
Also, don’t omit those math instructions you’ll sincerely become the use of in actual life.
2. Finding a 20% tip.
Did you experience your eating service? Leave your server a 20 percentage tip with this smooth math trick. According to Kate Snow, writer of The Math Facts That Stick series.
All you need to do is divide your test quantity via way of means of 5. For example, in case your test comes to $85, divide that via way of means of 5 and your 20 per cent tip may be $17.
For extra on tipping, here’s our manual on how an awful lot to tip in each situation.
3. Multiplying -digit numbers via way of means of eleven.
Multiplying -digit numbers via way of means of eleven is a breeze with this smart trick from math.hmc.edu. Simply upload the 2 digits collectively and location the sum withinside the middle.
For example, in case you’re multiplying 25 via way of means of eleven, upload the 2 and 5 collectively to get seven, and location the seven among the one’s numbers to locate the very last solution,
that is 275.
While we’re discussing math, attempt fixing this second-grade math trouble no person can parent out. Calculate cross product from the cross product calculator.
4. Doubling.
To double a massive wide variety, multiply every character wide variety via way of means of and upload them collectively. Snow indicates beginning from the left to make it less complicated to hold
the tune of the numbers.
“For example, to double 147, begin with the masses location. Double a hundred is two hundred. Double forty is eighty. Double seven is 14.
Add all of them up (two hundred plus eighty plus 14) and also you get 294,” says Snow. If you observed you’re a math expert, attempt out those mathematics problems.
5. Multiplying numbers that result in 0.
Even equations concerning massive, intimidating numbers that result in 0 may be without difficulty solved with this available trick. According to education.cu-portland.edu, simply exclude the zeros
from the equation, then upload them returned afterwards.
For example, in case you’re multiplying six hundred via way of means of four hundred, exclude the zeros and remedy for 6 instances 4, that is 24. After, remember the full wide variety of zeros that
had been withinside the unique equation and tack them onto the wide variety you solved for to locate your very last solution.
Since there had been 4 zeros withinside the unique equation, your very last solution for this case is 240,000. This is how intellectual math works.
6. Multiplying via way of means of 9.
If you don’t have your nines instances desk memorized, now no longer worry. According to Snow, to multiply via way of means of 9, absolutely multiply the wide variety via way of means of 10 first.
Then subtract the unique wide variety. For example, in case you want to multiply 9 via way of means of 23, multiply 23 via way of means of 10 to locate 230. Then, subtract 23 from 230 to locate your
very last solution of 207.
We know, that math may be a headache. However, those smart math jokes are even hilarious to non-mathematicians.
7. Dividing via way of means of 10, a hundred, or 1,000.
To divide various via way of means of 10, all you want to do is flow the decimal location one spot to the left of the unique wide variety to locate your solution, in keeping with Snow.
To divide via way of means of a hundred, the equal concept applies, besides, you’ll flow the decimal location spots to the left of the unique wide variety. As for dividing via way of means of 1,000,
flow the decimal location 3 spots to the left.
For example, in case you’re dividing 42. ninety-four via way of means of 10, you absolutely flow the decimal location one spot to the left to locate your very last solution of 4.294.
8. Multiplying via way of means of 10, a hundred, or 1,000.
Similar to the way you divide various via way of means of 10, a hundred, or 1,000, Snow says you’ll want to do the alternative to multiply via way of means of them.
To multiply various via way of means of 10, flow the decimal location one spot to the proper. To multiply various via way of means of a hundred, flow the decimal location spots to the proper.
To multiply various via way of means of 1,000, flow the decimal location 3 spots to the proper. For example, in case you want to multiply 366. seventy-eight via way of means of a hundred, shift the
decimal location spots to the proper to locate your very last solution of 36,678. By the manner.
If you could remedy this math trouble on the primary attempt, you are probably a genius.
9. Turning a repeating fraction right into a decimal.
According to businessinsider.com, there are 3 steps you’ll want to observe to without difficulty flip a repeating decimal right into a fraction. First, locate the wide variety this is repeating.
For example, withinside the wide variety 0.636363…, 63 is the repeating wide variety. Then, parent out what number of locations that wide variety has. In this case, 63 has locations. Finally, divide
the repeating wide variety via way of means of various that has the equal quantity of locations made up via way of means of nines, which might be 99 on this case. Reduce the fraction of 63/99 to ⅞
and done.
10. Multiplying via way of means of 25.
Not anyone has their 25 instances desk memorized. However, that’s now no longer trouble while you suppose of every institution of 25 as a sector. “If you had 17 quarters.
How an awful lot of cash might you have? Every 4 quarters make a dollar, so sixteen of the quarters equals 4 greenbacks or four hundred cents. The more sector provides 25 cents, for a complete of 425
cents,” Snow explains.
11. Squaring numbers that result in 5.
This math trick calls for steps, says Snow. In order to rectangular various that leads to 5, take the primary digit of the wide variety and multiply it via way of means of itself. After, upload the
end result to itself for the solution.
For example, in case you’re multiplying 35 via way of means of 35, take the 3 and multiply it via way of means of itself, that is 9, then upload 3 to that solution, that is 12.
Finally, upload the wide variety 25 to the give up of the wide variety you located and this is the very last solution: 1,225. Feeling like a math expert? Put your capabilities to the check and try
and skip this math check for 5th graders.
12. Subtracting via way of means of including.
If you locate addition to be a touch less complicated than subtraction, this trick is for you. If you’re handling subtracting numbers that might be pretty near collectively, attempt fixing the
equation via way of means of including instead. “Instead of seeking to remove 327 from 334, consider it as ‘327 plus what number of extra identical 334?’” says Snow. | {"url":"https://articleft.com/12-easy-math-tricks-youll-wish-youd-known-this-whole-time/","timestamp":"2024-11-03T19:00:59Z","content_type":"text/html","content_length":"127685","record_id":"<urn:uuid:e4d6eba9-d1f5-4466-a93d-611aba551099>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00494.warc.gz"} |
1920. Titan Ruins: the Infinite Power of Magic
`It seems that our efforts were not in vain,' Alba said after studying an ancient volume. `I've found here a method for accumulating a tremendous amount of energy. And the idea is so simple that I
wonder why nobody has discovered it. The usual problem is that magic energy is unstable and it's difficult to keep it in one place. But if we channel it along a closed path, it'll have no way out. We
only have to choose the right length of the path to keep the flow stable. Then we can pump in as much energy as we want. And when we break the path we'll release a magic flow of enormous power. It'll
be a real breakthrough in war magic!'
`Yes, but the path must be long enough and you can't drag a large device over a battlefield.'
`That's not a problem. The form of the path can be arbitrary. We can design a compact scheme of the device. Actually, the device is here already, it's there in the corner. We only have to adjust it.'
Indeed, there was a square grid with n × n nodes. At each node there was a prism that could be turned so as to either direct a flow of magic energy straightly or turn it by 90°. Soren and Alba had to
position L prisms so that a cyclic flow of magic energy of length L could be directed through them.
You are given the integers n and L (2 ≤ n ≤ 100; 4 ≤ L ≤ 20 000).
If it is impossible to organize a cycle of required length, output “Unsuitable device”. Otherwise, output “Overwhelming power of magic” in the first line. In each of the following L lines give two
integers in the range from 1 to n, which are the coordinates of the grid nodes through which energy should pass. The distances between two consecutive nodes and between the first and the last nodes
must be equal to 1. The energy mustn't pass more than once through the same node, because this may produce unpredictable and, most likely, lethal effects.
input output
2 6 Unsuitable device
Overwhelming power of magic
Problem Author: Alexey Samsonov (prepared by Dmitry Ivankov)
Problem Source: NEERC 2012, Eastern subregional contest | {"url":"https://timus.online/problem.aspx?space=1&num=1920","timestamp":"2024-11-13T01:55:53Z","content_type":"text/html","content_length":"7475","record_id":"<urn:uuid:121c8568-9ed2-4d8e-97e8-b110bc052387>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00523.warc.gz"} |
Minimum Boats Required to rescue people
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409. Minimum Boats Required to rescue people
Objective: Given N people need to be rescued by crossing the river by boat. Each boat can carry a maximum weight of given limit K. Each boat carries at most 2 people at the same time, provided the
sum of the weight of those people is at most limit K.
Write an algorithm to find the minimum number of boats required for N people to cross the river.
Input: people = [1,2], limit = 3
Output: 1
Explanation: 1 boat - with people weight - (1, 2)
Input: people = [5, 1, 4, 2], limit = 6
Output: 2
2 boats
First boat with people -(1, 5)
Second boat with people - (4, 2)
Input: people = [3, 4, 1, 2], limit = 4
Output: 3
3 boats
First boat with people - (1, 3)
Second boat with people - (4)
Third boat with people - (2)
1. Sort the array in ascending order.
2. Take two pointers, left pointer at the beginning of the array and right pointer at the most of array. Check persons at the left pointer and right pointer and add their weights
1. If the sum of weights is <= limit K then both persons can be in one boat. increment left pointer and decrement right pointer. add 1 to the total number of boats required.
2. else person at the right pointer has to go in one boat so decrement right pointer. add 1 to the total number of boats required.
Time Complexity: O(nlogn)
Total Number of boats required: 3
Source: Leetcode | {"url":"https://tutorialhorizon.com/algorithms/minimum-boats-required-to-rescue-people/","timestamp":"2024-11-12T22:49:09Z","content_type":"text/html","content_length":"78958","record_id":"<urn:uuid:2a4fb0e7-1aaf-46a2-8de8-1bffde05dcfb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00679.warc.gz"} |
st: RE: Re: Which random effect estimators use Gauss-Hermite
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st: RE: Re: Which random effect estimators use Gauss-Hermite
From "Cowell, Alexander J." <[email protected]>
To "'[email protected]'" <[email protected]>
Subject st: RE: Re: Which random effect estimators use Gauss-Hermite
Date Fri, 2 May 2003 15:21:25 -0400
Thanks, Scott
So long as xtnbreg is based on the famour Hausman et al., I think you're
dead on.
-----Original Message-----
From: Scott Merryman [mailto:[email protected]]
Sent: Saturday, April 26, 2003 5:44 PM
To: [email protected]
Subject: st: Re: Which random effect estimators use Gauss-Hermite
----- Original Message -----
From: "Cowell, Alexander J." <[email protected]>
To: <[email protected]>
Sent: Friday, April 25, 2003 11:59 AM
Subject: st: Which random effect estimators use Gauss-Hermite
> Hi there
> The manual points out that after running xtlogit with random effects, one
> should use quadchck (though I don't see why this isn't just the default in
> xtlogit). This is because the quadrature method of computing the log
> likelihood and the derivatives may give unstable estimates. This makes
> sense.
> Rather cryptically the manual (version 7.0) also says in the 'quadchck'
> entry "Some random-effects estimators in Stata use Gauss-Hermite
> quadrature...).
> My questions are:
> 1. Which estimators do and which don't use G-H quadrature?
At least in Stata 8 -quadchk- checks the quadrature approximation used in
the random-effects estimators of the following commands:
xtpoisson with the normal option
These estimators all assume a normal distribution for the random effect.
> 2. Or, if #1 is too much to answer, what does xtnbreg use?
Gaussian quadrature is not used to maximize the log-likelihood but to
approximate integrals that do not exist in closed form. Stata uses the
Newton-Raphson algorithm to maximize the likelihood function (or if
the -difficult- option is employed then steepest ascent is used in the
problem subspaces). For -xtnbreg , re- with random effect d(i), it is
assumed that 1(1+d(i)) is distributed as a Beta distribution. I believe the
integral has a closed form so quadrature approximation is not necessary.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2003-05/msg00025.html","timestamp":"2024-11-03T13:46:05Z","content_type":"text/html","content_length":"10459","record_id":"<urn:uuid:04c3482d-2207-45c3-8504-aa40ff0b1c4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00382.warc.gz"} |
Sigma Protocols
The previous 3-coloring example certainly works as a zero knowledge proof, but is quite slow, and requires a lot of interaction. There are efficient protocols for interactive proofs, we will study
sigma protocols.
Sigma Protocols
Definition. An effective relation is a binary relation $\mc{R} \subset \mc{X} \times \mc{Y}$, where $\mc{X}$, $\mc{Y}$, $\mc{R}$ are efficiently recognizable finite sets. Elements of $\mc{Y}$ are
called statements. If $(x, y) \in \mc{R}$, then $x$ is called a witness for $y$.
Definition. Let $\mc{R} \subset \mc{X} \times \mc{Y}$ be an effective relation. A sigma protocol for $\mc{R}$ is a pair of algorithms $(P, V)$ satisfying the following.
□ The prover $P$ is an interactive protocol algorithm, which takes $(x, y) \in \mc{R}$ as input.
□ The verifier $V$ is an interactive protocol algorithm, which takes $y \in \mc{Y}$ as input, and outputs $\texttt{accept}$ or $\texttt{reject}$.
The interaction goes as follows.^1
1. $P$ computes a commitment message $t$ and sends it to $V$.
2. $V$ chooses a random challenge $c \la \mc{C}$ from a challenge space and sends it to $P$.
3. $P$ computes a response $z$ and sends it to $V$.
4. $V$ outputs either $\texttt{accept}$ or $\texttt{reject}$, computed strictly as a function of the statement $y$ and the conversation $(t, c, z)$.
For all $(x, y) \in \mc{R}$, at the end of the interaction between $P(x, y)$ and $V(y)$, $V(y)$ always outputs $\texttt{accept}$.
• The verifier is deterministic except for choosing a random challenge $c \la \mc{C}$.
• If the output is $\texttt{accept}$, then the conversation $(t, c, z)$ is an accepting conversation for $y$.
• In most cases, the challenge space has to be super-poly. We say that the protocol has a large challenge space.
The soundness property says that it is infeasible for any prover to make the verifier accept a statement that is false.
Definition. Let $\Pi = (P, V)$ be a sigma protocol for $\mc{R} \subset \mc{X}\times \mc{Y}$. For a given adversary $\mc{A}$, the security game goes as follows.
1. The adversary chooses a statement $y^{\ast} \in \mc{Y}$ and gives it to the challenger.
2. The adversary interacts with the verifier $V(y^{\ast})$, where the challenger plays the role of verifier, and the adversary is a possibly cheating prover.
The adversary wins if $V(y^{\ast})$ outputs $\texttt{accept}$ but $y^{\ast} \notin L_\mc{R}$. The advantage of $\mc{A}$ with respect to $\Pi$ is denoted $\rm{Adv}_{\rm{Snd}}[\mc{A}, \Pi]$ and
defined as the probability that $\mc{A}$ wins the game.
If the advantage is negligible for all efficient adversaries $\mc{A}$, then $\Pi$ is sound.
Special Soundness
For sigma protocols, it suffices to require special soundness.
Definition. Let $(P, V)$ be a sigma protocol for $\mc{R} \subset \mc{X} \times \mc{Y}$. $(P, V)$ provides special soundness if there is an efficient deterministic algorithm $\rm{Ext}$, called a
knowledge extractor with the following property.
Given a statement $y \in \mc{Y}$ and two accepting conversations $(t, c, z)$ and $(t, c’, z’)$ with $c \neq c’$, $\rm{Ext}$ outputs a witness (proof) $x \in \mc{X}$ such that $(x, y) \in \mc{R}$.
The extractor efficiently finds a proof $x$ for $y \in \mc{Y}$. This means, if a possibly cheating prover $P^{\ast}$ makes $V$ accept $y$ with non-negligible probability, then $P^{\ast}$ must have
known a proof $x$ for $y$. Thus $P^{\ast}$ isn’t actually a dishonest prover, he already has a proof.
Note that the commitment $t$ is the same for the two accepting conversations. The challenge $c$ and $c’$ are chosen after the commitment, so if the prover can come up with $z$ and $z’$ so that $(t,
c, z)$ and $(t, c’, z’)$ are accepting conversations for $y$, then the prover must have known $x$.
We also require that the challenge space is large, the challenger shouldn’t be accepted by luck.
Special Soundness $\implies$ Soundness
Theorem. Let $\Pi$ be a sigma protocol with a large challenge space. If $\Pi$ provides special soundness, then $\Pi$ is sound.
For every efficient adversary $\mc{A}$,
\[\rm{Adv}_{\rm{Snd}}[\mc{A}, \Pi] \leq \frac{1}{N}\]
where $N$ is the size of the challenge space.
Proof. Suppose that $\mc{A}$ chooses a false statement $y^{\ast}$ and a commitment $t^{\ast}$. It suffices to show that there exists at most one challenge $c$ such that $(t^{\ast}, c, z)$ is an
accepting conversation for some response $z$.
If there were two such challenges $c, c’$, then there would be two accepting conversations for $y^{\ast}$, which are $(t^{\ast}, c, z)$ and $(t^{\ast}, c’, z’)$. Now by special soundness, there
exists a witness $x$ for $y^{\ast}$, which is a contradiction.
Special Honest Verifier Zero Knowledge
The conversation between $P$ and $V$ must not reveal anything.
Definition. Let $(P, V)$ be a sigma protocol for $\mc{R} \subset \mc{X} \times \mc{Y}$. $(P, V)$ is special honest verifier zero knowledge (special HVZK) if there exists an efficient
probabilistic algorithm $\rm{Sim}$ (simulator) that satisfies the following.
□ For all inputs $(y, c) \in \mc{Y} \times \mc{C}$, $\rm{Sim}(y, c)$ outputs a pair $(t, z)$ such that $(t, c, z)$ is always an accepting conversation for $y$.
□ For all $(x, y) \in \mc{R}$, let $c \la \mc{C}$ and $(t, z) \la \rm{Sim}(y, c)$. Then $(t, c, z)$ has the same distribution as the conversation between $P(x, y)$ and $V(y)$.
The difference is that the simulator takes an additional input $c$. Also, the simulator produces an accepting conversation even if the statement $y$ does not have a proof.
Also note that the simulator is free to generate the messages in any convenient order.
The Schnorr Identification Protocol Revisited
The Schnorr identification protocol is actually a sigma protocol. Refer to Schnorr identification protocol (Modern Cryptography) for the full description.
The pair $(P, V)$ is a sigma protocol for the relation $\mc{R} \subset \mc{X} \times \mc{Y}$ where
\[\mc{X} = \bb{Z}_q, \quad \mc{Y} = G, \quad \mc{R} = \left\lbrace (\alpha, u) \in \bb{Z}_q \times G : g^\alpha = u \right\rbrace.\]
The challenge space $\mc{C}$ is a subset of $\bb{Z}_q$.
The protocol provides special soundness. If $(u_t, c, \alpha_z)$ and $(u_t, c’, \alpha_z’)$ are two accepting conversations with $c \neq c’$, then we have
\[g^{\alpha_z} = u_t \cdot u^c, \quad g^{\alpha_z'} = u_t \cdot u^{c'},\]
so we have $g^{\alpha_z - \alpha_z’} = u^{c - c’}$. Setting $\alpha^{\ast} = (\alpha_z - \alpha_z’) /(c - c’)$ satisfies $g^{\alpha^{\ast}} = u$, solving the discrete logarithm and $\alpha^{\ast}$ is
a proof.
As for HVZK, the simulator chooses $\alpha_z \la \bb{Z}_q$, $c \la \mc{C}$ randomly and sets $u_t = g^{\alpha_z} \cdot u^{-c}$. Then $(u_t, c, \alpha_z)$ will be accepted. Note that the order doesn’t
matter. Also, the distribution is same, since $c$ and $\alpha_z$ are uniform over $\mc{C}$ and $\bb{Z}_q$ and the choice of $c$ and $\alpha_z$ determines $u_t$ uniquely. This is identical to the
distribution in the actual protocol.
Dishonest Verifier
In case of dishonest verifiers, $V$ may not follow the protocol. For example, $V$ may choose non-uniform $c \in \mc{C}$ depending on the commitment $u_t$. In this case, the conversation from the
actual protocol and the conversation generated by the simulator will have different distributions.
We need a different distribution. The simulator must also take the verifier’s actions as input, to properly simulate the dishonest verifier.
Modified Schnorr Protocol
The original protocol can be modified so that the challenge space $\mc{C}$ is smaller. Completeness property is obvious, and the soundness error grows, but we can always repeat the protocol.
As for zero knowledge, the simulator $\rm{Sim}_{V^{\ast}}(u)$ generates a verifier’s view $(u, c, z)$ as follows.
• Guess $c’ \la \mc{C}$. Sample $z’ \la \bb{Z}_q$ and set $u’ = g^{z’}\cdot u^{-c’}$. Send $u’$ to $V^{\ast}$.
• If the response from the verifier $V^{\ast}(u’)$ is $c$ and $c \neq c’$, restart.
□ $c = c’$ holds with probability $1 / \left\lvert \mc{C} \right\lvert$, since $c’$ is uniform.
• Otherwise, output $(u, c, z) = (u’, c’, z’)$.
Sending $u’$ to $V^{\ast}$ is possible because the simulator also takes the actions of $V^{\ast}$ as input. The final output conversation has distribution identical to the real protocol execution.
Overall, this modified protocol works for dishonest verifiers, at the cost of efficiency because of the increased soundness error. We have a security-efficiency tradeoff.
But in most cases, it is enough to assume honest verifiers, as we will see soon.
Other Sigma Protocol Examples
Okamoto’s Protocol
This one is similar to Schnorr protocol. This is used for proving the representation of a group element.
Let $G = \left\langle g \right\rangle$ be a cyclic group of prime order $q$, let $h \in G$ be some arbitrary group element, fixed as a system parameter. A representation of $u$ relative to $g$ and
$h$ is a pair $(\alpha, \beta) \in \bb{Z}_q^2$ such that $g^\alpha h^\beta = u$.
