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Average for Nested Bar instead of Total, or another option?
Good morning. We'd like to show the average of the two bars in this card, and ideally it would be using a Nested Bar chart but open to other suggestions (currently using a Grouped Bar chart).
The Value is a beast mode, which takes into consideration the actual sale amount less the valuation, divided by the valuation: (CASE WHEN `Valuation` = 0 THEN 0 ELSE (SUM(`SaleAmount` - `Valuation`))
/ SUM(`Valuation`) END)
The Series consists ot the two (2) stores, the bars in the cart would be Store 1 and Store 2.
Any thoughts on how we can achieve this, i.e. an average bar, or line, to show the average of both Valuation to Sale Amount Variances by Store? Thanks
• Have you tried utilizing the Line+nested bar chart type?
In this example I've created a simple beast mode for an average of my values for A1, A2, and A3 divided by 3 and plugged that into the Y Axis dimension.
**Say "Thanks" by clicking the heart in the post that helped you.
**Please mark the post that solves your problem by clicking on "Accept as Solution"
• Thanks, @JasonAltenburg, believe that got us in the right direction!
Wat we've done is separate the "stores" out into their own columns (Store 1 & Store 2), and you can see from image 1 that only the row of data that applies to that store appears in the store
specific column that was created with a beast mode.
However, we're still not able to properly calculate the individual stores so their bar shows the correct variance seen in image 2 (top portion). The total is correct on the existing card (top)
and the new card (bottom), but you'll see the difference in the store variances and the new card (bottom) is incorrect.
Believe it may have something to do with the null values in the newly created columns, and have tried several beast modes to account for them but haven't been able to get it so they reconcile
with the individual store variances from the existing card (top).
This beast mode, which separates the stores into new columns, is where we also have to calcualte the correct percentage of 2.95% for each, and this is what we've come up with so far that brings
us closest to what we need but still not accurate.
WHEN `ParentStore` = 'Store 1' THEN (SUM(IFNULL(`Valuation To Sale Amount Variance`,0)) / COUNT(IFNULL(`Valuation To Sale Amount Variance`,0)))
Any additional assistance would be greatly appreciated!
• Any ideas on why we're unable to get the individual stores to reconcile? Is there anything additional that we could provide to help understand the ask better? Thank you in advance for any
assistance on this!
• Could you try this as your beastmode?
WHEN `ParentStore` = 'Store 1' THEN IFNULL(`Valuation To Sale Amount Variance`,0) end)
when `ParentStore`='Store 1' and ABS(ifnull(`Valuation To Sale Amount Variance`,0))>0 then 1 else 0 end)
I'm pretty sure that this expression will count the 0's as a value as well:
COUNT(IFNULL(`Valuation To Sale Amount Variance`,0)))
which is why the average was getting thrown off.
“There is a superhero in all of us, we just need the courage to put on the cape.” -Superman
• I did a little bit of testing to confirm my previous post:
I think it really depends on your expected result. I used three different beastmodes to test the outcomes of each.
1. count ifnull
This was testing the denominator in your original field. This does, indeed, count a value for every row even if there was a null in the dataset.
2. count()
This will include a count for any row with any value (exlcudes null values)
3. sum()
sum(case when ABS(IFNULL(`count`,0))>0 then 1 else 0 end)
This will tell you the number of rows with a value (excludes 0's and nulls)
“There is a superhero in all of us, we just need the courage to put on the cape.” -Superman
• @ST_-Superman-_would it be ok to DM you, maybe provide some of the raw data we're working with via Excel?
Still unable to get this to work as expected, not sure if it has to do with WHAT we're doing or maybe related to the information provided in this thread not being thorough enough to get the
correct feedback on a solution. Thanks!
This discussion has been closed.
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For Less Assignments And Alternatives by Olaniyan Evelyn. | cmonionline.com
It is important to give students an assignment because it reinforces what they learn at school, and ultimately help them learn the material better. However, too much assignment is not helpful.
Excessive time spent on completing an assignment can take away a students’ social life, family time, and also limits the students’ participation in sports or other extra-curricular activities.