Okamoto’s protocol for the relation
\[\mc{R} = \bigg\lbrace \big( (\alpha, \beta), u \big) \in \bb{Z}_q^2 \times G : g^\alpha h^\beta = u \bigg\rbrace\]
goes as follows.
1. $P$ computes random $\alpha_t, \beta_t \la \bb{Z}_q$ and sends commitment $u_t \la g^{\alpha_t}h^{\beta_t}$ to $V$.
2. $V$ computes challenge $c \la \mc{C}$ and sends it to $P$.
3. $P$ computes $\alpha_z \la \alpha_t + \alpha c$, $\beta_z \la \beta_t + \beta c$ and sends $(\alpha_z, \beta_z)$ to $V$.
4. $V$ outputs $\texttt{accept}$ if and only if $g^{\alpha_z} h^{\beta_z} = u_t \cdot u^c$.
Completeness is obvious.
Theorem. Okamoto’s protocol provides special soundness and is special HVZK.
Proof. Very similar to the proof of Schnorr. Refer to Theorem 19.9.^2
The Chaum-Pedersen Protocol for DH-Triples
The Chaum-Pederson protocol is for convincing a verifier that a given triple is a DH-triple.
Let $G = \left\langle g \right\rangle$ be a cyclic group of prime order $q$. $(g^\alpha, g^\beta, g^\gamma)$ is a DH-triple if $\gamma = \alpha\beta$. Then, the triple $(u, v, w)$ is a DH-triple if
and only if $v = g^\beta$ and $w = u^\beta$ for some $\beta \in \bb{Z}_q$.
The Chaum-Pederson protocol for the relation
\[\mc{R} = \bigg\lbrace \big( \beta, (u, v, w) \big) \in \bb{Z}_q \times G^3 : v = g^\beta \land w = u^\beta \bigg\rbrace\]
goes as follows.
1. $P$ computes random $\beta_t \la \bb{Z}_q$ and sends commitment $v_t \la g^{\beta_t}$, $w_t \la u^{\beta_t}$ to $V$.
2. $V$ computes challenge $c \la \mc{C}$ and sends it to $P$.
3. $P$ computes $\beta_z \la \beta_t + \beta c$, and sends it to $V$.
4. $V$ outputs $\texttt{accept}$ if and only if $g^{\beta_z} = v_t \cdot v^c$ and $u^{\beta_z} = w_t \cdot w^c$.
Completeness is obvious.
Theorem. The Chaum-Pedersen protocol provides special soundness and is special HVZK.
Proof. Also similar. See Theorem 19.10.^2
This can be used to prove that an ElGamal ciphertext $c = (u, v) = (g^k, h^k \cdot m)$ is an encryption of $m$ with public key $h = g^\alpha$, without revealing the private key or the ephemeral key
$k$. If $(g^k, h^k \cdot m)$ is a valid ciphertext, then $(h, u, vm^{-1}) = (g^\alpha, g^k, g^{\alpha k})$ is a valid DH-triple.
Sigma Protocol for Arbitrary Linear Relations
Schnorr, Okamoto, Chaum-Pedersen protocols look similar. They are special cases of a generic sigma protocol for proving a linear relation among group elements. Read more in Section 19.5.3.^2
Sigma Protocol for RSA
Let $(n, e)$ be an RSA public key, where $e$ is prime. The Guillou-Quisquater (GQ) protocol is used to convince a verifier that he knows an $e$-th root of $y \in \bb{Z}_n^{\ast}$.
The Guillou-Quisquater protocol for the relation
\[\mc{R} = \bigg\lbrace (x, y) \in \big( \bb{Z}_n^{\ast} \big)^2 : x^e = y \bigg\rbrace\]
goes as follows.
1. $P$ computes random $x_t \la \bb{Z}_n^{\ast}$ and sends commitment $y_t \la x_t^e$ to $V$.
2. $V$ computes challenge $c \la \mc{C}$ and sends it to $P$.
3. $P$ computes $x_z \la x_t \cdot x^c$ and sends it to $V$.
4. $V$ outputs $\texttt{accept}$ if and only if $x_z^e = y_t \cdot y^c$.
Completeness is obvious.
Theorem. The GQ protocol provides special soundness and is special HVZK.
Proof. Also similar. See Theorem 19.13.^2
Combining Sigma Protocols
Using the basic sigma protocols, we can build sigma protocols for complex statements.
AND-Proof Construction
The construction is straightforward, since we can just prove both statements.
Given two sigma protocols $(P_0, V_0)$ for $\mc{R}0 \subset \mc{X}_0 \times \mc{Y}_0$ and $(P_1, V_1)$ for $\mc{R}_1 \subset \mc{X}_1 \times \mc{Y}_1$, we construct a sigma protocol for the relation
$\mc{R}\rm{AND}$ defined on $(\mc{X}_0 \times \mc{X}_1) \times (\mc{Y}_0 \times \mc{Y}_1)$ as
\[\mc{R}_\rm{AND} = \bigg\lbrace \big( (x_0, x_1), (y_0, y_1) \big) : (x_0, y_0) \in \mc{R}_0 \land (x_1, y_1) \in \mc{R}_1 \bigg\rbrace.\]
Given a pair of statements $(y_0, y_1) \in \mc{Y}_0 \times \mc{Y}_1$, the prover tries to convince the verifier that he knows a proof $(x_0, x_1) \in \mc{X}_0 \times \mc{X}_1$. This is equivalent to
proving the AND of both statements.
1. $P$ runs $P_i(x_i, y_i)$ to get a commitment $t_i$. $(t_0, t_1)$ is sent to $V$.
2. $V$ computes challenge $c \la C$ and sends it to $P$.
3. $P$ uses the challenge for both $P_0, P_1$, obtains response $z_0$, $z_1$, which is sent to $V$.
4. $V$ outputs $\texttt{accept}$ if and only if $(t_i, c, z_i)$ is an accepting conversation for $y_i$.
Completeness is clear.
Theorem. If $(P_0, V_0)$ and $(P_1, V_1)$ provide special soundness and are special HVZK, then the AND protocol $(P, V)$ defined above also provides special soundness and is special HVZK.
Proof. For special soundness, let $\rm{Ext}_0$, $\rm{Ext}_1$ be the knowledge extractor for $(P_0, V_0)$ and $(P_1, V_1)$, respectively. Then the knowledge extractor $\rm{Ext}$ for $(P, V)$ can be
constructed straightforward. For statements $(y_0, y_1)$, suppose that $\big( (t_0, t_1), c, (z_0, z_1) \big)$ and $\big( (t_0, t_1), c’, (z_0’, z_1’) \big)$ are two accepting conversations. Feed $\
big( y_0, (t_0, c, z_0), (t_0, c’, z_0’) \big)$ to $\rm{Ext}_0$, and feed $\big( y_1, (t_1, c, z_1), (t_1, c’, z_1’) \big)$ to $\rm{Ext}_1$.
For special HVZK, let $\rm{Sim}_0$ and $\rm{Sim}_1$ be simulators for each protocol. Then the simulator $\rm{Sim}$ for $(P, V)$ is built by using $(t_0, z_0) \la \rm{Sim}_0(y_0, c)$ and $(t_1, z_1) \
la \rm{Sim}_1(y_1, c)$. Set
\[\big( (t_0, t_1), (z_0, z_1) \big) \la \rm{Sim}\big( (y_0, y_1), c \big).\]
We have used the fact that the challenge is used for both protocols.
OR-Proof Construction
However, OR-proof construction is difficult. The prover must convince the verifier that either one of the statement is true, but should not reveal which one is true.
If the challenge is known in advance, the prover can cheat. We exploit this fact. For the proof of $y_0 \lor y_1$, do the real proof for $y_b$ and cheat for $y_{1-b}$.
Suppose we are given two sigma protocols $(P_0, V_0)$ for $\mc{R}_0 \subset \mc{X}_0 \times \mc{Y}_0$ and $(P_1, V_1)$ for $\mc{R}_1 \subset \mc{X}_1 \times \mc{Y}_1$. We assume that these both use
the same challenge space, and both are special HVZK with simulators $\rm{Sim}_0$ and $\rm{Sim}_1$.
We combine the protocols to form a sigma protocol for the relation $\mc{R}_\rm{OR}$ defined on ${} \big( \braces{0, 1} \times (\mc{X}_0 \cup \mc{X}_1) \big) \times (\mc{Y}_0\times \mc{Y}_1) {}$ as
\[\mc{R}_\rm{OR} = \bigg\lbrace \big( (b, x), (y_0, y_1) \big): (x, y_b) \in \mc{R}_b\bigg\rbrace.\]
Here, $b$ denotes the actual statement $y_b$ to prove. For $y_{1-b}$, we cheat.
$P$ is initialized with $\big( (b, x), (y_0, y_1) \big) \in \mc{R}_\rm{OR}$ and $V$ is initialized with $(y_0, y_1) \in \mc{Y}_0 \times \mc{Y}_1$. Let $d = 1 - b$.
1. $P$ computes $c_d \la \mc{C}$ and $(t_d, z_d) \la \rm{Sim}_d(y_d, c_d)$.
2. $P$ runs $P_b(x, y_b)$ to get a real commitment $t_b$ and sends $(t_0, t_1)$ to $V$.
3. $V$ computes challenge $c \la C$ and sends it to $P$.
4. $P$ computes $c_b \la c \oplus c_d$, feeds it to $P_b(x, y_b)$ obtains a response $z_b$.
5. $P$ sends $(c_0, z_0, z_1)$ to $V$.
6. $V$ computes $c_1 \la c \oplus c_0$, and outputs $\texttt{accept}$ if and only if $(t_0, c_0, z_0)$ is an accepting conversation for $y_0$ and $(t_1, c_1, z_1)$ is an accepting conversation
for $y_1$.
Step $1$ is the cheating part, where the prover chooses a challenge, and generates a commitment and a response from the simulator.
Completeness follows from the following.
• $c_b = c \oplus c_{1-b}$, so $c_1 = c \oplus c_0$ always holds.
• Both conversations $(t_0, c_0, z_0)$ and $(t_1, c_1, z_1)$ are accepted.
□ An actual proof is done for statement $y_b$.
□ For statement $y_{1-b}$, the simulator always outputs an accepting conversation.
$c_b = c \oplus c_d$ is random, so $P$ cannot manipulate the challenge. Also, $V$ checks $c_1 = c \oplus c_0$.
Theorem. If $(P_0, V_0)$ and $(P_1, V_1)$ provide special soundness and are special HVZK, then the OR protocol $(P, V)$ defined above also provides special soundness and is special HVZK.
Proof. For special soundness, suppose that $\rm{Ext}_0$ and $\rm{Ext}_1$ are knowledge extractors. Let
\[\big( (t_0, t_1), c, (c_0, z_0, z_1) \big), \qquad \big( (t_0, t_1), c', (c_0', z_0', z_1') \big)\]
be two accepting conversations with $c \neq c’$. Define $c_1 = c \oplus c_0$ and $c_1’ = c’ \oplus c_0’$. Since $c \neq c’$, it must be the case that either $c_0 \neq c_0’$ or $c_1 \neq c_1’$. Now $\
rm{Ext}$ will work as follows.
• If $c_0 \neq c_0’$, output $\bigg( 0, \rm{Ext}_0\big( y_0, (t_0, c_0, z_0), (t_0, c_0’, z_0’) \big) \bigg)$.
• If $c_1 \neq c_1’$, output $\bigg( 1, \rm{Ext}_1\big( y_1, (t_1, c_1, z_1), (t_1, c_1’, z_1’) \big) \bigg)$.
Then $\rm{Ext}$ will extract the knowledge.
For special HVZK, define $c_0 \la \mc{C}$, $c_1 \la c \oplus c_0$. Then run each simulator to get
\[(t_0, z_0) \la \rm{Sim}_0(y_0, c_0), \quad (t_1, z_1) \la \rm{Sim}_1(y_1, c_1).\]
Then the simulator for $(P, V)$ outputs
\[\big( (t_0, t_1), (c_0, z_0, z_1) \big) \la \rm{Sim}\big( (y_0, y_1), c \big).\]
The simulator just simulates for both of the statements and returns the messages as in the protocol. $c_b$ is random, and the remaining values have the same distribution since the original two
protocols were special HVZK.
Example: OR of Sigma Protocols with Schnorr Protocol
Let $G = \left\langle g \right\rangle$ be a cyclic group of prime order $q$. The prover wants to convince the verifier that he knows the discrete logarithm of either $h_0$ or $h_1$ in $G$.
Suppose that the prover knows $x_b \in \bb{Z}_q$ such that $g^{x_b} = h_b$.
1. Choose $c_{1-b} \la \mc{C}$ and call simulator of $1-b$ to obtain $(u_{1-b}, z_{1-b}) \la \rm{Sim}_{1-b}$.
2. $P$ sends two commitments $u_0, u_1$.
☆ For $u_b$, choose random $y \la \bb{Z}_q$ and set $u_b = g^y$.
☆ For $u_{1-b}$, use the value from the simulator.
3. $V$ sends a single challenge $c \la \mc{C}$.
4. Using $c_{1-b}$, split the challenge into $c_0$, $c_1$ so that they satisfy $c_0 \oplus c_1 = c$. Then send $(c_0, c_1, z_0, z_1)$ to $V$.
☆ For $z_b$, calculate $z_b \la y + c_b x$.
☆ For $z_{1-b}$, use the value from the simulator.
5. $V$ checks if $c = c_0 \oplus c_1$. $V$ accepts if and only if $(u_0, c_0, z_0)$ and $(u_1, c_1, z_1)$ are both accepting conversations.
• Since $c, c_{1-b}$ are random, $c_b$ is random. Thus one of the proofs must be valid.
Generalized Constructions
See Exercise 19.26 and 19.28.^2
Non-interactive Proof Systems
Sigma protocols are interactive proof systems, but we can convert them into non-interactive proof systems using the Fiat-Shamir transform.
First, the definition of non-interactive proof systems.
Definition. Let $\mc{R} \subset \mc{X} \times \mc{Y}$ be an effective relation. A non-interactive proof system for $\mc{R}$ is a pair of algorithms $(G, V)$ satisfying the following.
□ $G$ is an efficient probabilistic algorithm that generates the proof as $\pi \la G(x, y)$ for $(x, y) \in \mc{R}$. $\pi$ belongs to some proof space $\mc{PS}$.
□ $V$ is an efficient deterministic algorithm that verifies the proof as $V(y, \pi)$ where $y \in \mc{Y}$ and $\pi \in \mc{PS}$. $V$ outputs either $\texttt{accept}$ or $\texttt{reject}$. If
$V$ outputs $\texttt{accept}$, $\pi$ is a valid proof for $y$.
For all $(x, y) \in \mc{R}$, the output of $G(x, y)$ must be a valid proof for $y$.
Non-interactive Soundness
Intuitively, it is hard to create a valid proof of a false statement.
Definition. Let $\Phi = (G, V)$ be a non-interactive proof system for $\mc{R} \subset \mc{X} \times \mc{Y}$ with proof space $\mc{PS}$. An adversary $\mc{A}$ outputs a statement $y^{\ast} \in \mc
{Y}$ and a proof $\pi^{\ast} \in \mc{PS}$ to attack $\Phi$.
The adversary wins if $V(y^{\ast}, \pi^{\ast}) = \texttt{accept}$ and $y^{\ast} \notin L_\mc{R}$. The advantage of $\mc{A}$ with respect to $\Phi$ is defined as the probability that $\mc{A}$
wins, and is denoted as $\rm{Adv}_{\rm{niSnd}}[\mc{A}, \Phi]$.
If the advantage is negligible for all efficient adversaries $\mc{A}$, $\Phi$ is sound.
Non-interactive Zero Knowledge
The Fiat-Shamir Transform
The basic idea is using a hash function to derive a challenge, instead of a verifier. Now the only job of the verifier is checking the proof, requiring no interaction for the proof.
Definition. Let $\Pi = (P, V)$ be a sigma protocol for a relation $\mc{R} \subset \mc{X} \times \mc{Y}$. Suppose that conversations $(t, c, z) \in \mc{T} \times \mc{C} \times \mc{Z}$. Let $H : \
mc{Y} \times \mc{T} \rightarrow \mc{C}$ be a hash function.
Define the Fiat-Shamir non-interactive proof system $\Pi_\rm{FS} = (G_\rm{FS}, V_\rm{FS})$ with proof space $\mc{PS} = \mc{T} \times \mc{Z}$ as follows.
□ For input $(x, y) \in \mc{R}$, $G_\rm{FS}$ runs $P(x, y)$ to obtain a commitment $t \in \mc{T}$. Then computes the challenge $c = H(y, t)$, which is fed to $P(x, y)$, obtaining a response $z
\in \mc{Z}$. $G_\rm{FS}$ outputs $(t, z) \in \mc{T} \times \mc{Z}$.
□ For input $\big( y, (t, z) \big) \in \mc{Y} \times (\mc{T} \times \mc{Z})$, $V_\rm{FS}$ verifies that $(t, c, z)$ is an accepting conversation for $y$, where $c = H(y, t)$.
Any sigma protocol can be converted into a non-interactive proof system. Its completeness is automatically given by the completeness of the sigma protocol.
By modeling the hash function as a random oracle, we can show that:
• If the sigma protocol is sound, then so is the non-interactive proof system.^3
• If the sigma protocol is special HVZK, then running the non-interactive proof system does not reveal any information about the secret.
• No interactions are required, resulting in efficient protocols with lower round complexity.
• No need to consider dishonest verifiers, since prover chooses the challenge. The verifier only verifies.
• In distributed systems, a single proof can be used multiple times.
Soundness of the Fiat-Shamir Transform
Theorem. Let $\Pi$ be a sigma protocol for a relation $\mc{R} \subset \mc{X} \times \mc{Y}$, and let $\Pi_\rm{FS}$ be the Fiat-Shamir non-interactive proof system derived from $\Pi$ with hash
function $H$. If $\Pi$ is sound and $H$ is modeled as a random oracle, then $\Pi_\rm{FS}$ is also sound.
Let $\mc{A}$ be a $q$-query adversary attacking the soundness of $\Pi_\rm{FS}$. There exists an adversary $\mc{B}$ attacking the soundness of $\Pi$ such that
\[\rm{Adv}_{\rm{niSnd^{ro}}}[\mc{A}, \Pi_\rm{FS}] \leq (q + 1) \rm{Adv}_{\rm{Snd}}[\mc{B}, \Pi].\]
Proof Idea. Suppose that $\mc{A}$ produces a valid proof $(t^{\ast}, z^{\ast})$ on a false statement $y^{\ast}$. Without loss of generality, $\mc{A}$ queries the random oracle at $(y^{\ast}, t^{\
ast})$ within $q+1$ queries. Then $\mc{B}$ guesses which of the $q+1$ queries is the relevant one. If $\mc{B}$ guesses the correct query, the conversation $(t^{\ast}, c, z^{\ast})$ will be accepted
and $\mc{B}$ succeeds. The factor $q+1$ comes from the choice of $\mc{B}$.
Zero Knowledge of the Fiat-Shamir Transform
Omitted. Works…
The Fiat-Shamir Signature Scheme
Now we understand why the Schnorr signature scheme used hash functions. In general, the Fiat-Shamir transform can be used to convert sigma protocols into signature schemes.
We need $3$ building blocks.
• A sigma protocol $(P, V)$ with conversations of the form $(t, c, z)$.
• A key generation algorithm $G$ for $\mc{R}$, that outputs $pk = y$, $sk = (x, y) \in \mc{R}$.
• A hash function $H : \mc{M} \times \mc{T} \rightarrow \mc{C}$, modeled as a random oracle.
Definition. The Fiat-Shamir signature scheme derived from $G$ and $(P, V)$ works as follows.
□ Key generation: invoke $G$ so that $(pk, sk) \la G()$.
☆ $pk = y \in \mc{Y}$ and $sk = (x, y) \in \mc{R}$.
□ Sign: for message $m \in \mc{M}$
1. Start the prover $P(x, y)$ and obtain the commitment $t \in \mc{T}$.
2. Compute the challenge $c \la H(m, t)$.
3. $c$ is fed to the prover, which outputs a response $z$.
4. Output the signature $\sigma = (t, z) \in \mc{T} \times \mc{Z}$.
□ Verify: with the public key $pk = y$, compute $c \la H(m, t)$ and check that $(t, c, z)$ is an accepting conversation for $y$ using $V(y)$.
If an adversary can come up with a forgery, then the underlying sigma protocol is not secure.
Example: Voting Protocol
$n$ voters are casting a vote, either $0$ or $1$. At the end, all voters learn the sum of the votes, but we want to keep the votes secret for each party.
We can use the multiplicative ElGamal encryption scheme in this case. Assume that a trusted vote tallying center generates a key pair, keeps $sk = \alpha$ to itself and publishes $pk = g^\alpha$.
Each voter encrypts the vote $b_i$ and the ciphertext is
\[(u_i, v_i) = (g^{\beta_i}, h^{\beta_i} \cdot g^{b_i})\]
where $\beta_i \la\bb{Z}_q$. The vote tallying center aggregates all ciphertexts my multiplying everything. No need to decrypt yet. Then
\[(u^{\ast}, v^{\ast}) = \left( \prod_{i=1}^n g^{\beta_i}, \prod_{i=1}^n h^{\beta_i} \cdot g^{b_i} \right) = \big( g^{\beta^{\ast}}, h^{\beta^{\ast}} \cdot g^{b^{\ast}} \big),\]
where $\beta^{\ast} = \sum_{i=1}^n \beta_i$ and $b^{\ast} = \sum_{i=1}^n b_i$. Now decrypt $(u^{\ast}, v^{\ast})$ and publish the result $b^{\ast}$.^4
Since the ElGamal scheme is semantically secure, the protocol is also secure if all voters follow the protocol. But a dishonest voter can encrypt $b_i = -100$ or some arbitrary value.