Therefore, the amount of assignment a teacher can give to students should be restricted, and only assigned due to necessity. For example, a practical like estimating the number of houses in the
student’s street which should be done at home.
An author, Tamin Ansary, reports that since 1981, the number of assignments teachers give to students of the average sixth grader has increased by more than fifty percent. Though many teachers defend
large amounts of assignments helps prepare students for a competitive world but the truth is, it doesn’t give the students access to their families like they are supposed to have. The author of The
End of Homework, Dr. Kralovec, argues that doing homework during school has little or no effect on the successful study skills of students. As an educator, I empathize with the argument that
assignments are often random and can take unrealistic amounts of time. Having that in mind, I constantly consider the assignment to be given to my pupils. As each day approaches, I weigh the purpose
of assignments and consider if the assignments are making a positive impact not only on the students learning but also on their home school connection with their parents.
Many teachers tend to believe that assignment is their sacred cow, and believe me; I understand the value of assignments as once a student and also an educator. I firmly believe students should
practice on their own, this is a great help to their academic performance, but while I believe assignments are important, I have also understood that the number of assignments teachers should give is
Many a time’s teachers and even some parents believe busywork is a waste of everyone’s time, but if the assignments are given for busywork, then it is a waste of time indeed. It wastes the students’
time doing it, their parents’ time helping with the assignment, and the teachers’ time tracking and even grading it. I would say, just because a worksheet is part of the curriculum or part of the
Teachers lesson, doesn’t mean it should be assigned. If it doesn’t serve the purpose, then what is the essence of using it?
Another reason why fewer assignments should be given is that; more work doesn’t mean more learning. Sometimes, teachers give assignments because they feel like it is studious to do, but giving more
assignments doesn’t mean students will learn more, especially if the assignments are busywork, and if the students are overwhelmed. They might even know how to do it correctly.
Family time is valuable to students. Teachers teach students how to relate well with families even in school and giving them much assignment will take away the time they spend with their families. If
teachers need students to build a strong relationship, then they need not take up all the time meant to be spent with their families. It is true that for lots of students it is the TV that’s their
companion at night instead of spending time with their parents, but that’s not how all students have it. There a different family who loves to relax together in the evening but cannot because their
children are occupied with assignments and this has created bridges in many families.
Many assignments cause students in both high school and junior high school to stay up until midnight or even later. When extracurricular activities such as sport, cooking, etc.., are added to the
students’ activities, they may even have to wake up early the next morning, thereby cutting short their normal time of sleep. Some teachers and even parents argue that it is beneficial to reduce
students’ academic work after their school activities. However, reducing the few hours students are supposed to use for play and exercise could be a factor in increasing obesity rates.
The goal of school should be to teach students how to learn and most importantly to love learning as a lifelong process. No one wants to hand his/her students fish; the goal is to teach them how to
fish. Many times, teachers get overwhelmed and give assignments to cover up materials they didn’t have sufficient time to teach in class. Assignments should not be to teach the class, but instead, it
should be used to practice what is been taught in class.
There are great ways teachers can reduce assignments for effective learning and a good relationship between students, teachers and parents will be foster. Teachers should eliminate all busywork.
Before they give any assignment, they should ask themselves what point is the assignment for.
Teachers can ask students’ to write down how long it took them to complete an assignment. They can be asked to simply track this at the top of their assignment. Though their answers might not be
accurate, they should give a general idea. Teachers should endeavour to pay special attention to how long it takes a struggling student to complete their assignments. If it is taking too long, then
the teacher needs to be creative to figure out how s/he can shorten the assignment. The goal is to give as little as possible, not to add more even if the students are getting it done quickly.
Some assignments can be turned into classwork. Just because something is given as an assignment doesn’t mean it has to be an assignment. The teachers can find time for the students to do it in class.
I believe the quality of work will increase when this is done.
Olaniyan Evelyn, a graduate of Obafemi Awolowo University, Ile-ife is a God-lover and a professional educator passionate about seeing the best future of every student of the country through the best
educational system. She can be reached at olaniyanevelyn25@gmail.com | {"url":"https://cmonionline.com/2020/11/28/for-less-assignments-and-alternatives-by-olaniyan-evelyn/","timestamp":"2024-11-14T18:40:19Z","content_type":"text/html","content_length":"240667","record_id":"<urn:uuid:8d426809-f9c7-4924-9493-7ebb6bb98913>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00156.warc.gz"} |
Wave Visualizer
This is my final project for CSC 471 Introduction to Computer Graphics. For this project I wanted to create some sort of physics visualization tool, to combine my interests in physics and computer
science. I decided to make a tool that shows simple solutions to the wave equation. I settled on this idea beacause visualizing waves with their respective equations is both visually pleasing and
also educational. Having a good geometric intuition can make working with equations much easier, and this tool provides that link.
Waves from multiple sources interfering constructively and destructively (preset #8):
How to Use
• A, D, Up, Down: Rotate camera angle left, right, up, down
• W, S: Move camera forward, backward.
• 1-9: Select a solution to the wave equation.
• 0: Reset to z=0, also resets t to 0.
The equation in the top left is a billboard with a textue that is an image of the equation with a transparent background. I wrote the equations in LaTeX, and then converted them to images with
Roger's Online Equation Editor. The billboard also isn't passed any information about the view matrix, making it somewhat of a Heads-up Display. In the vertex shader its final position is simply:
gl_Position = vec4(vertPos,1);
The mesh is a 200x200 grid similar to the mesh used for the terrain landscape labs. All the points are moved closer together to make the wave seem more continuous. The height value of the mesh is
then just determined by which equatinon is selected. I left the mesh as a wireframe because I felt it provided more information about the wave while not taking away much of what makes it look nice.
It is easier to see exactly how the waves move in time, and since this was an educational tool, I felt it appropriate.
Here is a quick video of me flipping through all the presets of the program. The ball in the middle is shown to see how the space is being distorted. | {"url":"https://rnevils.dev/waves.html","timestamp":"2024-11-05T07:34:39Z","content_type":"text/html","content_length":"4474","record_id":"<urn:uuid:25a991db-68bd-4b7b-80a1-eebf30eda7be>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00252.warc.gz"} |
Statistics for Data Science and Machine Learning
Population vs. Sample
A population consists of the entire set of individuals or items that are the subject of a statistical study. It encompasses every member that fits the criteria of the research question.
• Characteristics:
□ Comprehensive: Includes all individuals or items of interest.
□ Parameters: Measurements that describe the entire population. Examples of parameters include:
☆ Population Mean (μ): The average of all values in the population.
☆ Population Standard Deviation (σ): A measure of the dispersion of values in the population.
Example: All students enrolled in a university.
A sample is a subset of the population selected for the purpose of analysis. It allows researchers to draw conclusions about the population without examining every individual.
• Characteristics:
□ Subset: A smaller, manageable group chosen from the population.
□ Statistics: Measurements that describe the sample. Examples of statistics include:
☆ Sample Mean (x̄): The average of all values in the sample.
☆ Sample Standard Deviation (s): A measure of the dispersion of values in the sample.
Example: A group of 200 students chosen randomly from a university’s total enrollment.
Mean, Median, and Mode
The mean, or average, is a measure of central tendency that is calculated by summing all the values in a dataset and then dividing by the number of values.
Mean (x̄) = (Σx) / N
• Σx is the sum of all values in the dataset.
• N is the number of values in the dataset.
For the dataset: 2, 4, 6, 8, 10
Mean (x̄) = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6
The median is the middle value of a dataset when it is ordered from least to greatest. If the dataset has an odd number of observations, the median is the middle value. If it has an even number of
observations, the median is the average of the two middle values.
For an odd number of observations: Median = middle value
For an even number of observations: Median = (middle value 1 + middle value 2) / 2
For the dataset (odd number): 1, 3, 3, 6, 7, 8, 9
Median = 6
For the dataset (even number): 1, 2, 3, 4, 5, 6, 8, 9
Median = (4 + 5) / 2 = 9 / 2 = 4.5
The mode is the value that appears most frequently in a dataset. A dataset can have more than one mode if multiple values have the same highest frequency, or no mode if all values are unique.