To fix this, we can make each voter prove that the vote is valid. Using the Chaum-Pedersen protocol for DH-triples and the OR-proof construction, the voter can submit a proof that the ciphertext is
either a encryption of $b_i = 0$ or $1$. We can also apply the Fiat-Shamir transform here for efficient protocols, resulting in non-interactive proofs. | {"url":"https://log.zxcvber.com/lecture-notes/modern-cryptography/2023-11-07-sigma-protocols/","timestamp":"2024-11-03T19:58:28Z","content_type":"text/html","content_length":"57783","record_id":"<urn:uuid:93d9f6e5-94f0-4c0f-a717-b6b5465336ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00697.warc.gz"} |
[GAP Forum] Maximal subgroups.
johnathon simons johnathonasimons at outlook.com
Thu Aug 31 10:15:51 BST 2017
Dear all,
Back again for another round of fun - ha!
According to a certain paper I've found online, it states that for the Mathieu group M12, the triple (2A,4A,8A) is not a rigid generator of M12: that is, G =/= <g1, g2, g3> where g1 is contained in 2A, g2 is contained in 4A, g3 is contained in 8A and g1*g2*g3 = 1. In particular, it says that (2A,4A,8A) generates a proper subgroup of M12 as can be seen from the character tables of the maximal subgroups.
However, when I run the below algorithm (which attempts to find the rigid generating triple) I end up "apparently" finding one.
findNiceTriple := function(G, cls1, cls2, cls3)
local g1, g2, g3;
g1 := Representative(cls1);
for g2 in cls2 do
g3 := (g1*g2)^-1;
if g3 in cls3 and M12 = Group(g1, g2) then
return [g1, g2, g3];
return fail;
Then for example:
gap> M12:=MathieuGroup(12);
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8),(5,9)(7,10) ])
gap> cc:=ConjugacyClasses(M12);;
gap> Length(cc);
So then to see if the triple (2A, 4A, 8A) can generate a rigid triple of elements (g1,g1,g3) with g1*g2*g3 = 1 we have:
gap> findNiceTriple(M12, cc[2], cc[6], cc[11]);
[ (1,8,10,12,11)(3,7,4,5,9), (2,10,7,5)(3,7,8,9), (1,11,12,10,2,4,7,6,8,9)(3,5)]
1) If I'm not mistaken, this implies that (2A, 4A, 8A) is such a rigid triple that generates M12? If not, could someone please clarify as to why this is not true (is there some issue with the above algorithm - I have simply taken the conjugacy classes assuming they appear in an ordered list as shown above (hence cc[6] corresponds to the sixth conjugacy class which is 4A).
2) Furthermore, if the above approach is hopeless in determining rigid triples, could someone please inform me as why that is the case and how one can determine whether such a triple generates a proper subgroup of M12 by simply looking at the characer tables of the maximal subgroups?
Thank you as always,
Sent from Outlook<http://aka.ms/weboutlook>
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stellar equation of state codes from cococubed
Astronomy Research
Radiative Opacity
2024 Neutrino Emission from Stars
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2021 White Dwarf Pulsations & ^22Ne
Software Instruments
Stellar equation of states
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Chemical potentials
Stellar atmospheres
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Stellar opacities
Neutrino energy loss rates
Ephemeris routines
Fermi-Dirac functions
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Plane - cube intersection
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Nuclear reaction networks
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Fitting to conic sections
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Other Stuff:
Bicycle Adventures
Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
Before using the software instruments below, perhaps glance at the articles that describe them. The first law of thermodynamics $$ {\rm dE = T \ dS + {P\over \rho^2} \ d\rho} \label{eq1} \tag{1} $$
is an exact differential, which requires that the thermodynamic relations $$ \eqalignno { {\rm P} \ & = \ {\rm \rho^2 \ \dfrac{\partial E}{\partial \rho} \Biggm|_T \ + \ T \ \dfrac{\partial P}{\
partial T} \Biggm|_{\rho} } & (2) \cr {\rm \dfrac{\partial E}{\partial T} \Biggm|_{\rho}} \ & = \ {\rm T \ \dfrac{\partial S}{\partial T} \Biggm|_{\rho} } & (3) \cr {\rm - \dfrac{\partial S}{\partial
\rho} \Biggm|_T } \ & = \ {\rm {1 \over \rho^2} \ \dfrac{\partial P}{\partial T} \Biggm|_{\rho} } & (4) \cr } $$ be satisfied. An equation of state is thermodynamically consistent if all three of
these identities are true. Thermodynamic inconsistency may manifest itself in the unphysical buildup (or decay) of the entropy (or temperature) during numerical simulations of what should be an
adiabatic flow.
When the temperature and density are the natural thermodynamic variables to use, the appropriate thermodynamic potential is the Helmholtz free energy $$ {\rm F = E - T \ S} \hskip 0.5in {\rm dF = -S
\ dT + {P \over \rho^2} \ d\rho} \label{eq5} $$ With the pressure defined as $$ {\rm P \ = \ \rho^2 \ \dfrac{\partial F}{\partial \rho} \Biggm|_T } \label{eq6} \tag{6} $$ the first of the Maxwell
relations (Eq. 2) is automatically satisfied, as substitution of Eq. (5) into Eq. (6) demonstrates. With the entropy defined as $$ {\rm S \ = \ -\dfrac{\partial F}{\partial T} \Biggm|_{\rho} } \label
{eq7} \tag{7} $$ the second of the Maxwell relations (Eq. 3) is automatically satisfied, as substitution of Eq. (5) into Eq. (7) demonstrates. The requirement that the mixed partial derivatives
commute $$ {\rm \dfrac{\partial^2 F}{\partial T \ \partial \rho} \ = \ \dfrac{\partial F}{\partial \rho \ \partial T} } \label{eq8} \tag{8} $$ ensures that the third of the thermodynamic identity
(Eq. 4) is satisfied, as substitution of Eq. (5) into Eq. (8) shows.
Consider any interpolating function for the Helmholtz free energy $F(\rho,{\rm T})$ which satisfies Eq. (8). Thermodynamic consistency is guaranteed as long as Eq. (6) is used first to evaluate the
pressure, Eq. (7) is used second to evaluate the entropy, and finally Eq. (5) is used to evaluate the internal energy. In fact, this procedure is almost too robust! The interpolated values may be a
horribly inaccurate but they will be thermodynamically consistent.
Here then are bzip2 tarballs of six stellar interior equations of state:
helmholtz.tbz nadyozhin.tbz iben.tbz
weaver.tbz arnett.tbz timmes.tbz
Also see Josiah Schwab's python-helmholtz for the Helmholtz equation of state, and Matt Coleman's port of the Helmholtz equation of state to python helmeos.
The Skye EOS = an improved Helmholtz EOS for the non-interacting parts + an improved Potekhin & Chabrier EOS for the Coulomb plama parts + auto-differentiation. Its the bees knees for ionized plasmas
as of 2021. Skye is avaliable at https://github.com/adamjermyn/Skye, and the article is described more on the thermodynamics research page.
The Helmholtz EOS implements the formalism above on a grid, executes the fastest (memory is faster than cpu), displays perfect thermodynamic consistency, and has a maximum error on the default grid
of 10$^{-6}$. Helmholtz is the stellar EOS of choice in the FLASH software instrument and a backplane of the EOS module in the MESA software instrument. The Helmholtz free energy data file provided
spans 10$^{-12}$ ≤ density (g cm$^{-3}$) ≤ 10$^{15}$ and 10$^{3}$ ≤ temperature (K) ≤ 10$^{13}$ at 20 points per decade. The Nadyozhin EOS is the fastest of the analytic routines, has very good
thermodynamic consistency, a maximum error of 10$^{-5}$, and is also avaliable in FLASH. The Timmes EOS is as slow as molasses during a North Dakota winter, but it computes the non-interacting
electron-positron equation of state with no approximations, is exact to machine precision in IEEE double precision arithmetic, has excellent thermodynamic consistency, and serves as the reference
point for comparisons to the other EOS routines. In fact, the Helmholtz free energy table used by the Helmholtz EOS is calculated from the Timmes EOS.
There are times when a simpler cold fermi gas EOS is a wonderful thing. Such an EOS is in cold_fermi_gas.tbz. One can see this equation of state in action on this cold white dwarf page. | {"url":"https://cococubed.com/code_pages/eos.shtml","timestamp":"2024-11-09T06:13:22Z","content_type":"text/html","content_length":"16652","record_id":"<urn:uuid:3a41111b-d921-4d9c-b1d7-2592fcce3a71>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00165.warc.gz"} |
© 2021 Griffin Chure & Manuel Razo-Mejia. This work is licensed under a Creative Commons Attribution License CC-BY 4.0. All code contained herein is licensed under an MIT license
# For scientific computing import numpy as np # For plotting import matplotlib.pyplot as plt import seaborn as sns sns.set_context("notebook") sns.set_theme(style="darkgrid")
In this tutorial, we will cover the basics of writing stochastic simulations and their application to biological phenomena ranging from the diffusion of molecules to genetic drift in populations.
In science, we are often more interested in the distribution of a set of outcomes rather than a single event. This may be the probability distribution of a molecule diffusing a specific distance as a
function of time, the distribution of mRNA molecules per cell produced from a constitutively expressing promoter, or the probability distribution of a model parameter given a collection of data.
Stochastic simulations allow us to generate a series of simulations of a system in which one step (such as the direction a molecule will diffuse) is governed by random chance. These simulations often
boil down to flipping a coin to dictate if said step will occur or not.
Of course, sitting in your office chair flipping a US quarter over and over again is not how one should do a simulation. To get a sense of the probability distribution of some outcome, we often have
to simulate the process thousands of times. This means that we need to know how to make our computers do the heavy lifting.
It's often easy to forget just how powerful modern computers can be. What once required a serious computational cluster only twenty years ago can now be done on a 10 mm thick compartment made of
rose-gold colored aluminium. In the following exercise, we will demonstate how you can learn about the behavior of biological systems from the comfort of your laptop in only half a screen of code.
Think of a molecule that moves either left or right with equal step probabilities at each subsequent time point. We can decide whether to walk left or right by flipping a coin and seeing if it comes
up 'heads' or 'tails'.
# Flip a coin three times. flip_1 = np.random.rand() flip_2 = np.random.rand() flip_3 = np.random.rand() print(flip_1, flip_2, flip_3)
Note that this will change every time that we run the code cell. How do we convert this to a 'heads' and 'tails' readout? We can assume that this is a totally fair coin. This means that the
probability of getting "heads" to come up $P_H$ is the same as flipping a "tails" $P_T$ such that $P_H + P_T = 1$. This means that for a fair coin, $P_H = P_T = 0.5$. To convert our coin flips above,
we simply have to test if the flip is above or below $0.5$. If it is below, we'll say that the coin was flipped "heads", otherwise, it is "tails".
# Convert our coinflips to heads and tails. flips = [flip_1, flip_2, flip_3] for flip in flips: if flip < 0.5: print("Heads") else: print("Tails")
Now imagine that we wanted to flip the coin one thousand times. Obviously, we shouldn't write out a thousand variables and then loop through them. We could go through a loop for one thousand times
and flip a coin at each step or flip one thousand coins at once and store them in an array. In the interest of simplicity, we'll go with option one. Let's flip a coin one thousand times and compute
the probability of getting "heads".
# Test that our coin flipping algorithm is fair. n_flips = 1000 # That's a lot of flips! p = 0.5 # Our anticipated probability of a heads. # Flip the coin n_flips times. flips = np.random.rand
(n_flips) # Compute the number of heads. heads_or_tails = flips < p # Will result in a True (1.0) if heads. n_heads = np.sum(heads_or_tails) # Gives the total number of heads. # Compute the
probability of a heads in our simulation. p_sim = n_heads / n_flips print('Predicted p = %s. Simulated p = %s.' %(p, p_sim))
In the above code cell, we've also introduced a way to format strings using the %s formatter. We can specify that a value should inserted at that position (%) as a string (s) by providing a tuple of
the values after the string in the order they should be inserted prefixed by a magic operator %. Note that these strings are inserted in the order in which they appear in the tuple.
We see that our simulated probability is very close to our imposed $P_H$, but not exactly. This is the nature of stochastic simulations. It's based on repeated random draws. If we were to continue to
flip a coin more times, our simulated $P_H$ would get closer and closer to $0.5$. This is why doing many repetitions of stochastic simulations is necessary to generate reliable statistics.
So how do we relate this to diffusion? We'll start at position zero and flip a coin at each time step. If it is less than 0.5, we'll take a step left. Otherwise, we'll take a step to the right. At
each time point, we'll keep track of our position and then plot our trajectory.
# Define our step probability and number of steps. step_prob = 0.5 # Can step left or right equally. n_steps = 1000 # Essentially time. # Set up a vector to store our positions. position = np.zeros
(n_steps) # Full of zeros. # Loop through each time step. for i in range(1, n_steps): # Flip a coin. flip = np.random.rand() # Figure out which way we should step. if flip < step_prob: step = -1 # To
the 'left'. else: step = 1 # to the 'right'. # Update our position based off of where we were in the last time point. position[i] = position[i-1] + step
Notice that at the beginning of our for loop, we specified our range to be from 1 to n_steps. This is because the first entry (index 0) of our position vector is our starting position. Since we
update our position at timepoint i based off of where we were at time step i - 1, we have to start at index 1.
Now that we've taken the random walk, let's plot it. We'll take a look at where our molecule was at each time point.
# Make a vector of time points. steps = np.arange(0, n_steps, 1) # Arange from 0 to n_steps taking intervals of 1. # Plot it! plt.plot(steps, position) plt.xlabel('number of steps') plt.ylabel
Again, since our steps are based on the generation of random numbers. This trajectory will change every time you run the code. As we discussed earlier, the power of stochastic simulation comes from
doing them many times over. Let's write our random walk code again one thousand times and plot all of the traces.
# Perform the random walk 1000 times. n_simulations = 1000 # Make a new position vector. This will include all simulations. position = np.zeros((n_simulations, n_steps)) # Redefine our step
probability just to be clear. step_prob = 0.5 # Loop through each simulation. for i in range(n_simulations): # Loop through each step. for j in range(1, n_steps): # Flip a coin. flip = np.random.rand
() # Figure out how to step. if flip < step_prob: step = -1 else: step = 1 # Update our position. position[i, j] = position[i, j-1] + step
You'll notice that this cell took a little bit longer to run than the previous one. This is because we are doing the simulation a thousand times over! To show the random walks, we'll plot all of the
trajectories over each other as thin lines.
# Plot all of the trajectories together. for i in range(n_simulations): # Remembering that `position` is just a two-dimensional matrix that is # n_simulations by n_steps, we can get each step for a
given simulation # by indexing as position[i, :]. plt.plot(steps, position[i, :], linewidth=1, alpha=0.5) # Add axis labels. plt.xlabel('number of steps') plt.ylabel('position');
Pretty cool! We can look at the distribution of positions at various steps in time by making histograms of the positions of each simulation. Let's take a look at the distribution of positions at $t =
200$ steps.
# Make a histogram of the positions. To look at t=200, we have to index at # 199 because indexing starts at 0 in Python. We'll also normalize the # histogram (normed=True) so we can get a measure of
probability. plt.hist(position[:, 199], bins=20, density=True) plt.xlabel('position') plt.ylabel('probability') # Set the xlimits to cover the entire range. plt.xlim([-100, 100]);
We see that this qualitatively appears to be Gaussian. If we had to guess, we could say that the mean looks like it is right at about zero. Let's take a look at the distribution of positions at the
last time point as well.
# Make a histogram of the position distribution at the last time step. We could # just index at 999, but indexing at -1 will always return the distribution at # the last time step, whatever that may
be. plt.hist(position[:, -1], bins=20, density=True) plt.xlabel('position') plt.ylabel('probability') plt.xlim([-100, 100]);
Again, this distribution looks somewhat Gaussian with a mean of approximately zero. We can actually compute the mean position from our simulation by iterating through each time step and simply
computing the mean. Let's plot the mean at each time point as a red line.
# Compute the mean position at each step and plot it. mean_position = np.zeros(n_steps) for i in range(n_steps): mean_position[i] = np.mean(position[:, i]) # Plot all of the simulations. for i in
range(n_simulations): plt.plot(steps, position[i, :], linewidth=1, alpha=0.5) # Plot the mean as a thick red line. plt.plot(steps, mean_position, 'r-') # Add the labels. plt.xlabel('number of steps')
As we will learn in a few weeks, this is exactly what we would expect. While the mean position is zero, the mean squared displacement is not quite so trivial. Let's compute this value and plot it as
a function of the number of steps.
# Compute the mean squared displacement. msd = np.zeros(n_steps) for i in range(n_steps): msd[i] = np.mean(position[:, i]**2) # Plot the mean squared as a function of the number of steps. plt.plot
(steps, msd) plt.xlabel('number of steps') plt.ylabel('mean square displacement') plt.ylim([0, 1100]);
That certainly looks like it scales linearly with the number of steps, just as we predicted.
In the first example we look at a series of unbounded random walkers. We even showed that the mean square displacement of the walkers grows linearly as time progresses. If we were to let the
simulation run for much longer times, this trend would continue indefinitely. But we know that the world is finite. Furthermore, for cells that range between 1 µm for a bacteria to 1 mm for a Xenopus
frog egg to ≈ 1 m for a long neuron axon, there is a limit of how far molecules can diffuse before running into a boundary.
Our simulations can include such boundaries. For our particular purpose we will consider reflective boundaries--as opposed to absorbing boundaries--such that when a molecule hits the wall, it is
simply reflected back to continue its random walk within this bounded region. The trick lies in keeping track of the position of the particle with respect to the boundary, and whenever the trajectory
exceeds the boundary, for the specific step in which this happens, we simply multiply the displacement by -1, implementing in this way the reflective nature of the boundary.
Let's work through an example. First we will define the size of the boundary (which we will consider to be $\pm$ the defined size.
Now we can use the exact same code as before for the single walker. The difference being that at each step we will check whether or not the particle went pass the boundary, and if so, we will reflect
the trajectory.
# Define our step probability and number of steps. step_prob = 0.5 # Can step left or right equally. n_steps = 5000 # Essentially time. # Set up a vector to store our positions. position = np.zeros
(n_steps) # Full of zeros. # Loop through each time step. for i in range(1, n_steps): # Flip a coin. flip = np.random.rand() # Figure out which way we should step. if flip < step_prob: step = -1 # To
the 'left'. else: step = 1 # to the 'right'. # Check if the position is pass the boundary if (position[i-1] + step > box) or (position[i-1] + step < -box): # If it is pass the boundary, reflect the
trajectory position[i] = position[i-1] - step # Otherwise add the regular step else: # Update our position based off of where we were in the last time point. position[i] = position[i-1] + step
And with this simple extra if statement we have implemented our reflective boundary! Let's take a look at the trajectory
# Make a vector of time points. steps = np.arange(0, n_steps, 1) # Arange from 0 to n_steps taking intervals of 1. # Plot it! plt.plot(steps, position) # Add lines defining boundary plt.axhline(box,
lw=2, linestyle="--", color="black") plt.axhline(-box, lw=2, linestyle="--", color="black") plt.xlabel('number of steps') plt.ylabel('position');
We can see that indeed the walker is bounded by the limits that we set. Just for fun let's run multiple trajectories.
# Redefine box size box = 20 # Define number of simulations n_simulations = 10 # Define number of steps n_steps = 1000 # Make a new position vector. This will include all simulations. position =
np.zeros((n_simulations, n_steps)) # Redefine our step probability just to be clear. step_prob = 0.5 # Loop through each simulation. for i in range(n_simulations): # Loop through each step. for j in
range(1, n_steps): # Flip a coin. flip = np.random.rand() # Figure out how to step. if flip < step_prob: step = -1 else: step = 1 # Check if the position is pass the boundary if (position[i, j-1] +
step > box) or (position[i, j-1] + step < -box): # If it is pass the boundary, reflect the trajectory position[i, j] = position[i, j-1] - step # Otherwise add the regular step else: # Update our
position based off of where we were in the last time point. position[i, j] = position[i, j-1] + step
and now we are ready to look at the multiple trajectories
# Make a vector of time points. steps = np.arange(0, n_steps, 1) # Arange from 0 to n_steps taking intervals of 1. # Plot all of the simulations. for i in range(n_simulations): plt.plot(steps,
position[i, :], linewidth=1, alpha=0.5) # Add lines defining boundary plt.axhline(box, lw=2, linestyle="--", color="black") plt.axhline(-box, lw=2, linestyle="--", color="black") # Add the labels.
plt.xlabel('number of steps') plt.ylabel('position'); | {"url":"http://www.rpgroup.caltech.edu/aph161/assets/tut/t3/t03_stochastic_simulations.html","timestamp":"2024-11-06T15:29:17Z","content_type":"text/html","content_length":"1049304","record_id":"<urn:uuid:a942916a-da1c-423e-a95a-e1d394fdc313>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00731.warc.gz"} |
Knowledge Distillation Tutorial
Click here to download the full example code
Knowledge Distillation Tutorial¶
Author: Alexandros Chariton
Knowledge distillation is a technique that enables knowledge transfer from large, computationally expensive models to smaller ones without losing validity. This allows for deployment on less powerful
hardware, making evaluation faster and more efficient.