Mode = value with the highest frequency
For the dataset: 1, 2, 2, 3, 4, 4, 4, 5, 5
Mode = 4
Variance and Standard Deviation
Variance measures the spread of a set of numbers. It represents the average of the squared differences from the mean, providing a sense of how much the values in a dataset deviate from the mean.
For a population:
Variance (σ²) = Σ (x - μ)² / N
For a sample:
Variance (s²) = Σ (x - x̄)² / (n - 1)
• Σ is the sum of all values.
• x is each individual value.
• μ is the population mean.
• x̄ is the sample mean.
• N is the total number of values in the population.
• n is the total number of values in the sample.
For the sample dataset: 2, 4, 4, 4, 5, 5, 7, 9
1. Calculate the sample mean (x̄): x̄ = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5 2. Calculate each (x - x̄)²: (2 - 5)² = 9 (4 - 5)² = 1 (4 - 5)² = 1 (4 - 5)² = 1 (5 - 5)² = 0 (5 - 5)² = 0 (7 - 5)² = 4 (9 - 5)² = 16 3. Sum of squared differences: Σ (x - x̄)² = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32 4. Calculate the variance: s² = 32 / (8 - 1) = 32 / 7 ≈ 4.57
Standard Deviation
Standard deviation is the square root of the variance. It provides a measure of the spread of the values in a dataset in the same units as the data, making it easier to interpret.
For a population:
Standard Deviation (σ) = √(Σ (x - μ)² / N)
For a sample:
Standard Deviation (s) = √(Σ (x - x̄)² / (n - 1))
Using the variance calculated above (s² ≈ 4.57):
Standard Deviation (s) = √4.57 ≈ 2.14
Correlation Coefficient
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
• r = 1: Perfect positive correlation
• r = -1: Perfect negative correlation
• r = 0: No correlation
The Pearson correlation coefficient (often denoted as r) is calculated using the following formula:
r = Σ((x - x̄)(y - ȳ)) / √(Σ(x - x̄)² * Σ(y - ȳ)²)
• x and y are the individual values of the two variables.
• x̄ and ȳ are the means of the two variables.
• Σ denotes the summation over all data points.
• r > 0: Positive correlation (as one variable increases, the other tends to increase).
• r < 0: Negative correlation (as one variable increases, the other tends to decrease).
• r = 0: No linear correlation.
• The closer r is to 1 or -1, the stronger the correlation.
Consider two variables, X (hours of study) and Y (exam scores), for a group of students:
Hours of Study (X) Exam Scores (Y)
• Calculate the means (x̄ and ȳ).
• Calculate the deviations from the means (x – x̄ and y – ȳ).
• Square the deviations and sum them.
• Multiply the deviations of X and Y, sum them, and divide by the product of the square roots of the sum of squared deviations.
r ≈ 0.98
The correlation coefficient r is approximately 0.98, indicating a strong positive linear relationship between hours of study and exam scores. As hours of study increase, exam scores tend to increase
as well.
Point Estimation
Point estimation is a statistical method used to estimate an unknown parameter of a population based on sample data. It involves using a single value, called a point estimate, to approximate the true
value of the parameter.
Key Concepts
• Population: The entire group of individuals, items, or events of interest in a statistical study.
• Parameter: A numerical characteristic of a population that is unknown and typically of interest in statistical analysis. Examples include the population mean, population proportion, population
variance, etc.
• Sample: A subset of the population from which data is collected.
• Point Estimate: A single value, calculated from sample data, that serves as the best guess for the true value of the population parameter. It is denoted by a specific symbol, such as “x̄” for a
point estimate of parameter “μ”.
Properties of Point Estimates
• Unbiasedness: A point estimate is unbiased if its expected value is equal to the true value of the parameter being estimated.
• Efficiency: An efficient point estimate has the smallest possible variance among all unbiased estimators of the parameter.
• Consistency: A consistent point estimate converges to the true value of the parameter as the sample size increases.