In this tutorial, we will run a number of experiments focused at improving the accuracy of a lightweight neural network, using a more powerful network as a teacher. The computational cost and the
speed of the lightweight network will remain unaffected, our intervention only focuses on its weights, not on its forward pass. Applications of this technology can be found in devices such as drones
or mobile phones. In this tutorial, we do not use any external packages as everything we need is available in torch and torchvision.
In this tutorial, you will learn:
• How to modify model classes to extract hidden representations and use them for further calculations
• How to modify regular train loops in PyTorch to include additional losses on top of, for example, cross-entropy for classification
• How to improve the performance of lightweight models by using more complex models as teachers
• 1 GPU, 4GB of memory
• PyTorch v2.0 or later
• CIFAR-10 dataset (downloaded by the script and saved in a directory called /data)
import torch
import torch.nn as nn
import torch.optim as optim
import torchvision.transforms as transforms
import torchvision.datasets as datasets
# Check if GPU is available, and if not, use the CPU
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
Loading CIFAR-10¶
CIFAR-10 is a popular image dataset with ten classes. Our objective is to predict one of the following classes for each input image.
The input images are RGB, so they have 3 channels and are 32x32 pixels. Basically, each image is described by 3 x 32 x 32 = 3072 numbers ranging from 0 to 255. A common practice in neural networks is
to normalize the input, which is done for multiple reasons, including avoiding saturation in commonly used activation functions and increasing numerical stability. Our normalization process consists
of subtracting the mean and dividing by the standard deviation along each channel. The tensors 《mean=[0.485, 0.456, 0.406]》 and 《std=[0.229, 0.224, 0.225]》 were already computed, and they
represent the mean and standard deviation of each channel in the predefined subset of CIFAR-10 intended to be the training set. Notice how we use these values for the test set as well, without
recomputing the mean and standard deviation from scratch. This is because the network was trained on features produced by subtracting and dividing the numbers above, and we want to maintain
consistency. Furthermore, in real life, we would not be able to compute the mean and standard deviation of the test set since, under our assumptions, this data would not be accessible at that point.
As a closing point, we often refer to this held-out set as the validation set, and we use a separate set, called the test set, after optimizing a model’s performance on the validation set. This is
done to avoid selecting a model based on the greedy and biased optimization of a single metric.
# Below we are preprocessing data for CIFAR-10. We use an arbitrary batch size of 128.
transforms_cifar = transforms.Compose([
transforms.Normalize(mean=[0.485, 0.456, 0.406], std=[0.229, 0.224, 0.225]),
# Loading the CIFAR-10 dataset:
train_dataset = datasets.CIFAR10(root='./data', train=True, download=True, transform=transforms_cifar)
test_dataset = datasets.CIFAR10(root='./data', train=False, download=True, transform=transforms_cifar)
Files already downloaded and verified
Files already downloaded and verified
This section is for CPU users only who are interested in quick results. Use this option only if you’re interested in a small scale experiment. Keep in mind the code should run fairly quickly using
any GPU. Select only the first num_images_to_keep images from the train/test dataset
#from torch.utils.data import Subset
#num_images_to_keep = 2000
#train_dataset = Subset(train_dataset, range(min(num_images_to_keep, 50_000)))
#test_dataset = Subset(test_dataset, range(min(num_images_to_keep, 10_000)))
train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=128, shuffle=True, num_workers=2)
test_loader = torch.utils.data.DataLoader(test_dataset, batch_size=128, shuffle=False, num_workers=2)
Defining model classes and utility functions¶
Next, we need to define our model classes. Several user-defined parameters need to be set here. We use two different architectures, keeping the number of filters fixed across our experiments to
ensure fair comparisons. Both architectures are Convolutional Neural Networks (CNNs) with a different number of convolutional layers that serve as feature extractors, followed by a classifier with 10
classes. The number of filters and neurons is smaller for the students.
# Deeper neural network class to be used as teacher:
class DeepNN(nn.Module):
def __init__(self, num_classes=10):
super(DeepNN, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 128, kernel_size=3, padding=1),
nn.Conv2d(128, 64, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(64, 64, kernel_size=3, padding=1),
nn.Conv2d(64, 32, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
self.classifier = nn.Sequential(
nn.Linear(2048, 512),
nn.Linear(512, num_classes)
def forward(self, x):
x = self.features(x)
x = torch.flatten(x, 1)
x = self.classifier(x)
return x
# Lightweight neural network class to be used as student:
class LightNN(nn.Module):
def __init__(self, num_classes=10):
super(LightNN, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(16, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
self.classifier = nn.Sequential(
nn.Linear(1024, 256),
nn.Linear(256, num_classes)
def forward(self, x):
x = self.features(x)
x = torch.flatten(x, 1)
x = self.classifier(x)
return x
We employ 2 functions to help us produce and evaluate the results on our original classification task. One function is called train and takes the following arguments:
• model: A model instance to train (update its weights) via this function.
• train_loader: We defined our train_loader above, and its job is to feed the data into the model.
• epochs: How many times we loop over the dataset.
• learning_rate: The learning rate determines how large our steps towards convergence should be. Too large or too small steps can be detrimental.
• device: Determines the device to run the workload on. Can be either CPU or GPU depending on availability.
Our test function is similar, but it will be invoked with test_loader to load images from the test set.
def train(model, train_loader, epochs, learning_rate, device):
criterion = nn.CrossEntropyLoss()
optimizer = optim.Adam(model.parameters(), lr=learning_rate)
for epoch in range(epochs):
running_loss = 0.0
for inputs, labels in train_loader:
# inputs: A collection of batch_size images
# labels: A vector of dimensionality batch_size with integers denoting class of each image
inputs, labels = inputs.to(device), labels.to(device)
outputs = model(inputs)
# outputs: Output of the network for the collection of images. A tensor of dimensionality batch_size x num_classes
# labels: The actual labels of the images. Vector of dimensionality batch_size
loss = criterion(outputs, labels)
running_loss += loss.item()
print(f"Epoch {epoch+1}/{epochs}, Loss: {running_loss / len(train_loader)}")
def test(model, test_loader, device):
correct = 0
total = 0
with torch.no_grad():
for inputs, labels in test_loader:
inputs, labels = inputs.to(device), labels.to(device)
outputs = model(inputs)
_, predicted = torch.max(outputs.data, 1)
total += labels.size(0)
correct += (predicted == labels).sum().item()
accuracy = 100 * correct / total
print(f"Test Accuracy: {accuracy:.2f}%")
return accuracy
Cross-entropy runs¶
For reproducibility, we need to set the torch manual seed. We train networks using different methods, so to compare them fairly, it makes sense to initialize the networks with the same weights. Start
by training the teacher network using cross-entropy:
nn_deep = DeepNN(num_classes=10).to(device)
train(nn_deep, train_loader, epochs=10, learning_rate=0.001, device=device)
test_accuracy_deep = test(nn_deep, test_loader, device)
# Instantiate the lightweight network:
nn_light = LightNN(num_classes=10).to(device)
Epoch 1/10, Loss: 1.3332555117204672
Epoch 2/10, Loss: 0.8745550930957355
Epoch 3/10, Loss: 0.6840274586244617
Epoch 4/10, Loss: 0.5346566625415822
Epoch 5/10, Loss: 0.41133782926880186
Epoch 6/10, Loss: 0.30730283751969445
Epoch 7/10, Loss: 0.22276928332989174
Epoch 8/10, Loss: 0.16607769419584434
Epoch 9/10, Loss: 0.13346084133933878
Epoch 10/10, Loss: 0.11983871535228952
Test Accuracy: 75.12%
We instantiate one more lightweight network model to compare their performances. Back propagation is sensitive to weight initialization, so we need to make sure these two networks have the exact same
new_nn_light = LightNN(num_classes=10).to(device)
To ensure we have created a copy of the first network, we inspect the norm of its first layer. If it matches, then we are safe to conclude that the networks are indeed the same.
# Print the norm of the first layer of the initial lightweight model
print("Norm of 1st layer of nn_light:", torch.norm(nn_light.features[0].weight).item())
# Print the norm of the first layer of the new lightweight model
print("Norm of 1st layer of new_nn_light:", torch.norm(new_nn_light.features[0].weight).item())
Norm of 1st layer of nn_light: 2.327361822128296
Norm of 1st layer of new_nn_light: 2.327361822128296
Print the total number of parameters in each model:
total_params_deep = "{:,}".format(sum(p.numel() for p in nn_deep.parameters()))
print(f"DeepNN parameters: {total_params_deep}")
total_params_light = "{:,}".format(sum(p.numel() for p in nn_light.parameters()))
print(f"LightNN parameters: {total_params_light}")
DeepNN parameters: 1,186,986
LightNN parameters: 267,738
Train and test the lightweight network with cross entropy loss:
train(nn_light, train_loader, epochs=10, learning_rate=0.001, device=device)
test_accuracy_light_ce = test(nn_light, test_loader, device)
Epoch 1/10, Loss: 1.4686825074198302
Epoch 2/10, Loss: 1.1565482149953428
Epoch 3/10, Loss: 1.0267558780777486
Epoch 4/10, Loss: 0.92361319263268
Epoch 5/10, Loss: 0.8484364450740083
Epoch 6/10, Loss: 0.7808415658028839
Epoch 7/10, Loss: 0.7161723332636801
Epoch 8/10, Loss: 0.6607577864013975
Epoch 9/10, Loss: 0.6044853925704956
Epoch 10/10, Loss: 0.55508375297422
Test Accuracy: 70.50%
As we can see, based on test accuracy, we can now compare the deeper network that is to be used as a teacher with the lightweight network that is our supposed student. So far, our student has not
intervened with the teacher, therefore this performance is achieved by the student itself. The metrics so far can be seen with the following lines:
print(f"Teacher accuracy: {test_accuracy_deep:.2f}%")
print(f"Student accuracy: {test_accuracy_light_ce:.2f}%")
Teacher accuracy: 75.12%
Student accuracy: 70.50%
Knowledge distillation run¶
Now let’s try to improve the test accuracy of the student network by incorporating the teacher. Knowledge distillation is a straightforward technique to achieve this, based on the fact that both
networks output a probability distribution over our classes. Therefore, the two networks share the same number of output neurons. The method works by incorporating an additional loss into the
traditional cross entropy loss, which is based on the softmax output of the teacher network. The assumption is that the output activations of a properly trained teacher network carry additional
information that can be leveraged by a student network during training. The original work suggests that utilizing ratios of smaller probabilities in the soft targets can help achieve the underlying
objective of deep neural networks, which is to create a similarity structure over the data where similar objects are mapped closer together. For example, in CIFAR-10, a truck could be mistaken for an
automobile or airplane, if its wheels are present, but it is less likely to be mistaken for a dog. Therefore, it makes sense to assume that valuable information resides not only in the top prediction
of a properly trained model but in the entire output distribution. However, cross entropy alone does not sufficiently exploit this information as the activations for non-predicted classes tend to be
so small that propagated gradients do not meaningfully change the weights to construct this desirable vector space.
As we continue defining our first helper function that introduces a teacher-student dynamic, we need to include a few extra parameters:
• T: Temperature controls the smoothness of the output distributions. Larger T leads to smoother distributions, thus smaller probabilities get a larger boost.
• soft_target_loss_weight: A weight assigned to the extra objective we’re about to include.
• ce_loss_weight: A weight assigned to cross-entropy. Tuning these weights pushes the network towards optimizing for either objective.
def train_knowledge_distillation(teacher, student, train_loader, epochs, learning_rate, T, soft_target_loss_weight, ce_loss_weight, device):
ce_loss = nn.CrossEntropyLoss()
optimizer = optim.Adam(student.parameters(), lr=learning_rate)
teacher.eval() # Teacher set to evaluation mode
student.train() # Student to train mode
for epoch in range(epochs):
running_loss = 0.0
for inputs, labels in train_loader:
inputs, labels = inputs.to(device), labels.to(device)
# Forward pass with the teacher model - do not save gradients here as we do not change the teacher's weights
with torch.no_grad():
teacher_logits = teacher(inputs)
# Forward pass with the student model
student_logits = student(inputs)
#Soften the student logits by applying softmax first and log() second
soft_targets = nn.functional.softmax(teacher_logits / T, dim=-1)
soft_prob = nn.functional.log_softmax(student_logits / T, dim=-1)
# Calculate the soft targets loss. Scaled by T**2 as suggested by the authors of the paper "Distilling the knowledge in a neural network"
soft_targets_loss = torch.sum(soft_targets * (soft_targets.log() - soft_prob)) / soft_prob.size()[0] * (T**2)
# Calculate the true label loss
label_loss = ce_loss(student_logits, labels)
# Weighted sum of the two losses
loss = soft_target_loss_weight * soft_targets_loss + ce_loss_weight * label_loss
running_loss += loss.item()
print(f"Epoch {epoch+1}/{epochs}, Loss: {running_loss / len(train_loader)}")
# Apply ``train_knowledge_distillation`` with a temperature of 2. Arbitrarily set the weights to 0.75 for CE and 0.25 for distillation loss.
train_knowledge_distillation(teacher=nn_deep, student=new_nn_light, train_loader=train_loader, epochs=10, learning_rate=0.001, T=2, soft_target_loss_weight=0.25, ce_loss_weight=0.75, device=device)
test_accuracy_light_ce_and_kd = test(new_nn_light, test_loader, device)
# Compare the student test accuracy with and without the teacher, after distillation
print(f"Teacher accuracy: {test_accuracy_deep:.2f}%")
print(f"Student accuracy without teacher: {test_accuracy_light_ce:.2f}%")
print(f"Student accuracy with CE + KD: {test_accuracy_light_ce_and_kd:.2f}%")
Epoch 1/10, Loss: 2.404144715775004
Epoch 2/10, Loss: 1.890097956218378
Epoch 3/10, Loss: 1.6662996138453179
Epoch 4/10, Loss: 1.5062101808045527
Epoch 5/10, Loss: 1.3777181861345724
Epoch 6/10, Loss: 1.2645151011474298
Epoch 7/10, Loss: 1.163147846451196
Epoch 8/10, Loss: 1.0817804188679552
Epoch 9/10, Loss: 1.0054900307789483
Epoch 10/10, Loss: 0.937642643823648
Test Accuracy: 70.87%
Teacher accuracy: 75.12%
Student accuracy without teacher: 70.50%
Student accuracy with CE + KD: 70.87%
Cosine loss minimization run¶
Feel free to play around with the temperature parameter that controls the softness of the softmax function and the loss coefficients. In neural networks, it is easy to include to include additional
loss functions to the main objectives to achieve goals like better generalization. Let’s try including an objective for the student, but now let’s focus on their hidden states rather than their
output layers. Our goal is to convey information from the teacher’s representation to the student by including a naive loss function, whose minimization implies that the flattened vectors that are
subsequently passed to the classifiers have become more similar as the loss decreases. Of course, the teacher does not update its weights, so the minimization depends only on the student’s weights.
The rationale behind this method is that we are operating under the assumption that the teacher model has a better internal representation that is unlikely to be achieved by the student without
external intervention, therefore we artificially push the student to mimic the internal representation of the teacher. Whether or not this will end up helping the student is not straightforward,
though, because pushing the lightweight network to reach this point could be a good thing, assuming that we have found an internal representation that leads to better test accuracy, but it could also
be harmful because the networks have different architectures and the student does not have the same learning capacity as the teacher. In other words, there is no reason for these two vectors, the
student’s and the teacher’s to match per component. The student could reach an internal representation that is a permutation of the teacher’s and it would be just as efficient. Nonetheless, we can
still run a quick experiment to figure out the impact of this method. We will be using the CosineEmbeddingLoss which is given by the following formula:
Obviously, there is one thing that we need to resolve first. When we applied distillation to the output layer we mentioned that both networks have the same number of neurons, equal to the number of
classes. However, this is not the case for the layer following our convolutional layers. Here, the teacher has more neurons than the student after the flattening of the final convolutional layer. Our
loss function accepts two vectors of equal dimensionality as inputs, therefore we need to somehow match them. We will solve this by including an average pooling layer after the teacher’s
convolutional layer to reduce its dimensionality to match that of the student.
To proceed, we will modify our model classes, or create new ones. Now, the forward function returns not only the logits of the network but also the flattened hidden representation after the
convolutional layer. We include the aforementioned pooling for the modified teacher.
class ModifiedDeepNNCosine(nn.Module):
def __init__(self, num_classes=10):
super(ModifiedDeepNNCosine, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 128, kernel_size=3, padding=1),
nn.Conv2d(128, 64, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(64, 64, kernel_size=3, padding=1),
nn.Conv2d(64, 32, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
self.classifier = nn.Sequential(
nn.Linear(2048, 512),
nn.Linear(512, num_classes)
def forward(self, x):
x = self.features(x)
flattened_conv_output = torch.flatten(x, 1)
x = self.classifier(flattened_conv_output)
flattened_conv_output_after_pooling = torch.nn.functional.avg_pool1d(flattened_conv_output, 2)
return x, flattened_conv_output_after_pooling
# Create a similar student class where we return a tuple. We do not apply pooling after flattening.
class ModifiedLightNNCosine(nn.Module):
def __init__(self, num_classes=10):
super(ModifiedLightNNCosine, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(16, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
self.classifier = nn.Sequential(
nn.Linear(1024, 256),
nn.Linear(256, num_classes)
def forward(self, x):
x = self.features(x)
flattened_conv_output = torch.flatten(x, 1)
x = self.classifier(flattened_conv_output)
return x, flattened_conv_output
# We do not have to train the modified deep network from scratch of course, we just load its weights from the trained instance
modified_nn_deep = ModifiedDeepNNCosine(num_classes=10).to(device)
# Once again ensure the norm of the first layer is the same for both networks
print("Norm of 1st layer for deep_nn:", torch.norm(nn_deep.features[0].weight).item())
print("Norm of 1st layer for modified_deep_nn:", torch.norm(modified_nn_deep.features[0].weight).item())
# Initialize a modified lightweight network with the same seed as our other lightweight instances. This will be trained from scratch to examine the effectiveness of cosine loss minimization.
modified_nn_light = ModifiedLightNNCosine(num_classes=10).to(device)
print("Norm of 1st layer:", torch.norm(modified_nn_light.features[0].weight).item())
Norm of 1st layer for deep_nn: 7.509754657745361
Norm of 1st layer for modified_deep_nn: 7.509754657745361
Norm of 1st layer: 2.327361822128296
Naturally, we need to change the train loop because now the model returns a tuple (logits, hidden_representation). Using a sample input tensor we can print their shapes.
# Create a sample input tensor
sample_input = torch.randn(128, 3, 32, 32).to(device) # Batch size: 128, Filters: 3, Image size: 32x32
# Pass the input through the student
logits, hidden_representation = modified_nn_light(sample_input)
# Print the shapes of the tensors
print("Student logits shape:", logits.shape) # batch_size x total_classes
print("Student hidden representation shape:", hidden_representation.shape) # batch_size x hidden_representation_size
# Pass the input through the teacher
logits, hidden_representation = modified_nn_deep(sample_input)
# Print the shapes of the tensors
print("Teacher logits shape:", logits.shape) # batch_size x total_classes
print("Teacher hidden representation shape:", hidden_representation.shape) # batch_size x hidden_representation_size
Student logits shape: torch.Size([128, 10])
Student hidden representation shape: torch.Size([128, 1024])
Teacher logits shape: torch.Size([128, 10])
Teacher hidden representation shape: torch.Size([128, 1024])
In our case, hidden_representation_size is 1024. This is the flattened feature map of the final convolutional layer of the student and as you can see, it is the input for its classifier. It is 1024
for the teacher too, because we made it so with avg_pool1d from 2048. The loss applied here only affects the weights of the student prior to the loss calculation. In other words, it does not affect
the classifier of the student. The modified training loop is the following:
def train_cosine_loss(teacher, student, train_loader, epochs, learning_rate, hidden_rep_loss_weight, ce_loss_weight, device):
ce_loss = nn.CrossEntropyLoss()
cosine_loss = nn.CosineEmbeddingLoss()
optimizer = optim.Adam(student.parameters(), lr=learning_rate)
teacher.eval() # Teacher set to evaluation mode
student.train() # Student to train mode
for epoch in range(epochs):
running_loss = 0.0
for inputs, labels in train_loader:
inputs, labels = inputs.to(device), labels.to(device)
# Forward pass with the teacher model and keep only the hidden representation
with torch.no_grad():
_, teacher_hidden_representation = teacher(inputs)
# Forward pass with the student model
student_logits, student_hidden_representation = student(inputs)
# Calculate the cosine loss. Target is a vector of ones. From the loss formula above we can see that is the case where loss minimization leads to cosine similarity increase.
hidden_rep_loss = cosine_loss(student_hidden_representation, teacher_hidden_representation, target=torch.ones(inputs.size(0)).to(device))
# Calculate the true label loss
label_loss = ce_loss(student_logits, labels)
# Weighted sum of the two losses
loss = hidden_rep_loss_weight * hidden_rep_loss + ce_loss_weight * label_loss
running_loss += loss.item()
print(f"Epoch {epoch+1}/{epochs}, Loss: {running_loss / len(train_loader)}")
We need to modify our test function for the same reason. Here we ignore the hidden representation returned by the model.
def test_multiple_outputs(model, test_loader, device):
correct = 0
total = 0
with torch.no_grad():
for inputs, labels in test_loader:
inputs, labels = inputs.to(device), labels.to(device)
outputs, _ = model(inputs) # Disregard the second tensor of the tuple
_, predicted = torch.max(outputs.data, 1)
total += labels.size(0)
correct += (predicted == labels).sum().item()
accuracy = 100 * correct / total
print(f"Test Accuracy: {accuracy:.2f}%")
return accuracy
In this case, we could easily include both knowledge distillation and cosine loss minimization in the same function. It is common to combine methods to achieve better performance in teacher-student
paradigms. For now, we can run a simple train-test session.
# Train and test the lightweight network with cross entropy loss
train_cosine_loss(teacher=modified_nn_deep, student=modified_nn_light, train_loader=train_loader, epochs=10, learning_rate=0.001, hidden_rep_loss_weight=0.25, ce_loss_weight=0.75, device=device)
test_accuracy_light_ce_and_cosine_loss = test_multiple_outputs(modified_nn_light, test_loader, device)
Epoch 1/10, Loss: 1.3059198783181818
Epoch 2/10, Loss: 1.0764762134198338
Epoch 3/10, Loss: 0.9798224342753515
Epoch 4/10, Loss: 0.9039881669956705
Epoch 5/10, Loss: 0.848508194889254
Epoch 6/10, Loss: 0.8017997186811988
Epoch 7/10, Loss: 0.7592443207950543
Epoch 8/10, Loss: 0.7258237780207564
Epoch 9/10, Loss: 0.6868311371034979
Epoch 10/10, Loss: 0.6615204779845675
Test Accuracy: 69.69%
Intermediate regressor run¶
Our naive minimization does not guarantee better results for several reasons, one being the dimensionality of the vectors. Cosine similarity generally works better than Euclidean distance for vectors
of higher dimensionality, but we were dealing with vectors with 1024 components each, so it is much harder to extract meaningful similarities. Furthermore, as we mentioned, pushing towards a match of
the hidden representation of the teacher and the student is not supported by theory. There are no good reasons why we should be aiming for a 1:1 match of these vectors. We will provide a final
example of training intervention by including an extra network called regressor. The objective is to first extract the feature map of the teacher after a convolutional layer, then extract a feature
map of the student after a convolutional layer, and finally try to match these maps. However, this time, we will introduce a regressor between the networks to facilitate the matching process. The
regressor will be trainable and ideally will do a better job than our naive cosine loss minimization scheme. Its main job is to match the dimensionality of these feature maps so that we can properly
define a loss function between the teacher and the student. Defining such a loss function provides a teaching 《path,》 which is basically a flow to back-propagate gradients that will change the
student’s weights. Focusing on the output of the convolutional layers right before each classifier for our original networks, we have the following shapes:
# Pass the sample input only from the convolutional feature extractor
convolutional_fe_output_student = nn_light.features(sample_input)
convolutional_fe_output_teacher = nn_deep.features(sample_input)
# Print their shapes
print("Student's feature extractor output shape: ", convolutional_fe_output_student.shape)
print("Teacher's feature extractor output shape: ", convolutional_fe_output_teacher.shape)
Student's feature extractor output shape: torch.Size([128, 16, 8, 8])
Teacher's feature extractor output shape: torch.Size([128, 32, 8, 8])
We have 32 filters for the teacher and 16 filters for the student. We will include a trainable layer that converts the feature map of the student to the shape of the feature map of the teacher. In
practice, we modify the lightweight class to return the hidden state after an intermediate regressor that matches the sizes of the convolutional feature maps and the teacher class to return the
output of the final convolutional layer without pooling or flattening.
class ModifiedDeepNNRegressor(nn.Module):
def __init__(self, num_classes=10):
super(ModifiedDeepNNRegressor, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 128, kernel_size=3, padding=1),
nn.Conv2d(128, 64, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(64, 64, kernel_size=3, padding=1),
nn.Conv2d(64, 32, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
self.classifier = nn.Sequential(
nn.Linear(2048, 512),
nn.Linear(512, num_classes)
def forward(self, x):
x = self.features(x)
conv_feature_map = x
x = torch.flatten(x, 1)
x = self.classifier(x)
return x, conv_feature_map
class ModifiedLightNNRegressor(nn.Module):
def __init__(self, num_classes=10):
super(ModifiedLightNNRegressor, self).__init__()
self.features = nn.Sequential(
nn.Conv2d(3, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(16, 16, kernel_size=3, padding=1),
nn.MaxPool2d(kernel_size=2, stride=2),
# Include an extra regressor (in our case linear)
self.regressor = nn.Sequential(
nn.Conv2d(16, 32, kernel_size=3, padding=1)
self.classifier = nn.Sequential(
nn.Linear(1024, 256),
nn.Linear(256, num_classes)
def forward(self, x):
x = self.features(x)
regressor_output = self.regressor(x)
x = torch.flatten(x, 1)
x = self.classifier(x)
return x, regressor_output
After that, we have to update our train loop again. This time, we extract the regressor output of the student, the feature map of the teacher, we calculate the MSE on these tensors (they have the
exact same shape so it’s properly defined) and we back propagate gradients based on that loss, in addition to the regular cross entropy loss of the classification task.
def train_mse_loss(teacher, student, train_loader, epochs, learning_rate, feature_map_weight, ce_loss_weight, device):
ce_loss = nn.CrossEntropyLoss()
mse_loss = nn.MSELoss()
optimizer = optim.Adam(student.parameters(), lr=learning_rate)
teacher.eval() # Teacher set to evaluation mode
student.train() # Student to train mode
for epoch in range(epochs):
running_loss = 0.0
for inputs, labels in train_loader:
inputs, labels = inputs.to(device), labels.to(device)
# Again ignore teacher logits
with torch.no_grad():
_, teacher_feature_map = teacher(inputs)
# Forward pass with the student model
student_logits, regressor_feature_map = student(inputs)
# Calculate the loss
hidden_rep_loss = mse_loss(regressor_feature_map, teacher_feature_map)
# Calculate the true label loss
label_loss = ce_loss(student_logits, labels)
# Weighted sum of the two losses
loss = feature_map_weight * hidden_rep_loss + ce_loss_weight * label_loss
running_loss += loss.item()
print(f"Epoch {epoch+1}/{epochs}, Loss: {running_loss / len(train_loader)}")
# Notice how our test function remains the same here with the one we used in our previous case. We only care about the actual outputs because we measure accuracy.
# Initialize a ModifiedLightNNRegressor
modified_nn_light_reg = ModifiedLightNNRegressor(num_classes=10).to(device)
# We do not have to train the modified deep network from scratch of course, we just load its weights from the trained instance
modified_nn_deep_reg = ModifiedDeepNNRegressor(num_classes=10).to(device)
# Train and test once again
train_mse_loss(teacher=modified_nn_deep_reg, student=modified_nn_light_reg, train_loader=train_loader, epochs=10, learning_rate=0.001, feature_map_weight=0.25, ce_loss_weight=0.75, device=device)
test_accuracy_light_ce_and_mse_loss = test_multiple_outputs(modified_nn_light_reg, test_loader, device)
Epoch 1/10, Loss: 1.7593588859528837
Epoch 2/10, Loss: 1.3730355287756761
Epoch 3/10, Loss: 1.2217909899514046
Epoch 4/10, Loss: 1.1249605873051811
Epoch 5/10, Loss: 1.0448348395659794
Epoch 6/10, Loss: 0.9812982126574992
Epoch 7/10, Loss: 0.9246600782474899
Epoch 8/10, Loss: 0.8732889065962008
Epoch 9/10, Loss: 0.8284777469952088
Epoch 10/10, Loss: 0.7902423676932254
Test Accuracy: 70.69%
It is expected that the final method will work better than CosineLoss because now we have allowed a trainable layer between the teacher and the student, which gives the student some wiggle room when
it comes to learning, rather than pushing the student to copy the teacher’s representation. Including the extra network is the idea behind hint-based distillation.
print(f"Teacher accuracy: {test_accuracy_deep:.2f}%")
print(f"Student accuracy without teacher: {test_accuracy_light_ce:.2f}%")
print(f"Student accuracy with CE + KD: {test_accuracy_light_ce_and_kd:.2f}%")
print(f"Student accuracy with CE + CosineLoss: {test_accuracy_light_ce_and_cosine_loss:.2f}%")
print(f"Student accuracy with CE + RegressorMSE: {test_accuracy_light_ce_and_mse_loss:.2f}%")
Teacher accuracy: 75.12%
Student accuracy without teacher: 70.50%
Student accuracy with CE + KD: 70.87%
Student accuracy with CE + CosineLoss: 69.69%
Student accuracy with CE + RegressorMSE: 70.69%
None of the methods above increases the number of parameters for the network or inference time, so the performance increase comes at the little cost of calculating gradients during training. In ML
applications, we mostly care about inference time because training happens before the model deployment. If our lightweight model is still too heavy for deployment, we can apply different ideas, such
as post-training quantization. Additional losses can be applied in many tasks, not just classification, and you can experiment with quantities like coefficients, temperature, or number of neurons.
Feel free to tune any numbers in the tutorial above, but keep in mind, if you change the number of neurons / filters chances are a shape mismatch might occur.
For more information, see:
Total running time of the script: ( 2 minutes 54.454 seconds) | {"url":"https://tutorials.pytorch.kr/beginner/knowledge_distillation_tutorial.html","timestamp":"2024-11-03T06:06:20Z","content_type":"text/html","content_length":"171857","record_id":"<urn:uuid:e7eef3ce-d6ac-4234-9eea-b68f33f76d15>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00208.warc.gz"} |
Analysis I - Real Analysis
Construction of the field of real numbers and the least upper-bound property. Review of sets, countable & uncountable sets. Metric Spaces: topological properties, the topology of Euclidean space.
Sequences and series. Continuity: definition and basic theorems, uniform continuity, the Intermediate Value Theorem. Differentiability on the real line: definition, the Mean Value Theorem. The
Riemann-Stieltjes integral: definition and examples, the Fundamental Theorem of Calculus. Sequences and series of functions, uniform convergence, the Weierstrass Approximation Theorem.
Differentiability in higher dimensions: motivations, the total derivative, and basic theorems. Partial derivatives, characterization of continuously-differentiable functions. The Inverse and Implicit
Function Theorems. Higher-order derivatives. | {"url":"https://math.iisc.ac.in/all-courses/ma221.html","timestamp":"2024-11-02T10:47:00Z","content_type":"text/html","content_length":"16772","record_id":"<urn:uuid:3b3516d5-177d-4aa3-b77b-d489c9a9c923>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00045.warc.gz"} |
reflection calculator x axis
And we want this positive 3 In standard reflections, we reflect over a line, like the y-axis or the x-axis. And so in general, that flips it over the y-axis. Conceptually, a reflection is basically a
'flip' of a shape over the line Direct link to David Severin's post It helps me to compare it, Posted 6 years ago. X-axis goes left and right, when reflecting you will need to go up or down depending
on the quadrant. Any points on the y-axis stay on the y-axis; it's the points off the axis that switch sides. equal to? flip it over the x-axis. like this. Check whether the coordinates are working
or not by plugging them into the equation of the reflecting line. Let's check our answer. It demands a time commitment which makes it integral to professional development. to happen when I do that?
Pick your course now. I'm just switching to this of everywhere you saw an x before you replaced We don't have to do this just Step 1: Know that we're reflecting across the x-axis. Some of the common
examples include the reflection of light, sound, and water waves. Direct link to Hecretary Bird's post As far as I know, most ca, Posted 3 years ago. just like that. 6 comma negative 7 is reflec--
this should say And we can represent it by Direct link to Anant Sogani's post We need an _m x n_ matrix, Posted 9 years ago. Only one step away from your solution of order no. $. say it's mapped to
if you want to use the language that I used okay, well let's up take to see if we could take (-3, -4 ) \rightarrow (-3 , \red{4}) Whatever you'd gotten for x-values on the positive (or right-hand)
side of the graph, you're now getting for x-values on the negative (or left-hand) side of the graph, and vice versa. For this transformation, I'll switch to a cubic function, being g(x) = x3 + x2 3x
1. So no surprise there, g of x was graphed right on top of f of x. So when you flip it, it looks like this. it'll be twice as tall, so it'll look like this. We can reflect the graph of y=f(x) over
the x-axis by graphing y=-f(x) and over the y-axis by graphing y=f(-x). What , Posted 4 years ago. to the negative of f of x and we get that. Then the new graph, being the graph of h(x), looks like
this: Flipping a function upside-down always works this way: you slap a "minus" on the whole thing. $, $ Find the vertices of triangle A'B'C' after a reflection across the x-axis. So If I were to
flip a polynomial over the y-axis say x^4+2x^3-4x^2+3x+4 it would become -x^4-2x^3+4x^2-3x+4 correct? Reflection-in-action includes the power of observation, analysis, and touch or feel the problem
to fix. It traces out f of x. m \overline{C'A'} = 5 And I think you're already And I kind of switch $, $ Each individual number in the matrix is called an element or entry. evaluate the principle
root of and we know that the And we know that the set in R2 f(x b) shifts the function b units to the right. Now, how would I flip it over the x-axis? The general rule for a reflection over the
x-axis: $ Unlock more options the more you use StudyPug. comparing between g(x) and y = -x^2, the y value is -1 as opposed to -4, and -1 is 1/4 of -4 so that's the scale. The incident light ray which
touches the plane is said to be reflected off the surface. Now! That's going to be equal to e to the, instead of putting an x there, we will put a negative x. reflection across the y-axis. That does
not apply when, let's say, an nth (i.e a square) root or an absolute value is in between it, like for k(x). So if you apply the If you're seeing this message, it means we're having trouble loading
external resources on our website. transformation, T, becomes minus 3, 4. Get $30 referral bonus and Earn 10% COMMISSION on all your friend's order for life! outside the radical sign, and then, I'm
gonna take the square root, and I'm gonna put a negative So what we want is, this point, When we graph this function, we get the line shown in the following graph: Now, we can perform two different
transformations on the function $latex f(x)$ to obtain the following functions: If we plot functions (i) and (ii) together with the original function $latex f(x)$, we have: In case (i), the graph of
the original function $latex f(x)$ has been reflected over the x-axis. That is when they're multiplied directly against each other. And notice, it did exactly what we expect. So as we just talk
through doing to the x1 term. So we've plotted Just like that. Choose 1 answer: A A A A A B B B B B C C C C C D D D D D E E E E E Stuck? So let's take our transformation We can describe it as a
following transformation r(y=x)? This fixed line is called the line of reflection. Conic Sections: Parabola and Focus. What I just drew here. you imagine that this is some type of a lake, How are
they related to each other? that was a minus 3 in the x-coordinate right there, we Now, we can see that the graph of $latex f(x)=\cos(2x)$ has symmetry about the y-axis. The transformation of
functions is the changes that we can apply to a function to modify its graph. Direct link to Bernardo Hagen's post why is a function f(-x) a. This aspect of reflections is helpful because you can
often tell if your transformation is correct based on how it looks. Follow the below-mentioned procedures for the necessary guidance: If you face difficulties in understanding this phenomenon, feel
free to connect with our experts having sound knowledge of reflection calculator geometry. For having access to more examples, resort to the expert assignment writers of MyAssignmenthelp.com. of some
vector, x, y. Its formula is: r=i. If I had multiple terms, if this 2 is just 0. we have here-- so this next step here is whatever It would get you to How would you reflect a point over the line y=
-x? 3, 2. the point 8 comma 5. formed by the points, let's say the first point Multiply all inputs by -1 for a horizontal reflection. The reflection law states that the angle of reflection is always
the same as the angle of incidence. And we know that if we take On our green function, And when all else fails, just fold the sheet of paper along the mirror line and then hold it up to the light !
Times x, y. both the x and y-axis. to be equal to-- I want to take minus 1 times the x, so It is common to label each corner with letters, and to use a little dash (called a Prime) to mark each
corner of the reflected image. negative 7, so we're going to go 6 to the you can basically just take g(1) divided by f(1) (-1 divided by 4) and it'll be the scale (-1/4). Minus 3, 2. The previous
reflection was a reflection in the x -axis. Here the original is ABC and the reflected image is A'B'C', When the mirror line is the y-axis Direct link to zjleon2010's post at 4:45, the script say ',
Posted 4 years ago. Reflections are opposite isometries, something we will look below. A matrix is a rectangular array of numbers arranged in rows and columns. So the next thing I want to do How to
Find the Axis of Symmetry: Finding the axis of symmetry, like plotting the reflections themselves, is also a simple process. en. negative 6 comma 5, and then reflect across the y. We can do a lot
with equations. They can either shrink Specifies the points that Let's say that f of x, let's give it a nice, You can tell, Posted 3 years ago. First, lets start with a reflection geometry
definition: A reflection of a point, a line, or a figure in the X axis involved reflecting the image over the x axis to create a mirror image. Highly reflection across the y-axis. Point reflection
calculator : This calculator enables you to find the reflection point for the given coordinates. diagonal matrices. We've talked a lot about Henceforth, it demands a lot of clinical reasoning, as in
the patient interaction. up matrix-vector product. And so, that's why this is now defined. what if you were reflecting over a line like y = 3. A negative a reflects it, and if 01, it vertically
stretches the parabola. A reflection over the x-axis can be seen in the picture below in which point A is reflected to its image A'. Let dis equal the horizontal distance covered by the light between
reflections off either mirror. See how well your practice sessions are going over time. coordinate, but we're used to dealing with the y coordinate It will help you to develop the slope-intercept
form for the equation of the line. Notice that the y-coordinate for both points did not change, but the value of the x-coordinate changed from 5 to -5. If you're behind a web filter, please make sure
that the domains *.kastatic.org and *.kasandbox.org are unblocked. For example, in this video, y1 (when x = 1) = 1 and y2 = -1/4, so -1/4/1 gives -1/4. I want to make it 2 times specified by a set of
vectors. Now, by counting the distance between these two points, you should get the answer of 2 units. 2, times this point right here, which is 3, minus 2. So I'll do each of these. So you start off
with the Direct link to vtx's post comparing between g(x) an. First up, I'll put a "minus" on the argument of the function: Putting a "minus" on the argument reflects the graph in the y-axis. In this
activity, students explore reflections over the x-axis and y-axis, with an emphasis on how the coordinates of the pre-image and image are related. In real life, we think of a reflection as a mirror
image, like when we look at own reflection in the mirror. So let's do these in steps. A point reflection is just a type of reflection. \\ Reflect the triangle over the x-axis and then over the y-axis
1. To see how this works, take a look at the graph of h(x) = x2 + 2x 3. Posted 5 years ago. How can you solve the problem if you don't have the graph to help you? videos ago. Shouldn't -f(x) the
inverse of f(x) be y = -(x^2) instead of -x^2 because -2^2 = 2^2 (so if x = 2 | x = -2, y = 4 in both cases). Finding the Coordinates of a Point Reflected Across an Axis. Some simple reflections can
be performed easily in the coordinate plane using the general rules below. Y when is X is equal to negative two instead of Y being equal to four, it would now be equal to negative four. been
legitimate if we said the y-axis We flipped it over, so that we Lesson 13: Transforming quadratic functions. The same is true at 4 which is down 4 (which is 1/4 of the parent function which would be
at 16 (4^2=16). It is one unit up from the line, so go over one unit on the x-axis and drop down one unit. Where/How did he get 1/4? (A,B) \rightarrow (-A, B) Whenever we gaze at a mirror or blink at
the sunlight glinting from a lake, we see a reflection. negative of f of negative x and you would've gotten But that by itself does everything else is 0's all the way down. Yes, MyAssignmenthelp.com
experts possess a solid understanding of the intricacies associated with reflection rules in geometry. When x is one, instead of one now, you're taking the negative of it so you're gonna get negative
one. When the function of f(x) and -f(x) were plotted on the same graph and f(x) was equal to sqrt(x),a parabola formed. Learning about the reflection of functions over the x-axis and y-axis. You can
still navigate around the site and check out our free content, but some functionality, such as sign up, will not work. get the opposite of it. So first let's flip over, flip over the x-axis. However,
you need to understand its usage at the beginning. height we have here-- I want it to be 2 times as much. This is 3, 4. Since the inputs switched sides, so also does the graph. And so essentially you
just see if we scale by 1/4, does that do the trick? This is always true: g(x) is the mirror image of g(x); plugging in the "minus" of the argument gives you a graph that is the original reflected in
the y-axis. So the y-coordinate point across the y-axis, it would go all the Direct link to hdalaq's post I have a question, how do, Posted 11 years ago. 2. So you may see a form such as y=a (bx-c)^2
+ d. The parabola is translated (c,d) units, b reflects across y, but this just reflects it across the axis of symmetry, so it would look the same. And this is a really useful matrix. So what I
envision, we're instead of squaring one and getting one, you then Without necessarily And then if I reflected that Now on our green function, For this transformation, I'll switch to a cubic function,
being g(x) = x3 + x2 3x 1. Like other functions, f (x) = a g (bx), if a is negative (outside) it reflects across x axis and if b is negative it reflects across the y axis. You give an example of a
reflection over an axis - can you work through an example reflecting a shape (using linear algebra) over a non-axis line, please? You have to draw a normal line that is perpendicular to the
reflecting surface for calculating the angle of incidence and the angle of reflection. because it's negative, and then we've gone 5 up, the set of all of the positions or all of the position With our
services in place, you can be assured of getting the solutions within the stipulated time frame. So let's start with some I could call that our x2 7 is right there. A reflection over the x-axis can
be seen in the picture below in which point A is reflected to its image A'. The reflected ray is the one that bounces back. it over the x-axis. function would've taken on at a given value of x, Let's
look at this point right that it works. want this point to have its same y-coordinate.