Point Estimate Symbols
• Population Mean: “μ”
• Sample Mean: “x̄”
• Population Variance: “σ²”
• Sample Variance: “s²”
• Population Standard Deviation: “σ”
• Sample Standard Deviation: “s”
Suppose we want to estimate the mean income of all households in a city. We collect a random sample of 100 households and calculate the mean income of the sample (“x̄”). We use “x̄” as our point
estimate of the population mean income (“μ”).
An estimator is a statistical function or rule used to estimate an unknown parameter of a population based on sample data. It calculates a point estimate, which serves as the best guess for the true
value of the parameter.
Types of Estimators
1. Unbiased Estimator: An estimator whose expected value is equal to the true value of the parameter being estimated.
2. Consistent Estimator: An estimator that converges to the true value of the parameter as the sample size increases.
3. Efficient Estimator: An estimator with the smallest possible variance among all unbiased estimators of the parameter.
Biased and Unbiased Estimators
Unbiased Estimator
An estimator is unbiased if its expected value is equal to the true value of the population parameter it is estimating. In other words, an unbiased estimator does not systematically overestimate or
underestimate the parameter.
Example: Sample Mean as an Unbiased Estimator of Population Mean
• The sample mean (“x̄”) is an unbiased estimator of the population mean (“μ”). This means that, on average, the sample mean will equal the population mean when taken over many samples.
Formula for Sample Mean:
x̄ = Σx / n
• Σx is the sum of all sample values.
• n is the number of sample values.
Biased Estimator
An estimator is biased if its expected value is not equal to the true value of the population parameter it is estimating. A biased estimator systematically overestimates or underestimates the
Example: Sample Variance as a Biased Estimator of Population Variance
• The sample variance calculated using the formula with “n” in the denominator (instead of “n-1”) is a biased estimator of the population variance (“σ²”). This formula tends to underestimate the
true population variance, especially for small sample sizes.
Biased Formula for Sample Variance:
s²_biased = Σ(x - x̄)² / n
• Σ(x – x̄)² is the sum of squared deviations from the sample mean.
• n is the number of sample values.
To correct this bias, we use Bessel’s correction, replacing “n” with “n-1” in the denominator, which provides an unbiased estimator of the population variance.
Unbiased Formula for Sample Variance:
s²_unbiased = Σ(x - x̄)² / (n - 1)
• Σ(x – x̄)² is the sum of squared deviations from the sample mean.
• n is the number of sample values.
Hypothesis Testing
Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves making an initial assumption (the null hypothesis) and
determining whether the sample data provides sufficient evidence to reject this assumption in favor of an alternative hypothesis.
Key Concepts
• Null Hypothesis (H₀): The statement being tested, typically representing no effect or no difference. It is assumed to be true unless the data provides strong evidence against it.
• Alternative Hypothesis (H₁ or Ha): The statement we want to test for, representing an effect or a difference. It is accepted if the null hypothesis is rejected.
• Significance Level (α): The threshold for determining whether the evidence is strong enough to reject the null hypothesis. Common significance levels are 0.05, 0.01, and 0.10.
• Test Statistic: A standardized value calculated from sample data, used to determine whether to reject the null hypothesis. Different tests have different test statistics, such as the t-statistic
or z-statistic.
• p-value: The probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than or equal to the significance level, we
reject the null hypothesis.
• Type I Error (α): The error made when the null hypothesis is wrongly rejected (false positive).
• Type II Error (β): The error made when the null hypothesis is not rejected when it is false (false negative).
Steps in Hypothesis Testing
1. State the Hypotheses:
□ Null Hypothesis (H₀): Example – The population mean is equal to a specified value (μ = μ₀).
□ Alternative Hypothesis (H₁): Example – The population mean is not equal to the specified value (μ ≠ μ₀).
2. Choose the Significance Level (α):
□ Common choices are 0.05, 0.01, or 0.10.
3. Select the Appropriate Test and Calculate the Test Statistic:
□ Depending on the sample size and whether the population standard deviation is known, choose a test (e.g., z-test, t-test).
□ Calculate the test statistic using the sample data.