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reflection calculator x axis | {"url":"https://mudontheshoes.de/dsupkw3c/reflection-calculator-x-axis","timestamp":"2024-11-12T01:09:15Z","content_type":"text/html","content_length":"84936","record_id":"<urn:uuid:e8508a72-bd31-4b6c-96ad-65ffdec3be6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00678.warc.gz"} |
Johannes Kepler
Johannes Kepler
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Johannes Kepler
A 1610 portrait of Johannes Kepler by an unknown artist
Born December 27, 1571
Free Imperial City of Weil der Stadt near Stuttgart, HRE (now part of the Stuttgart Region of Baden-Württemberg, Germany)
Died November 15, 1630 (aged 58)
Regensburg, Electorate of Bavaria, HRE (now Germany)
Residence Germany
Nationality German
Fields Astronomy, astrology, mathematics and natural philosophy
Institutions University of Linz
Alma mater University of Tübingen
Known for Kepler's laws of planetary motion
Kepler conjecture
Johannes Kepler (German: [ˈkʰɛplɐ]; December 27 1571 – November 15 1630) was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best
known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided
one of the foundations for Isaac Newton's theory of universal gravitation.
During his career, Kepler was a mathematics teacher at a seminary school in Graz, Austria, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to astronomer
Tycho Brahe, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He was also a mathematics teacher in Linz, Austria, and an adviser to
General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting telescope (the Keplerian Telescope), and mentioned the telescopic
discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy (a branch of mathematics within the liberal arts) and
physics (a branch of natural philosophy). Kepler also incorporated religious arguments and reasoning into his work, motivated by the religious conviction and belief that God had created the world
according to an intelligible plan that is accessible through the natural light of reason. Kepler described his new astronomy as "celestial physics", as "an excursion into Aristotle's Metaphysics",
and as "a supplement to Aristotle's On the Heavens", transforming the ancient tradition of physical cosmology by treating astronomy as part of a universal mathematical physics.
Early years
Johannes Kepler was born on December 27, the feast day of St. John the Evangelist, 1571, at the Free Imperial City of Weil der Stadt (now part of the Stuttgart Region in the German state of
Baden-Württemberg, 30 km west of Stuttgart's centre). His grandfather, Sebald Kepler, had been Lord Mayor of that town but, by the time Johannes was born, he had two brothers and one sister and the
Kepler family fortune was in decline. His father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in
the Eighty Years' War in the Netherlands. His mother Katharina Guldenmann, an inn-keeper's daughter, was a healer and herbalist who was later tried for witchcraft. Born prematurely, Johannes claimed
to have been weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfather's inn with his phenomenal mathematical faculty.
He was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, writing that he "was taken by [his]
mother to a high place to look at it." At age nine, he observed another astronomical event, a lunar eclipse in 1580, recording that he remembered being "called outdoors" to see it and that the moon
"appeared quite red". However, childhood smallpox left him with weak vision and crippled hands, limiting his ability in the observational aspects of astronomy.
In 1589, after moving through grammar school, Latin school, and seminary at Maulbronn, Kepler attended Tübinger Stift at the University of Tübingen. There, he studied philosophy under Vitus Müller
and theology under Jacob Heerbrand (a student of Philipp Melanchthon at Wittenberg), who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved
himself to be a superb mathematician and earned a reputation as a skillful astrologer, casting horoscopes for fellow students. Under the instruction of Michael Maestlin, Tübingen's professor of
mathematics from 1583 to 1631, he learned both the Ptolemaic system and the Copernican system of planetary motion. He became a Copernican at that time. In a student disputation, he defended
heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the principal source of motive power in the universe. Despite his desire to become a minister, near the
end of his studies Kepler was recommended for a position as teacher of mathematics and astronomy at the Protestant school in Graz (later the University of Graz). He accepted the position in April
1594, at the age of 23.
Graz (1594–1600)
Mysterium Cosmographicum
Johannes Kepler's first major astronomical work, Mysterium Cosmographicum (The Cosmographic Mystery), was the first published defense of the Copernican system. Kepler claimed to have had an epiphany
on July 19, 1595, while teaching in Graz, demonstrating the periodic conjunction of Saturn and Jupiter in the zodiac; he realized that regular polygons bound one inscribed and one circumscribed
circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe. After failing to find a unique arrangement of polygons that fit known astronomical observations (even
with extra planets added to the system), Kepler began experimenting with 3-dimensional polyhedra. He found that each of the five Platonic solids could be uniquely inscribed and circumscribed by
spherical orbs; nesting these solids, each encased in a sphere, within one another would produce six layers, corresponding to the six known planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn.
By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, cube—Kepler found that the spheres could be placed at intervals corresponding (within the accuracy limits of
available astronomical observations) to the relative sizes of each planet’s path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet’s orb to the length
of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not
precise enough.
As he indicated in the title, Kepler thought he had revealed God’s geometrical plan for the universe. Much of Kepler’s enthusiasm for the Copernican system stemmed from his theological convictions
about the connection between the physical and the spiritual; the universe itself was an image of God, with the Sun corresponding to the Father, the stellar sphere to the Son, and the intervening
space between to the Holy Spirit. His first manuscript of Mysterium contained an extensive chapter reconciling heliocentrism with biblical passages that seemed to support geocentrism.
With the support of his mentor Michael Maestlin, Kepler received permission from the Tübingen university senate to publish his manuscript, pending removal of the Bible exegesis and the addition of a
simpler, more understandable description of the Copernican system as well as Kepler’s new ideas. Mysterium was published late in 1596, and Kepler received his copies and began sending them to
prominent astronomers and patrons early in 1597; it was not widely read, but it established Kepler’s reputation as a highly skilled astronomer. The effusive dedication, to powerful patrons as well as
to the men who controlled his position in Graz, also provided a crucial doorway into the patronage system.
Though the details would be modified in light of his later work, Kepler never relinquished the Platonist polyhedral-spherist cosmology of Mysterium Cosmographicum. His subsequent main astronomical
works were in some sense only further developments of it, concerned with finding more precise inner and outer dimensions for the spheres by calculating the eccentricities of the planetary orbits
within it. In 1621 Kepler published an expanded second edition of Mysterium, half as long again as the first, detailing in footnotes the corrections and improvements he had achieved in the 25 years
since its first publication.
In terms of the impact of Mysterium, it can be seen as an important first step in modernizing Copernicus' theory. There is no doubt that Copernicus' "De Revolutionibus" seeks to advance a
sun-centered system, but in this book he had to resort to Ptolemaic devices (viz., epicycles and eccentric circles) in order to explain the change in planets' orbital speed. Furthermore, Copernicus
continued to use as a point of reference the centre of the earth's orbit rather than that of the sun, as he says, "as an aid to calculation and in order not to confuse the reader by diverging too
much from Ptolemy." Therefore, although the thesis of the "Mysterium Cosmographicum" was in error, modern astronomy owes much to this work "since it represents the first step in cleansing the
Copernican system of the remnants of the Ptolemaic theory still clinging to it."
Marriage to Barbara Müller
In December 1595, Kepler was introduced to Barbara Müller, a 23-year-old widow (twice over) with a young daughter, Gemma van Dvijneveldt, and he began courting her. Müller, heiress to the estates of
her late husbands, was also the daughter of a successful mill owner. Her father Jobst initially opposed a marriage despite Kepler's nobility; though he had inherited his grandfather's nobility,
Kepler's poverty made him an unacceptable match. Jobst relented after Kepler completed work on Mysterium, but the engagement nearly fell apart while Kepler was away tending to the details of
publication. However, church officials—who had helped set up the match—pressured the Müllers to honour their agreement. Barbara and Johannes were married on April 27, 1597.
In the first years of their marriage, the Keplers had two children (Heinrich and Susanna), both of whom died in infancy. In 1602, they had a daughter (Susanna); in 1604, a son (Friedrich); and in
1607, another son (Ludwig).
Other research
Following the publication of Mysterium and with the blessing of the Graz school inspectors, Kepler began an ambitious program to extend and elaborate his work. He planned four additional books: one
on the stationary aspects of the universe (the Sun and the fixed stars); one on the planets and their motions; one on the physical nature of planets and the formation of geographical features
(focused especially on Earth); and one on the effects of the heavens on the Earth, to include atmospheric optics, meteorology and astrology.
He also sought the opinions of many of the astronomers to whom he had sent Mysterium, among them Reimarus Ursus (Nicolaus Reimers Bär)—the imperial mathematician to Rudolph II and a bitter rival of
Tycho Brahe. Ursus did not reply directly, but republished Kepler's flattering letter to pursue his priority dispute over (what is now called) the Tychonic system with Tycho. Despite this black mark,
Tycho also began corresponding with Kepler, starting with a harsh but legitimate critique of Kepler's system; among a host of objections, Tycho took issue with the use of inaccurate numerical data
taken from Copernicus. Through their letters, Tycho and Kepler discussed a broad range of astronomical problems, dwelling on lunar phenomena and Copernican theory (particularly its theological
viability). But without the significantly more accurate data of Tycho's observatory, Kepler had no way to address many of these issues.
Instead, he turned his attention to chronology and "harmony," the numerological relationships among music, mathematics and the physical world, and their astrological consequences. By assuming the
Earth to possess a soul (a property he would later invoke to explain how the sun causes the motion of planets), he established a speculative system connecting astrological aspects and astronomical
distances to weather and other earthly phenomena. By 1599, however, he again felt his work limited by the inaccuracy of available data—just as growing religious tension was also threatening his
continued employment in Graz. In December of that year, Tycho invited Kepler to visit him in Prague; on January 1, 1600 (before he even received the invitation), Kepler set off in the hopes that
Tycho's patronage could solve his philosophical problems as well as his social and financial ones. When he was an old man, he was allowed to continue his work in his home alone.
Prague (1600–1612)
Work for Tycho Brahe
On February 4, 1600, Kepler met Tycho Brahe and his assistants Franz Tengnagel and Longomontanus at Benátky nad Jizerou (35 km from Prague), the site where Tycho's new observatory was being
constructed. Over the next two months he stayed as a guest, analyzing some of Tycho's observations of Mars; Tycho guarded his data closely, but was impressed by Kepler's theoretical ideas and soon
allowed him more access. Kepler planned to test his theory from Mysterium Cosmographicum based on the Mars data, but he estimated that the work would take up to two years (since he was not allowed to
simply copy the data for his own use). With the help of Johannes Jessenius, Kepler attempted to negotiate a more formal employment arrangement with Tycho, but negotiations broke down in an angry
argument and Kepler left for Prague on April 6. Kepler and Tycho soon reconciled and eventually reached an agreement on salary and living arrangements, and in June, Kepler returned home to Graz to
collect his family.
Political and religious difficulties in Graz dashed his hopes of returning immediately to Tycho; in hopes of continuing his astronomical studies, Kepler sought an appointment as mathematician to
Archduke Ferdinand. To that end, Kepler composed an essay—dedicated to Ferdinand—in which he proposed a force-based theory of lunar motion: "In Terra inest virtus, quae Lunam ciet" ("There is a force
in the earth which causes the moon to move"). Though the essay did not earn him a place in Ferdinand's court, it did detail a new method for measuring lunar eclipses, which he applied during the July
10 eclipse in Graz. These observations formed the basis of his explorations of the laws of optics that would culminate in Astronomiae Pars Optica.
On August 2, 1600, after refusing to convert to Catholicism, Kepler and his family were banished from Graz. Several months later, Kepler returned, now with the rest of his household, to Prague.
Through most of 1601, he was supported directly by Tycho, who assigned him to analyzing planetary observations and writing a tract against Tycho's (by then deceased) rival, Ursus. In September, Tycho
secured him a commission as a collaborator on the new project he had proposed to the emperor: the Rudolphine Tables that should replace the Prutenic Tables of Erasmus Reinhold. Two days after Tycho's
unexpected death on October 24, 1601, Kepler was appointed his successor as imperial mathematician with the responsibility to complete his unfinished work. The next 11 years as imperial mathematician
would be the most productive of his life.
Advisor to Emperor Rudolph II
Kepler's primary obligation as imperial mathematician was to provide astrological advice to the emperor. Though Kepler took a dim view of the attempts of contemporary astrologers to precisely predict
the future or divine specific events, he had been casting well-received detailed horoscopes for friends, family and patrons since his time as a student in Tübingen. In addition to horoscopes for
allies and foreign leaders, the emperor sought Kepler's advice in times of political trouble (though Kepler's recommendations were based more on common sense than the stars). Rudolph was actively
interested in the work of many of his court scholars (including numerous alchemists) and kept up with Kepler's work in physical astronomy as well.
Officially, the only acceptable religious doctrines in Prague were Catholic and Utraquist, but Kepler's position in the imperial court allowed him to practice his Lutheran faith unhindered. The
emperor nominally provided an ample income for his family, but the difficulties of the over-extended imperial treasury meant that actually getting hold of enough money to meet financial obligations
was a continual struggle. Partly because of financial troubles, his life at home with Barbara was unpleasant, marred with bickering and bouts of sickness. Court life, however, brought Kepler into
contact with other prominent scholars ( Johannes Matthäus Wackher von Wackhenfels, Jost Bürgi, David Fabricius, Martin Bachazek, and Johannes Brengger, among others) and astronomical work proceeded
Astronomiae Pars Optica
As he slowly continued analyzing Tycho's Mars observations—now available to him in their entirety—and began the slow process of tabulating the Rudolphine Tables, Kepler also picked up the
investigation of the laws of optics from his lunar essay of 1600. Both lunar and solar eclipses presented unexplained phenomena, such as unexpected shadow sizes, the red colour of a total lunar
eclipse, and the reportedly unusual light surrounding a total solar eclipse. Related issues of atmospheric refraction applied to all astronomical observations. Through most of 1603, Kepler paused his
other work to focus on optical theory; the resulting manuscript, presented to the emperor on January 1, 1604, was published as Astronomiae Pars Optica (The Optical Part of Astronomy). In it, Kepler
described the inverse-square law governing the intensity of light, reflection by flat and curved mirrors, and principles of pinhole cameras, as well as the astronomical implications of optics such as
parallax and the apparent sizes of heavenly bodies. He also extended his study of optics to the human eye, and is generally considered by neuroscientists to be the first to recognize that images are
projected inverted and reversed by the eye's lens onto the retina. The solution to this dilemma was not of particular importance to Kepler as he did not see it as pertaining to optics, although he
did suggest that the image was later corrected "in the hollows of the brain" due to the "activity of the Soul." Today, Astronomiae Pars Optica is generally recognized as the foundation of modern
optics (though the law of refraction is conspicuously absent). With respect to the beginnings of projective geometry, Kepler introduced the idea of continuous change of a mathematical entity in this
work. He argued that if a focus of a conic section were allowed to move along the line joining the foci, the geometric form would morph or degenerate, one into another. In this way, an ellipse
becomes a parabola when a focus moves toward infinity, and when two foci of an ellipse merge into one another, a circle is formed. As the foci of a hyperbola merge into one another, the hyperbola
becomes a pair of straight lines. He also assumed that if a straight line is extended to infinity it will meet itself at a single point at infinity, thus having the properties of a large circle. This
idea was later utilized by Pascal, Leibniz, Monge and Poncelet, among others, and became known as geometric continuity and as the Law or Principle of Continuity.
The Supernova of 1604
In October 1604, a bright new evening star ( SN 1604) appeared, but Kepler did not believe the rumors until he saw it himself. Kepler began systematically observing the nebula. Astrologically, the
end of 1603 marked the beginning of a fiery trigon, the start of the ca. 800-year cycle of great conjunctions; astrologers associated the two previous such periods with the rise of Charlemagne (ca.
800 years earlier) and the birth of Christ (ca. 1600 years earlier), and thus expected events of great portent, especially regarding the emperor. It was in this context, as the imperial mathematician
and astrologer to the emperor, that Kepler described the new star two years later in his De Stella Nova. In it, Kepler addressed the star's astronomical properties while taking a skeptical approach
to the many astrological interpretations then circulating. He noted its fading luminosity, speculated about its origin, and used the lack of observed parallax to argue that it was in the sphere of
fixed stars, further undermining the doctrine of the immutability of the heavens (the idea accepted since Aristotle that the celestial spheres were perfect and unchanging). The birth of a new star
implied the variability of the heavens. In an appendix, Kepler also discussed the recent chronology work of the Polish historian Laurentius Suslyga; he calculated that, if Suslyga was correct that
accepted timelines were four years behind, then the Star of Bethlehem—analogous to the present new star—would have coincided with the first great conjunction of the earlier 800-year cycle.
Astronomia nova
The extended line of research that culminated in Astronomia nova (A New Astronomy)—including the first two laws of planetary motion—began with the analysis, under Tycho's direction, of Mars' orbit.
Kepler calculated and recalculated various approximations of Mars' orbit using an equant (the mathematical tool that Copernicus had eliminated with his system), eventually creating a model that
generally agreed with Tycho's observations to within two arcminutes (the average measurement error). But he was not satisfied with the complex and still slightly inaccurate result; at certain points
the model differed from the data by up to eight arcminutes. The wide array of traditional mathematical astronomy methods having failed him, Kepler set about trying to fit an ovoid orbit to the data.
Within Kepler's religious view of the cosmos, the Sun (a symbol of God the Father) was the source of motive force in the solar system. As a physical basis, Kepler drew by analogy on William Gilbert's
theory of the magnetic soul of the Earth from De Magnete (1600) and on his own work on optics. Kepler supposed that the motive power (or motive species) radiated by the Sun weakens with distance,
causing faster or slower motion as planets move closer or farther from it. Perhaps this assumption entailed a mathematical relationship that would restore astronomical order. Based on measurements of
the aphelion and perihelion of the Earth and Mars, he created a formula in which a planet's rate of motion is inversely proportional to its distance from the Sun. Verifying this relationship
throughout the orbital cycle, however, required very extensive calculation; to simplify this task, by late 1602 Kepler reformulated the proportion in terms of geometry: planets sweep out equal areas
in equal times—Kepler's second law of planetary motion.
He then set about calculating the entire orbit of Mars, using the geometrical rate law and assuming an egg-shaped ovoid orbit. After approximately 40 failed attempts, in early 1605 he at last hit
upon the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked. Finding that an elliptical orbit fit the Mars data, he immediately
concluded that all planets move in ellipses, with the sun at one focus—Kepler's first law of planetary motion. Because he employed no calculating assistants, however, he did not extend the
mathematical analysis beyond Mars. By the end of the year, he completed the manuscript for Astronomia nova, though it would not be published until 1609 due to legal disputes over the use of Tycho's
observations, the property of his heirs.
Dioptrice, Somnium manuscript and other work
In the years following the completion of Astronomia Nova, most of Kepler's research was focused on preparations for the Rudolphine Tables and a comprehensive set of ephemerides (specific predictions
of planet and star positions) based on the table (though neither would be completed for many years). He also attempted (unsuccessfully) to begin a collaboration with Italian astronomer Giovanni
Antonio Magini. Some of his other work dealt with chronology, especially the dating of events in the life of Jesus, and with astrology, especially criticism of dramatic predictions of catastrophe
such as those of Helisaeus Roeslin.
Kepler and Roeslin engaged in series of published attacks and counter-attacks, while physician Philip Feselius published a work dismissing astrology altogether (and Roeslin's work in particular). In
response to what Kepler saw as the excesses of astrology on the one hand and overzealous rejection of it on the other, Kepler prepared Tertius Interveniens (Third-party Interventions). Nominally this
work—presented to the common patron of Roeslin and Feselius—was a neutral mediation between the feuding scholars, but it also set out Kepler's general views on the value of astrology, including some
hypothesized mechanisms of interaction between planets and individual souls. While Kepler considered most traditional rules and methods of astrology to be the "evil-smelling dung" in which "an
industrious hen" scrapes, there was an "occasional grain-seed, indeed, even a pearl or a gold nugget" to be found by the conscientious scientific astrologer.
In the first months of 1610, Galileo Galilei—using his powerful new telescope—discovered four satellites orbiting Jupiter. Upon publishing his account as Sidereus Nuncius (Starry Messenger), Galileo
sought the opinion of Kepler, in part to bolster the credibility of his observations. Kepler responded enthusiastically with a short published reply, Dissertatio cum Nuncio Sidereo (Conversation with
the Starry Messenger). He endorsed Galileo's observations and offered a range of speculations about the meaning and implications of Galileo's discoveries and telescopic methods, for astronomy and
optics as well as cosmology and astrology. Later that year, Kepler published his own telescopic observations of the moons in Narratio de Jovis Satellitibus, providing further support of Galileo. To
Kepler's disappointment, however, Galileo never published his reactions (if any) to Astronomia Nova.:(
After hearing of Galileo's telescopic discoveries, Kepler also started a theoretical and experimental investigation of telescopic optics using a telescope borrowed from Duke Ernest of Cologne. The
resulting manuscript was completed in September 1610 and published as Dioptrice in 1611. In it, Kepler set out the theoretical basis of double-convex converging lenses and double-concave diverging
lenses—and how they are combined to produce a Galilean telescope—as well as the concepts of real vs. virtual images, upright vs. inverted images, and the effects of focal length on magnification and
reduction. He also described an improved telescope—now known as the astronomical or Keplerian telescope—in which two convex lenses can produce higher magnification than Galileo's combination of
convex and concave lenses.