4. Determine the p-value or Critical Value:
□ Compare the test statistic to a critical value from statistical tables or calculate the p-value.
5. Make a Decision:
□ If the p-value ≤ α, reject the null hypothesis (H₀).
□ If the p-value > α, do not reject the null hypothesis (H₀).
6. Interpret the Results:
□ Draw conclusions based on the decision made in the previous step.
Example: t-Test
The t-test is a statistical test used to determine whether there is a significant difference between the means of two groups or between a sample mean and a known population mean. It is particularly
useful when the sample size is small and the population standard deviation is unknown.
Types of t-Tests
1. One-Sample t-Test: Tests whether the mean of a single sample is significantly different from a known population mean.
2. Independent Two-Sample t-Test: Tests whether the means of two independent samples are significantly different.
3. Paired Sample t-Test: Tests whether the means of two related groups (e.g., measurements before and after treatment) are significantly different.
Key Concepts
• Null Hypothesis (H₀): The hypothesis that there is no effect or no difference. It assumes that any observed difference is due to sampling variability.
• Alternative Hypothesis (H₁ or Ha): The hypothesis that there is an effect or a difference. It suggests that the observed difference is real and not due to chance.
• Degrees of Freedom (df): The number of independent values or quantities that can vary in the analysis. It is used to determine the critical value from the t-distribution table.
• Significance Level (α): The threshold for rejecting the null hypothesis. Common significance levels are 0.05, 0.01, and 0.10.
• Test Statistic: A value calculated from the sample data that is used to make a decision about the null hypothesis.
One-Sample t-Test
Purpose: To determine if the sample mean is significantly different from a known population mean.
t = (x̄ - μ) / (s / √n)
• x̄ is the sample mean.
• μ is the population mean.
• s is the sample standard deviation.
• n is the sample size.
1. State the hypotheses.
2. Choose the significance level (α).
3. Calculate the test statistic (t).
4. Determine the critical value or p-value.
5. Make a decision and interpret the results.
Independent Two-Sample t-Test
Purpose: To determine if the means of two independent samples are significantly different.
t = (x̄₁ - x̄₂) / √[(s₁² / n₁) + (s₂² / n₂)]
• x̄₁ and x̄₂ are the sample means.
• s₁² and s₂² are the sample variances.
• n₁ and n₂ are the sample sizes.
1. State the hypotheses.
2. Choose the significance level (α).
3. Calculate the test statistic (t).
4. Determine the degrees of freedom (df).
5. Determine the critical value or p-value.
6. Make a decision and interpret the results.
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Tangents Archives - Stumbling Robot
Consider the cubic curve
Find values for
There is a second line passing through the point
First, we sketch the graph of
We compute the derivative of
So, the slope of the tangent line at the point
Then, since
Therefore, the line is
Next, if there is another line, say
Since it must pass through the point
So the line
Finally, since the point
Since these are both equal to
Find the point at which a given line is tangent to a given curve
Consider the curve given by the cubic equation
Show that the line
First, we compute the derivative of the curve,
For the line
Next, for the line to be tangent to the curve the point must actually be on the curve. So, we test out the two possible values of
This tangent line also intersects the curve at
Find values of constants so two polynomials intersect with the same slope at a point
Find values of
Since we want
Next, we compute the derivatives so that we can find the slope of the tangent lines at this point,
Since these must be the same at the point
Then from above we know
Therefore, the values of the constants are
Find constants so a quadratic has a given tangent at a particular point
Find values for
First, we compute the derivative
So, if the line
Next, the point
So, the values are
Find the points at which the given function has slope zero
Consider the function
Find the points
First, the derivative is given by
Then, the requirement that the graph of
Find points of a function at which the tangent line has specified values
Consider the function
Find the points at which the slope of
1. 0;
2. -1;
3. 5.
The derivative is given by
1. The slope is 0 means
2. The slope is -1 means
3. The slope is 5 means
Find points at which the tangent to a given function is zero
Find the points at which the tangent line to the function
is horizontal.