Around 1611, Kepler circulated a manuscript of what would eventually be published (posthumously) as Somnium (The Dream). Part of the purpose of Somnium was to describe what practicing astronomy would
be like from the perspective of another planet, to show the feasibility of a non-geocentric system. The manuscript, which disappeared after changing hands several times, described a fantastic trip to
the moon; it was part allegory, part autobiography, and part treatise on interplanetary travel (and is sometimes described as the first work of science fiction). Years later, a distorted version of
the story may have instigated the witchcraft trial against his mother, as the mother of the narrator consults a demon to learn the means of space travel. Following her eventual acquittal, Kepler
composed 223 footnotes to the story—several times longer than the actual text—which explained the allegorical aspects as well as the considerable scientific content (particularly regarding lunar
geography) hidden within the text.
Work in mathematics and physics
As a New Year's gift that year, he also composed for his friend and some-time patron Baron Wackher von Wackhenfels a short pamphlet entitled Strena Seu de Nive Sexangula (A New Year's Gift of
Hexagonal Snow). In this treatise, he published the first description of the hexagonal symmetry of snowflakes and, extending the discussion into a hypothetical atomistic physical basis for the
symmetry and posed what later became known as the Kepler conjecture, a statement about the most efficient arrangement for packing spheres. Kepler was one of the pioneers of the mathematical
applications of infinitesimals, see Law of Continuity.
Personal and political troubles
In 1611, the growing political-religious tension in Prague came to a head. Emperor Rudolph—whose health was failing—was forced to abdicate as King of Bohemia by his brother Matthias. Both sides
sought Kepler's astrological advice, an opportunity he used to deliver conciliatory political advice (with little reference to the stars, except in general statements to discourage drastic action).
However, it was clear that Kepler's future prospects in the court of Matthias were dim.
Also in that year, Barbara Kepler contracted Hungarian spotted fever, then began having seizures. As Barbara was recovering, Kepler's three children all fell sick with smallpox; Friedrich, 6, died.
Following his son's death, Kepler sent letters to potential patrons in Württemberg and Padua. At the University of Tübingen in Württemberg, concerns over Kepler's perceived Calvinist heresies in
violation of the Augsburg Confession and the Formula of Concord prevented his return. The University of Padua—on the recommendation of the departing Galileo—sought Kepler to fill the mathematics
professorship, but Kepler, preferring to keep his family in German territory, instead travelled to Austria to arrange a position as teacher and district mathematician in Linz. However, Barbara
relapsed into illness and died shortly after Kepler's return.
Kepler postponed the move to Linz and remained in Prague until Rudolph's death in early 1612, though between political upheaval, religious tension, and family tragedy (along with the legal dispute
over his wife's estate), Kepler could do no research. Instead, he pieced together a chronology manuscript, Eclogae Chronicae, from correspondence and earlier work. Upon succession as Holy Roman
Emperor, Matthias re-affirmed Kepler's position (and salary) as imperial mathematician but allowed him to move to Linz.
Linz and elsewhere (1612–1630)
In Linz, Kepler's primary responsibilities (beyond completing the Rudolphine Tables) were teaching at the district school and providing astrological and astronomical services. In his first years
there, he enjoyed financial security and religious freedom relative to his life in Prague—though he was excluded from Eucharist by his Lutheran church over his theological scruples. His first
publication in Linz was De vero Anno (1613), an expanded treatise on the year of Christ's birth; he also participated in deliberations on whether to introduce Pope Gregory's reformed calendar to
Protestant German lands; that year he also wrote the influential mathematical treatise Nova stereometria doliorum vinariorum, on measuring the volume of containers such as wine barrels, published in
Second marriage
On October 30, 1613, Kepler married the 24-year-old Susanna Reuttinger. Following the death of his first wife Barbara, Kepler had considered 11 different matches. He eventually returned to Reuttinger
(the fifth match) who, he wrote, "won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren." The first three children of this marriage (Margareta
Regina, Katharina, and Sebald) died in childhood. Three more survived into adulthood: Cordula (b. 1621); Fridmar (b. 1623); and Hildebert (b. 1625). According to Kepler's biographers, this was a much
happier marriage than his first.
Epitome of Copernican Astronomy, calendars and the witch trial of his mother
Since completing the Astronomia nova, Kepler had intended to compose an astronomy textbook. In 1615, he completed the first of three volumes of Epitome astronomiae Copernicanae (Epitome of Copernican
Astronomy); the first volume (books I-III) was printed in 1617, the second (book IV) in 1620, and the third (books V-VII) in 1621. Despite the title, which referred simply to heliocentrism, Kepler's
textbook culminated in his own ellipse-based system. The Epitome became Kepler's most influential work. It contained all three laws of planetary motion and attempted to explain heavenly motions
through physical causes. Though it explicitly extended the first two laws of planetary motion (applied to Mars in Astronomia nova) to all the planets as well as the Moon and the Medicean satellites
of Jupiter, it did not explain how elliptical orbits could be derived from observational data.
As a spin-off from the Rudolphine Tables and the related Ephemerides, Kepler published astrological calendars, which were very popular and helped offset the costs of producing his other
work—especially when support from the Imperial treasury was withheld. In his calendars—six between 1617 and 1624—Kepler forecast planetary positions and weather as well as political events; the
latter were often cannily accurate, thanks to his keen grasp of contemporary political and theological tensions. By 1624, however, the escalation of those tensions and the ambiguity of the prophecies
meant political trouble for Kepler himself; his final calendar was publicly burned in Graz.
In 1615, Ursula Reingold, a woman in a financial dispute with Kepler's brother Christoph, claimed Kepler's mother Katharina had made her sick with an evil brew. The dispute escalated, and in 1617,
Katharina was accused of witchcraft; witchcraft trials were relatively common in central Europe at this time. Beginning in August 1620 she was imprisoned for fourteen months. She was released in
October 1621, thanks in part to the extensive legal defense drawn up by Kepler. The accusers had no stronger evidence than rumors, along with a distorted, second-hand version of Kepler's Somnium, in
which a woman mixes potions and enlists the aid of a demon. Katharina was subjected to territio verbalis, a graphic description of the torture awaiting her as a witch, in a final attempt to make her
confess. Throughout the trial, Kepler postponed his other work to focus on his "harmonic theory". The result, published in 1619, was Harmonices Mundi ("Harmony of the World").
Harmonices Mundi
Kepler was convinced "that the geometrical things have provided the Creator with the model for decorating the whole world." In Harmony, he attempted to explain the proportions of the natural
world—particularly the astronomical and astrological aspects—in terms of music. The central set of "harmonies" was the musica universalis or "music of the spheres," which had been studied by
Pythagoras, Ptolemy and many others before Kepler; in fact, soon after publishing Harmonices Mundi, Kepler was embroiled in a priority dispute with Robert Fludd, who had recently published his own
harmonic theory.
Kepler began by exploring regular polygons and regular solids, including the figures that would come to be known as Kepler's solids. From there, he extended his harmonic analysis to music,
meteorology and astrology; harmony resulted from the tones made by the souls of heavenly bodies—and in the case of astrology, the interaction between those tones and human souls. In the final portion
of the work (Book V), Kepler dealt with planetary motions, especially relationships between orbital velocity and orbital distance from the Sun. Similar relationships had been used by other
astronomers, but Kepler—with Tycho's data and his own astronomical theories—treated them much more precisely and attached new physical significance to them.
Among many other harmonies, Kepler articulated what came to be known as the third law of planetary motion. He then tried many combinations until he discovered that (approximately) "The square of the
periodic times are to each other as the cubes of the mean distances." Although he gives the date of this epiphany (March 8, 1618), he does not give any details about how he arrived at this
conclusion. However, the wider significance for planetary dynamics of this purely kinematical law was not realized until the 1660s. For when conjoined with Christian Huygens' newly discovered law of
centrifugal force it enabled Isaac Newton, Edmund Halley and perhaps Christopher Wren and Robert Hooke to demonstrate independently that the presumed gravitational attraction between the Sun and its
planets decreased with the square of the distance between them. This refuted the traditional assumption of scholastic physics that the power of gravitational attraction remained constant with
distance whenever it applied between two bodies, such as was assumed by Kepler and also by Galileo in his mistaken universal law that gravitational fall is uniformly accelerated, and also by
Galileo's student Borrelli in his 1666 celestial mechanics. William Gilbert, after experimenting with magnets decided that the centre of the Earth was a huge magnet. His theory led Kepler to think
that a magnetic force from the Sun drove planets in their own orbits. It was an interesting explanation for planetary motion, but it was wrong. Before scientists could find the right answer, they
needed to know more about motion.
Rudolphine Tables and his last years
In 1623, Kepler at last completed the Rudolphine Tables, which at the time was considered his major work. However, due to the publishing requirements of the emperor and negotiations with Tycho
Brahe's heir, it would not be printed until 1627. In the meantime religious tension—the root of the ongoing Thirty Years' War—once again put Kepler and his family in jeopardy. In 1625, agents of the
Catholic Counter-Reformation placed most of Kepler's library under seal, and in 1626 the city of Linz was besieged. Kepler moved to Ulm, where he arranged for the printing of the Tables at his own
In 1628, following the military successes of the Emperor Ferdinand's armies under General Wallenstein, Kepler became an official advisor to Wallenstein. Though not the general's court astrologer per
se, Kepler provided astronomical calculations for Wallenstein's astrologers and occasionally wrote horoscopes himself. In his final years, Kepler spent much of his time traveling, from the imperial
court in Prague to Linz and Ulm to a temporary home in Sagan, and finally to Regensburg. Soon after arriving in Regensburg, Kepler fell ill. He died on November 15, 1630, and was buried there; his
burial site was lost after the Swedish army destroyed the churchyard. Only Kepler's self-authored poetic epitaph survived the times:
Mensus eram coelos, nunc terrae metior umbras
Mens coelestis erat, corporis umbra iacet.
I measured the skies, now the shadows I measure
Skybound was the mind, earthbound the body rests.
Reception of his astronomy
Kepler's laws were not immediately accepted. Several major figures such as Galileo and René Descartes completely ignored Kepler's Astronomia nova. Many astronomers, including Kepler's teacher,
Michael Maestlin, objected to Kepler's introduction of physics into his astronomy. Some adopted compromise positions. Ismael Boulliau accepted elliptical orbits but replaced Kepler's area law with
uniform motion in respect to the empty focus of the ellipse while Seth Ward used an elliptical orbit with motions defined by an equant.
Several astronomers tested Kepler's theory, and its various modifications, against astronomical observations. Two transits of Venus and Mercury across the face of the sun provided sensitive tests of
the theory, under circumstances when these planets could not normally be observed. In the case of the transit of Mercury in 1631, Kepler had been extremely uncertain of the parameters for Mercury,
and advised observers to look for the transit the day before and after the predicted date. Pierre Gassendi observed the transit on the date predicted, a confirmation of Kepler's prediction. This was
the first observation of a transit of Mercury. However, his attempt to observe the transit of Venus just one month later, was unsuccessful due to inaccuracies in the Rudolphine Tables. Gassendi did
not realize that it was not visible from most of Europe, including Paris. Jeremiah Horrocks, who observed the 1639 Venus transit, had used his own observations to adjust the parameters of the
Keplerian model, predicted the transit, and then built apparatus to observe the transit. He remained a firm advocate of the Keplerian model.
Epitome of Copernican Astronomy was read by astronomers throughout Europe, and following Kepler's death it was the main vehicle for spreading Kepler's ideas. Between 1630 and 1650, it was the most
widely used astronomy textbook, winning many converts to ellipse-based astronomy. However, few adopted his ideas on the physical basis for celestial motions. In the late 17th century, a number of
physical astronomy theories drawing from Kepler's work—notably those of Giovanni Alfonso Borelli and Robert Hooke—began to incorporate attractive forces (though not the quasi-spiritual motive species
postulated by Kepler) and the Cartesian concept of inertia. This culminated in Isaac Newton's Principia Mathematica (1687), in which Newton derived Kepler's laws of planetary motion from a
force-based theory of universal gravitation.
Historical and cultural legacy
Beyond his role in the historical development of astronomy and natural philosophy, Kepler has loomed large in the philosophy and historiography of science. Kepler and his laws of motion were central
to early histories of astronomy such as Jean Etienne Montucla’s 1758 Histoire des mathématiques and Jean-Baptiste Delambre's 1821 Histoire de l’astronomie moderne. These and other histories written
from an Enlightenment perspective treated Kepler's metaphysical and religious arguments with skepticism and disapproval, but later Romantic-era natural philosophers viewed these elements as central
to his success. William Whewell, in his influential History of the Inductive Sciences of 1837, found Kepler to be the archetype of the inductive scientific genius; in his Philosophy of the Inductive
Sciences of 1840, Whewell held Kepler up as the embodiment of the most advanced forms of scientific method. Similarly, Ernst Friedrich Apelt—the first to extensively study Kepler's manuscripts, after
their purchase by Catherine the Great—identified Kepler as a key to the " Revolution of the sciences". Apelt, who saw Kepler's mathematics, aesthetic sensibility, physical ideas, and theology as part
of a unified system of thought, produced the first extended analysis of Kepler's life and work.
Modern translations of a number of Kepler's books appeared in the late-nineteenth and early-twentieth centuries, the systematic publication of his collected works began in 1937 (and is nearing
completion in the early 21st century), and Max Caspar's Kepler biography was published in 1948. However, Alexandre Koyré's work on Kepler was, after Apelt, the first major milestone in historical
interpretations of Kepler's cosmology and its influence. In the 1930s and 1940s Koyré, and a number of others in the first generation of professional historians of science, described the " Scientific
Revolution" as the central event in the history of science, and Kepler as a (perhaps the) central figure in the revolution. Koyré placed Kepler's theorization, rather than his empirical work, at the
centre of the intellectual transformation from ancient to modern world-views. Since the 1960s, the volume of historical Kepler scholarship has expanded greatly, including studies of his astrology and
meteorology, his geometrical methods, the role of his religious views in his work, his literary and rhetorical methods, his interaction with the broader cultural and philosophical currents of his
time, and even his role as an historian of science.
The debate over Kepler's place in the Scientific Revolution has also produced a wide variety of philosophical and popular treatments. One of the most influential is Arthur Koestler's 1959 The
Sleepwalkers, in which Kepler is unambiguously the hero (morally and theologically as well as intellectually) of the revolution. Influential philosophers of science—such as Charles Sanders Peirce,
Norwood Russell Hanson, Stephen Toulmin, and Karl Popper—have repeatedly turned to Kepler: examples of incommensurability, analogical reasoning, falsification, and many other philosophical concepts
have been found in Kepler's work. Physicist Wolfgang Pauli even used Kepler's priority dispute with Robert Fludd to explore the implications of analytical psychology on scientific investigation. A
well-received, if fanciful, historical novel by John Banville, Kepler (1981), explored many of the themes developed in Koestler's non-fiction narrative and in the philosophy of science. Somewhat more
fanciful is a recent work of nonfiction, Heavenly Intrigue (2004), suggesting that Kepler murdered Tycho Brahe to gain access to his data. Kepler has acquired a popular image as an icon of scientific
modernity and a man before his time; science popularizer Carl Sagan described him as "the first astrophysicist and the last scientific astrologer."
The German composer Paul Hindemith wrote an opera about Kepler entitled Die Harmonie der Welt, and a symphony of the same name was derived from music for the opera.
In Austria, Kepler left behind such a historical legacy that he was one of the motifs of a silver collector's coin: the 10-euro Johannes Kepler silver coin, minted on September 10, 2002. The reverse
side of the coin has a portrait of Kepler, who spent some time teaching in Graz and the surrounding areas. Kepler was acquainted with Prince Hans Ulrich von Eggenberg personally, and he probably
influenced the construction of Eggenberg Castle (the motif of the obverse of the coin). In front of him on the coin is the model of nested spheres and polyhedra from Mysterium Cosmographicum.
In 2009, NASA named the Kepler Mission for Kepler's contributions to the field of astronomy.
In New Zealand's Fiordland National Park there is also a range of Mountains Named after Kepler, called the Kepler Mountains and a Three Day Walking Trail known as the Kepler Track through the
Mountains of the same name.
Kepler is honored together with Nicolaus Copernicus with a feast day on the liturgical calendar of the Episcopal Church (USA) on May 23.
• Mysterium cosmographicum (The Sacred Mystery of the Cosmos) (1596)
• De Fundamentis Astrologiae Certioribus On Firmer Fundaments of Astrology (1601)
• Astronomiae Pars Optica (The Optical Part of Astronomy) (1604)
• De Stella nova in pede Serpentarii (On the New Star in Ophiuchus's Foot) (1604)
• Astronomia nova (New Astronomy) (1609)
• Tertius Interveniens (Third-party Interventions) (1610)
• Dissertatio cum Nuncio Sidereo (Conversation with the Starry Messenger) (1610)
• Dioptrice (1611)
• De nive sexangula (On the Six-Cornered Snowflake) (1611)
• De vero Anno, quo aeternus Dei Filius humanam naturam in Utero benedictae Virginis Mariae assumpsit (1613)
• Eclogae Chronicae (1615, published with Dissertatio cum Nuncio Sidereo)
• Nova stereometria doliorum vinariorum (New Stereometry of Wine Barrels) (1615)
• Epitome astronomiae Copernicanae (Epitome of Copernican Astronomy) (published in three parts from 1618–1621)
• Harmonice Mundi (Harmony of the Worlds) (1619)
• Mysterium cosmographicum (The Sacred Mystery of the Cosmos) 2nd Edition (1621)
• Tabulae Rudolphinae (Rudolphine Tables) (1627)
• Somnium (The Dream) (1634) | {"url":"https://www.valeriodistefano.com/en/wp/j/Johannes_Kepler.htm","timestamp":"2024-11-08T21:20:09Z","content_type":"text/html","content_length":"150184","record_id":"<urn:uuid:29b12b37-7fb4-4288-bcea-0a9f62920462>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00130.warc.gz"} |
Spacetime-dependent Lagrangians and electrogravity duality
We apply the spacetime dependent lagrangian formalism [1] to the action in general relativity. We obtain a Barriola-Vilenkin type monopole solution by exploiting theelectrogravity duality of the
vacuum Einstein equations and using a modified definition of empty space. An {\it upper bound} is obtained on the monopole mass ${\tt M}$, ${\tt M}\leq e^{(1-\alpha)/\alpha}/(1-\alpha)^{2}{\tt G}$
where $\alpha = 2k $ is the global monopole charge. Keywords: global monopole, electrogravity duality, holographic principle. PACS: 11.15.-q, 11.27.+d, 14.80.Hv, 04.
Gravitation and Cosmology
Pub Date:
December 2007
□ General Relativity and Quantum Cosmology;
□ Astrophysics;
□ High Energy Physics - Theory
4 pages, latex | {"url":"https://ui.adsabs.harvard.edu/abs/2007GrCo...13..285G/abstract","timestamp":"2024-11-01T20:18:33Z","content_type":"text/html","content_length":"36501","record_id":"<urn:uuid:002aefb1-1d86-469d-ae0c-313a97c38ed6>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00469.warc.gz"} |
Problem or bug with MeijerG function (sharp jump)?
2766 Views
3 Replies
5 Total Likes
Problem or bug with MeijerG function (sharp jump)?
Hi, I run into the MeijerG function with the specific case
ListLinePlot[{Table[{x, 0.31830988618379064* MeijerG[{{0, 1/2}, {}} , {{0, 1/2}, {-1/2}} , x*x]}, {x, -24, 0, 1}]}, PlotRange -> All]
However, the curve is not continuous. Can anybody help to give some hints about the problem?
3 Replies
A developer I asked said this cliff comes from cancellations while summing the series.
His suggestions were higher Working Precision (which David showed nicely) and using a simpler-function version of the expression.
In[1]:= fsfe=FullSimplify[
FunctionExpand[ MeijerG[{{0,1/2},{}},{{0,1/2},{-1/2}},x*x] ], {x<0} ]
Out[1]= ( Pi (-1+E^x^2 (1+Erf[x] )))/x
In[2]:= qq = Rationalize[0.318309886183790640, 0];
In[3]:= Plot[ qq * fsfe, {x,-24,0}, PlotRange->All, WorkingPrecision->20, AxesOrigin->{0,0}]
Here, the WorkingPrecision->20 eliminated one artifact in the curve.
The original code can be modified simply. Note the N[.. ,20] around the MeijerG call.
That was enough extra precision to prevent cancellations at larger x's.
In[7]:= ListLinePlot[{ Table[{x, qq* N[MeijerG[{{0, 1/2}, {}} , {{0, 1/2}, {-1/2}} , x*x], 20] }, {x, -24, 0, 1}] }, PlotRange -> All]
You have encountered the limitations of using machine precision calculations in the internal numerical algorithms for MeijerG. Though I am not sure why you are using ListLinePlot, here is an example
using Plot with its WorkingPrecision option set to a higher non-MachinePrecision value. Also note that I changed your floating point value to an exact number so that the floating point value would
not conflict with the higher precision computations specified by the WorkingPrecision option.
Plot[27235615/85563208 MeijerG[{{0, 1/2}, {}}, {{0, 1/2}, {-(1/2)}}, x^2],
{x, -24, 0},
PlotRange -> All,
WorkingPrecision -> 50]
However, the behavior of where the "cliff" discontinuity appears in the function is a bit odd and depends on what the value of the WorkingPrecision option is. So it appears not to be a branch cut
crossing, but it is unclear what is happening in the internal algorithm that is causing it to truncate and return a value of 0 to the left of the WorkingPrecision-dependent discontinuity.