First, we compute the derivative,
Setting this equal to zero we have, | {"url":"https://www.stumblingrobot.com/tag/tangents/","timestamp":"2024-11-09T13:34:34Z","content_type":"text/html","content_length":"99778","record_id":"<urn:uuid:89ff5540-bf47-4a32-86df-b617422650fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00206.warc.gz"} |
tests/web_2/instantiation_stub_test.dart - sdk.git - Git at Google
// Copyright (c) 2018, the Dart project authors. Please see the AUTHORS file
// for details. All rights reserved. Use of this source code is governed by a
// BSD-style license that can be found in the LICENSE file.
// @dart = 2.7
// dart2jsOptions=--strong
import 'package:expect/expect.dart';
// This needs one-arg instantiation.
T f1a<T>(T t) => t;
// This needs no instantiation because it is not closurized.
T f1b<T>(T t1, T t2) => t1;
class Class {
// This needs two-arg instantiation.
bool f2a<T, S>(T t, S s) => t == s;
// This needs no instantiation because it is not closurized.
bool f2b<T, S>(T t, S s1, S s2) => t == s1;
int method1(int i, int Function(int) f) => f(i);
bool method2(int a, int b, bool Function(int, int) f) => f(a, b);
int method3(int a, int b, int c, int Function(int, int, int) f) => f(a, b, c);
main() {
// This needs three-arg instantiation.
T local1<T, S, U>(T t, S s, U u) => t;
// This needs no instantiation because but a local function is always
// closurized so we assume it does.
T local2<T, S, U>(T t, S s, U u1, U u2) => t;
Expect.equals(42, method1(42, f1a));
Expect.equals(f1b(42, 87), 42);
Class c = new Class();
Expect.isFalse(method2(0, 1, c.f2a));
Expect.isFalse(c.f2b(42, 87, 123));
Expect.equals(0, method3(0, 1, 2, local1));
Expect.equals(42, local2(42, 87, 123, 256)); | {"url":"https://dart.googlesource.com/sdk.git/+/refs/tags/2.18.0-243.0.dev/tests/web_2/instantiation_stub_test.dart","timestamp":"2024-11-10T11:56:23Z","content_type":"text/html","content_length":"24034","record_id":"<urn:uuid:b83f0c10-a535-4245-b3e6-97daddcbcbae>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00614.warc.gz"} |
Data Science is Here
Oh great, now we have to teach another new class? Exhausted teachers everywhere
A lot of people are starting to talk about Data Science.
Universities around the globe are creating specialized Masters of Data Science programs, and job prospects for Data Scientists are even better than for Computer Scientists. It's only a matter of time
before undergraduate majors in Data Science start cropping up, and within 3-5 years we'll see calls for Middle and High Schools to start teaching it too.
Even Code.org is starting to think about Data Science, and their curriculum manager asked four terrific questions about the goals for a Data Science course on Facebook. We've been been actively
supporting our Data Science course for a year now (and have been thinking about this problem a year before that!), so we answered him and shared what we've learned. We'd like to share our knowledge
on this topic publicly for anyone who's thinking about making a K12 Data Science course.
1. What should students learn? What should they produce?
Data Science uses math and programming, but it can't be a math and programming course. Sure, students will make use of datatypes, functions, iteration/loops, linear regression, measures of center and
variation, etc. — but Data Science is all about turning questions into programs and making meaning from the results! Students should discuss threats to validity, learn to think carefully about
outliers, and do a ton of talking and writing about their analysis.
Companies don't want just "coders". Every Engineer or CS major knows someone who can write a thousand lines of horrible, bug-riddled code that compiles and runs, but can't document their code or read
a spec. Likewise, it would be be terrible mistake teaching kids to master Excel, R or Python, but not how to interpret results and write about their findings.
A good Data Science class should be a good mix of math and programming, but the final project can't just be a program or a good-looking chart. Students should choose real research questions and write
real research papers, using appropriate language and precision to explain their thinking.
2. Can I just use spreadsheets? If not, what's the best coding language? Is it okay if students just make charts?
Spreadsheets aren't enough. We could write a lot on this subject, but Jesse Adler already did. Ideally, you'd want a course that lets you start with spreadsheets and then seamlessly transition into
programing. But don't worry about the tools right now: pick your learning goals first, then find the tools that help you get there.