Take a look at the following to see how the discontinuity is depending on the value of WorkingPrecision :
Plot[27235615/85563208 MeijerG[{{0, 1/2}, {}}, {{0, 1/2}, {-(1/2)}},
x^2], {x, -30, 0}, PlotRange -> All, WorkingPrecision -> wp],
{wp, 30, 200}, ContinuousAction -> False]
Here are two screenshots of this manipulate at different values of the WorkingPrecision slider:
I think this needs a special-functions expert. The function does go to zero abruptly and stay there.
In[1]:= MeijerG[{{0,1/2},{}},{{0,1/2},{-1/2}},x*x] /. {{x->-5.9},{x->-6},{x->-6.1},{x->-6.2},{x->-6.3},{x->-6.4}} //N
Out[1]= {0.490874,0.392699,1.5708,0.,0.,0.}
In[4]:= qq = 0.3183098861837906400000000000000000000000000;
In[5]:=Plot[{qq*MeijerG[{{0, 1/2}, {}}, {{0, 1/2}, {-1/2}}, x*x]}, {x, -24, 0} , PlotRange -> All, WorkingPrecision -> 25]
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/274799","timestamp":"2024-11-14T13:45:05Z","content_type":"text/html","content_length":"107289","record_id":"<urn:uuid:8328d191-9143-45f1-9527-ef127083984b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00669.warc.gz"} |
Simple Interest 9APPS
• 1. Mary invests $2,500 at an annual interest rate of 4%. Calculate the simple interest after 3 years.
• 2. James borrows $1,200 at a simple interest rate of 6% per annum. How much interest will he pay after 2.5 years?
• 3. If you deposit $800 in a savings account with an annual interest rate of 3.5%, how much interest will you earn after 4 years?
• 4. Tom takes a loan of $5,000 at a simple interest rate of 8% for 2 years.
Calculate the total amount he needs to repay.
• 5. Alex borrowed a sum of money at an annual interest rate of 8%. After 2 years, he repaid $3,600 in total. Calculate the original loan amount.
• 6. A savings account with a principal of $10,000 earned $1,200 in simple interest after a certain period. If the annual interest rate was 4%, calculate the time the money was invested.
• 7. Alex takes a loan of $7,500 at an annual interest rate of 6%.
Calculate the interest he earns each year.
• 8. Emily borrows $5,000 at an interest rate of 5% for 3 years.
How much is her total repayment?
• 9. Sarah took a loan of $5,000 and repaid a total of $6,200 after 2 years. If the loan was at a simple interest rate, calculate the annual interest rate.
• 10. Emma borrows $9,000 at a simple interest rate of 7% per annum. If she keeps the loan for 10 months, how much interest will she pay?
• 11. Michael invests $8,000 in a fixed deposit with an annual interest rate of 4.5%. After 5 months, withdraws the money.
How much interest had the investment earned?
• 12. Nicole deposits $2,000 in a savings account with an annual interest rate of 3.8%. After 9 months, she decides to close the account.
Calculate the total amount she receives.
• 13. Lisa invested a certain amount of money at an annual interest rate of 6%, and after 3 years, she received $720 as simple interest. Calculate the initial amount she invested.
• 14. Choose the word to complete the sentence: When investing money you can compare banks to find the ...... interest rate so you get the most interest.
A) Highest
B) Lowest
• 15. Choose the correct formula to calculate Simple Interest
A) R = PTI
B) I = PRT
C) P = IRT
D) T = PRI
• 16. Abdul invest $5000 for 3 years at 6.75% pa. His total interest in 3 years is $1012.50.
What is the value of his investment at the end of the term?
• 17. Khaled invested $2000 for 3 years at 10% pa simple interest.
Calculate the total interest earned.
• 18. Kylie invested $1000 for 4 years at 5% pa.
Find the value of the investment at the end of the term.
• 19. Amy borrows $4,500 at a simple interest rate of 8% per annum. After 1.5 years, she decides to repay the loan.
Calculate the total amount she repays.
• 20. A sum of $10,000 is invested at a simple interest rate of 6%. If the interest earned is $900, calculate the time the money was invested. | {"url":"https://www.thatquiz.org/tq/preview?c=p5njlhhs&s=s2r1gw","timestamp":"2024-11-10T15:38:12Z","content_type":"text/html","content_length":"12188","record_id":"<urn:uuid:fb075f2e-a4a3-4db2-84f5-b025cef3071a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00297.warc.gz"} |
Collin Prather - Using contextual MAB’s to maximize community census engagement
class MAB:
Base Multi-Armed Bandit class.
def __init__(self, slots: List[scipy.stats.bernoulli]):
self.slots = slots
self.k = len(self.slots)
self.choices = [list(0 for _ in range(self.k))]
self.rewards = [list(0 for _ in range(self.k))]
# initialize estimated probs uniformly
self.estimated_probs = [[1 / len(slots) for _ in range(self.k)]]
def pull(self, t):
# needs to be implemented individually for each type of MAB
def update_history(self, arm, reward):
choice = [1 if arm == i else 0 for i in range(self.k)]
r = [reward if arm == i else 0 for i in range(self.k)]
return None
def update_probs(self, t):
estimated_prob = []
for slot in range(self.k):
slot_choices = [c[slot] for c in self.choices]
slot_rewards = [r[slot] for r in self.rewards]
estimated_prob.append((np.sum(slot_rewards) / (np.sum(slot_choices) + 1)))
return None
def play(self, n):
for t in range(n):
arm, reward = self.pull(t) | {"url":"https://collinprather.com/posts/2020-06-08-contextual-mabs-census.html","timestamp":"2024-11-10T23:39:28Z","content_type":"application/xhtml+xml","content_length":"709764","record_id":"<urn:uuid:07254a3e-f182-4925-879c-ce8884f35c55>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00650.warc.gz"} |
Expansion Tank Sizing Formulas - Equipments - HVAC/R & Solar
Expansion tanks are a necessary part of all closed hydronic systems to control both minimum and maximum pressure throughout the system. Expansion tanks are provided in closed hydronic systems to (1)
accept changes in system water volume as water density changes with temperature to keep system pressures below equipment and piping system component pressure rating limits. Also, (2) maintain a
positive gauge pressure in all parts of the system to prevent air from leaking into the system. (3) Maintain sufficient pressures in all parts of the system to prevent boiling, including cavitation
at control valves and similar constrictions. (4) Maintain net positive suction head required (NPSHR) at the suction of pumps.
Bladder expansion tank
The latter two points generally apply only to high temperature (greater than approximately 210°F [99°C]) hot water systems. For most HVAC applications, only the first two points need to be
Tank Styles
There are four basic styles of expansion tanks:
Vented or open steel tanks
Since they are vented, open tanks must be located at the highest point of the system. Water temperature cannot be above 212°F (100°C), and the open air/water contact results in a constant migration
of air into the system, causing corrosion. Accordingly, this design is almost never used anymore.
Closed steel tanks
Also called plain steel tanks or compression tanks by some manufacturers.
This is the same tank style as the vented tank, but with the vent capped. This allows the tank to be located anywhere in the system and work with higher temperatures. But they still have the air/
water contact that allows for corrosion, and sometimes a gradual loss of air from the tank as it is absorbed into the water.
Unless precharged to the minimum operating pressure prior to connection to the system, this style of tank also must be larger than precharged tanks. Accordingly, this design is also almost never used
Diaphragm tanks
This was the first design of a compression tank that included an air/water barrier (a flexible membrane, to eliminate air migration) and that was designed to be precharged (to reduce tank size). The
flexible diaphragm typically is attached to the side of the tank near the middle and is not field replaceable; if the diaphragm ruptures, the tank must be replaced.
Bladder tanks
Bladder tanks use a balloon-like bladder to accept the expanded water. Bladders are often sized for the entire tank volume, called a “full acceptance” bladder, to avoid damage to the bladder in case
they become waterlogged. Bladders are gener ally field replaceable. This is now the most common type of large commercial expansion tank.
Sizing Formulas
The general formula for tank sizing, Equation 1 (with variable names adjusted to match those used in this article), from basic principles assuming perfect gas laws:
$$V_t = \frac{V_s(E_w – E_p)}{(P_s T_c / P_i T_s) – (P_s T_h / P_{max} T_s) – E_{wt}[1 – (P_s T_c / P_{max} T_s)] + E_t} – 0.02 V_s$$
V[t] = tank total volume
V[s] = system volume
P[s] = starting pressure when water first starts to enter the tank, absolute
P[i] = initial (precharge) pressure, absolute
P[max] = maximum pressure, absolute
E[w] = unit expansion ratio of the water in the system due to temperature rise = (ν[h]/ν[c]-1)
v[h] = the specific volume of water at the maximum temperature, Th.
v[c] = the specific volume of water at the minimum temperature, Tc .
E[p] = unit expansion ratio of the piping and other system components in the system due to temperature rise = 3α(T[h]-T[c] )
α = coefficient of expansion of piping and other system components, per degree
T[h] = maximum average water temperature in the system, degrees absolute
T[c] = minimum average water temperature in the system, degrees absolute
T[s] = starting air temperature in the tank prior to fill, degrees absolute
E[wt] = unit expansion ratio of water in the tank due to temperature rise
E[t] = unit expansion ratio of the expansion tank due to temperature rise
The last term (0.02 Vs ) accounts for additional air from desorption from dissolved air in the water. This equation can be simplified to Equation below by ignoring small terms and assuming tank
temperature stays close to the initial fill temperature (typically a good assumption, assuming no insulation on the tank or piping to it, which is a common, and recommended, practice):
$$V_t = \frac{V_s\left(\frac{v_h}{v_c} – 1 – 3\alpha(T_h – T_c)\right)}{\frac{P_s}{P_i} – \frac{P_s}{P_{max}}}$$
This equation includes the credit for the expansion of the piping system. This term is also relatively small and the expansion coefficients are hard to determine given the various materials in the
system, but it is included in Equation above since it is included in the ASHRAE Handbook sizing equations. This term is also included in some, but not most, expansion tank manufacturers’ selection
software. Most manufacturers conservatively ignore this term since it is small and no larger than the terms already ignored in the above Equation. Ignoring this term results in Equation below:
$$V_t = \frac{(((v_h/v_c) – 1) V_s)}{(P_s/P_i) – (P_s/P_{max})}$$
The numerator is the volume of the expanded water, V[e] , as it warms from minimum to maximum temperatures, so the equation can be written:
$$V_t = \frac{V_e}{\frac{P_s}{P_i} – \frac{P_s}{P_{max}}}$$
$$V_e = (v_h/v_c – 1) V_s$$
The equation can be further simplified based on the style of tank used.
Vented tank
For vented tanks, the pressures are all the same and the dominator limits to 1, so the tank size is simply the volume of expanded water:
$$V_t = V_e$$
Closed Tank (no precharge)
For unvented plain steel tanks, the starting pressure is typically atmospheric pressure with the tank empty (no precharge). The tank is then connected to the makeup water, which pressurizes the tank
to the fill pressure by displacing air in the system, essentially wasting part of the tank volume. So the sizing equation is:
$$V_l = \frac{V_e}{\frac{P_a}{P_i} – \frac{P_a}{P_{max}}}$$
Where, P[a] = atmospheric pressure
Precharged Tank
For any tank that is precharged to the required initial pressure, including properly charged diaphragm and bladder tanks, but also including closed plain steel tanks if precharged, P[s] is equal to P
[i] so the sizing equation reduces to:
$$V_t = \frac{V_e}{1 – \frac{P_i}{P_{max}}}$$
Note that this equation only applies when the tank is precharged to the required P[i] . Tanks are factory charged to a standard precharge of 12 psig (83 kPag).
Closed Tank
For higher desired precharge pressures, either a special order can be made from the factory or the contractor must increase the pressure with compressed air or a hand pump. But it is not uncommon for
this to be overlooked. This oversight can be compensated for by sizing the tank using Equation below (assuming atmospheric pressure at sea level):
$$V_t = \frac{V_e}{\frac{26.7}{P_i} – \frac{26.7}{P_{max}}}$$
(12 psig/26.7 psia [83 kPag/184 kPaa] precharge). This will increase the tank size vs. a properly precharged tank.
ASME Boiler and Pressure Vessel Code-2015, Section VI
ASME Boiler and Pressure Vessel Code-2015, Section VI, includes sizing equations (as do the UMC and IMC, which extract the equations verbatim), as shown in Equation below, with variables revised to
match those used in this article:
$$V_t = \frac{V_s(0.00041T_h – 0.0466)}{\frac{P_a}{P_i} – \frac{P_a}{P_{max}}}$$
Comparing the denominator of this Equation to Equation for Closed Tank (no precharge), this formula is clearly for sizing a nonprecharged tank; it will overestimate the size of a precharged tank. The
numerator is a curve fit of V[e] ; it assumes a minimum temperature of 65°F (18°C) and is only accurate in the range of about 170°F to 230°F (77°C to 110°C) average operating temperature. Therefore,
this equation cannot be used for very high temperature hot water (e.g. 350°F [177°C]), closed-circuit condenser water, or chilled water systems.
Author: Steven T. Taylor, PE
What are the primary functions of an expansion tank in a closed hydronic system?
An expansion tank in a closed hydronic system serves four primary functions: (1) to accept changes in system water volume as water density changes with temperature, (2) to maintain a positive gauge
pressure in all parts of the system to prevent air from leaking into the system, (3) to maintain sufficient pressures in all parts of the system to prevent boiling, including cavitation at control
valves and similar constrictions, and (4) to maintain net positive suction head required (NPSHR) at the suction of pumps. These functions are crucial to ensure the safe and efficient operation of the
What are the consequences of undersizing an expansion tank in a closed hydronic system?
Undersizing an expansion tank can lead to several consequences, including increased system pressure, reduced system efficiency, and potential equipment damage. Insufficient tank capacity can cause
the system to exceed the pressure rating of equipment and piping components, leading to premature failure or even catastrophic failure. Additionally, undersizing can result in inadequate pressure
maintenance, allowing air to enter the system and causing corrosion, erosion, and other issues.
How do I determine the required expansion tank size for my closed hydronic system?
To determine the required expansion tank size, you need to calculate the total volume of the system, including the volume of water in the pipes, radiators, and other components. You should also
consider the maximum expected temperature change in the system, as well as the pressure rating of the equipment and piping components. Using formulas such as the one provided in the ASHRAE Handbook
or other industry resources, you can calculate the required tank size based on these factors. It’s essential to consult with a qualified engineer or technician to ensure accurate calculations and
proper tank sizing.
What are the differences between open and closed expansion tanks, and when would I use each?
Open expansion tanks are vented to the atmosphere and are typically used in open systems where the tank is not pressurized. Closed expansion tanks, on the other hand, are pressurized and used in
closed systems where the tank is subjected to system pressure. Closed tanks are more common in modern hydronic systems due to their ability to maintain a positive pressure and prevent air from
entering the system. Open tanks are often used in older systems or in applications where the system pressure is relatively low. The choice between open and closed tanks depends on the specific system
requirements and design.
Can I use a standard formula to calculate the expansion tank size, or are there other factors to consider?
While standard formulas can provide a good starting point for calculating expansion tank size, there are other factors to consider, such as system complexity, piping layout, and equipment
specifications. For example, systems with multiple loops or zones may require larger tanks to accommodate the additional volume changes. Additionally, the type of fluid used in the system, such as
water or glycol, can affect the tank sizing calculation. It’s essential to consider these factors and consult with industry resources or a qualified engineer to ensure accurate tank sizing.
How often should I inspect and maintain my expansion tank to ensure optimal system performance?
Regular inspection and maintenance of the expansion tank are crucial to ensure optimal system performance and prevent potential issues. It’s recommended to inspect the tank at least annually,
checking for signs of corrosion, damage, or leakage. Additionally, the tank should be drained and cleaned periodically to remove sediment and debris that can affect its performance. The frequency of
maintenance may vary depending on the system design, operating conditions, and local regulations. Consult with a qualified technician or the tank manufacturer’s guidelines for specific maintenance | {"url":"https://hvac-eng.com/expansion-tank-sizing-formulas/","timestamp":"2024-11-08T09:15:09Z","content_type":"text/html","content_length":"263493","record_id":"<urn:uuid:8a3d7045-724c-4d83-9368-98f5a0647759>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00753.warc.gz"} |
Research type and sample size: Is there a correlation? - forms.app
The sample size refers to the number of participants in a sample size in market research. The sample size considers a group of participants chosen from the general population representing the study’s
target population.
When determining the target sample size, it is essential to avoid the tendency to choose an unjustifiably small population or the inclination to select an unjustifiably large population. Several
factors affect sample size. The foremost of these factors is the research type.
When you determine the sample size, you may consider the research type. This article will explain the correlation between the kind of research and the sample size, the standard sample size, how to
determine the correct sample, and the sample size formula in all detail.
Is there a standard sample size for some research studies?
There are resource and statistical considerations when determining sample sizes. When the population is large, 100 participants are typically considered the minimum sample size. However, the proposed
data analysis type and the anticipated response rate are influential determinants of the sample size in most studies.
The majority of statisticians agree that a sample size of 100 is necessary to obtain any kind of significant results. If your population is less than 100, you should survey every single person.
You can consider some factors when determining the appropriate sample size, such as research type, confidence level, or standard deviation. The types of research methods have a strong relation to
sample size. Before you determine the sample size, you should understand your research type.
Choosing a sample size for a correlational study
The correlational study aims to find the relationship between two or more variables and their causes and effects. Due to the study’s use of numerical data and lack of variable manipulation, it can be
said that it falls under the category of nonexperimental quantitative research.
According to Fraenkel Wallen, a correlational study's minimum acceptable sample size is at least 30. Additionally, they state that data from samples smaller than 30 may not reflect the degree of
Choosing a sample size for a clinical study
A crucial step in planning a clinical study is determining the sample size. The sample size must be carefully planned to avoid wasting money, staff time, and research resources. A study’s inability
to identify significant treatment effects due to inadequate sample size is not uncommon. The specified statistical hypotheses and a few study design factors determine an appropriate sample size.
These include the desired statistical power, the significance level, the desired statistical power and the minimal meaningful detectable difference (effect size).
Choosing a sample size for studies with repeated measures
Collecting repeated measures can lower study costs while simultaneously boosting statistical power for spotting changes. The correlations between repeated measurements from the same participant must
be considered when determining the proper sample size.
How to determine the correct sample size
Factors to consider for the correct sample size
You should understand the statistics and consider several factors affecting your research to select the appropriate sample size. After that, you can use a sample size formula to put everything
together and sample with assurance because you will know that there is a good chance your survey is statistically accurate.
The following steps are appropriate to find a sample size for data whose continuous data is counted numerically. You should not apply it to categorical data. Determine a few details about the target
population and the required level of accuracy before you can calculate a sample size:
1 - Population size: You should define who belongs and who doesn’t belong in your group to determine the total number of people you refer to. For example, if you research cars and want to learn about
car owners, you will include anyone who has owned at least one car at some point. If you cannot calculate the exact number, don’t worry. It is typical to have an undetermined total or an estimated
2 - The margin of error: Errors are unavoidable. You can decide how much variance between the means of your sample and the population you want to allow.
3 - Confidence level: Confidence level refers to your assurance that the actual mean will fall within your margin of error. 90%, 95%, and 99% confidence levels are the most typical confidence
4 - Standard deviation: The formal derivation asks you to predict the degree to which the responses you receive will differ from one another and the mean value.
The values all be grouped around the mean if the standard derivation is low, as opposed to high formal derivation. This indicates that the matter is dispersed over a much more comprehensive range
with extremely small and large outlying figures.
Since your survey hasn’t been administered yet, a safe option is a standard derivation of 0.5, which will help ensure that your sample size is adequate.
The sample size formula
The sample size formula uses the distinction between the population and sample to help you determine the appropriate sample size. This sample size is the number of observations within a particular
sample population. Since it is impossible to screen the entire population, a sample is taken, and a survey or surveys are conducted.
The sample size formula is calculated in two steps. First, the sample is determined for the entire population, and then the sample size is disturbed among the necessary people. The formula is as
1 - Find the Z-score
The Z-score displays how far a given ratio deviates from the mean by standard deviation. You should convert your level of confidence into a Z-sore. The Z-scores for the most typical confidence levels
are listed below:
• 90% Z-score = 1.645
• 95% – Z Score = 1.96
• 99% – Z Score = 2.576
2 - Utilize the sample size formula
After calculating your Z-score, standard deviation, and confidence interval for the sample size, you can start using the formula. You can use the following formula to perform the calculation
• [z2 X p(1-p)] / e2 / 1 + [z2 X p(1-p)] / e2 X N]
• N = population size
• e = Margin of error
• z = z-score
• p = standard of deviation
Utilize the sample size formula
In conclusion, the sample size is crucial for accurate, statistically significant results and a successful study. The research is more effective when the sample is appropriate because the results are
trustworthy, and the use of resources is kept to a minimum while upholding ethical standards. There is a correlation between the research type and sample size. You can consider the research type and
other essential factors mentioned above when determining the right sample size.
Sample size calculations directly impact research results. Very small sample sizes compromise a study’s external and internal validity. In contrast, conducting the fundamental research will be
difficult, expensive, and time-consuming if the sample is too large. To choose the correct sample size for your research, you should understand the underlying statistics and the research design types
and consider several variations. | {"url":"https://forms.app/es/blog/correlation-between-research-type-and-sample-size","timestamp":"2024-11-05T16:02:14Z","content_type":"text/html","content_length":"196324","record_id":"<urn:uuid:9137f261-9f6c-4231-92e3-fd1cafd17b00>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00404.warc.gz"} |