Charts and Plots aren't enough. They're usually necessary, but they're never sufficient. Students need to touch the data — this is where things get messy, interesting, and important! Suppose
disaggregating a dataset by gender shows strong correlations, where none were found to begin with; what does that mean? Maybe some outliers are confounding an analysis, but when we look at those
outliers we find something essential that we overlooked! When we train teachers in Data Science, touching the data is where things get real: this is where it all comes down to being able to defend
and explain their own thinking. That's a powerful realization!
The best tools get out of the way quickly, so you can teach the concepts instead of the language. A language for an introductory Data Science class should make it easy for students to dig into data,
without spending a lot of time on syntax or special libraries. Does your language require that students learn about for-loops just to filter a dataset? Do you need to spend a week or so on "intro
programming" before you can write your first query on real data? Are the error messages designed with young learners in mind? How much time are you spending teaching a language (Python, Snap, C, or
Java...) vs. teaching Data Science?
A good Data Science class should use appropriate tools to do real analysis and manipulation. There's research out there on the importance of authenticity. Kids (and adults) need to get their hands
dirty! A good language makes it possible to teach students how to sort, filter, and extend a dataset, visualize data in multiple representations, and do some simple programming.
3. Should we focus more on technical skills or the impact on society?
We think this is a false dichotomy. Focusing on impact without teaching a specific skillset risks becoming an empty, feel-good class in which we all talk about data (and maybe make some charts), but
don't actually do anything. Focusing on a skillset without connecting it to real impact will result in tons of kids learning how to load a CSV file, and which commands create graphs...and they'll
forget it by the end of next summer. And even if they don't, who cares? Inert knowledge is where CS Education goes to die.
A good Data Science class should teach a good mix of skills, grounded in engaging, authentic projects kids care about.
4. Is this more CS or Math? Won't the math scare students away?
Yes, it's CS. Teaching real, rigorous programming helps make this clear — yet another reason they need to touch the data, and why spreadsheets alone aren't enough.
But yes, it's also math. And that's ok! Math Education also has a problem with inert knowledge: people dislike math because it's rarely situated in real projects. They think "I'll never use this!",
and it becomes an inauthentic exercise in symbol pushing. Rigor isn't the problem, but some folks in CS think that the solution is to push rigor as far away as possible for fear of "spoiling the
But Math is a fundamental part of Data Science! It's not possible without it. Rather than tuck our tails between our legs, let's embrace Data Science as the ultimate answer to "when am I ever going
to use math?" and be proud of it. At Bootstrap, we reach nearly 25,000 students every year — primarily in underserved schools where math phobia is high — and we've found that rigor is a very good
thing when it's tied to a project that matters. Ask any child who just learned how to make toast or beat a videogame: they figured it out; they solved it; they know how to do it. That's what makes it
fun! If there's no rigor, there's nothing to solve. Nothing to crack, and no feeling of "YES!!!!" when it all comes together. We'd argue that you can't have fun without rigor.
Is the earth warming? Are Tom Brady or Lionel Messi the G.O.A.T? Does the school I go to matter more than the grades I get? Is stop-and-frisk racist? Who's got the best pizza in town? These are all
questions that kids care about, which can be answered with rigorous analysis and mountains of publicly-available data. This is awesome stuff, and we shouldn't shy away from the rigorous math.
A good Data Science class should fully embrace rigor, be it on the CS side or the math side of the equation. It should make it clear life is messy, and that rigorous (and repeatable, and
explainable!) analysis is how we get the answers to things that matter.
5. So where does this new course fit?
At Bootstrap, we know that there are a finite number of hours in the day and rooms in the building. A curriculum that has to be a standalone course will forever be relegated to "opt-in" status. Maybe
the kids with the means and inclination will take it, but that's all. We think every child is a data scientist, so Data Science is for everyone, which is why we've created a curricular module that
can be flexibly adapted and integrated into:
• Computer Science teachers are always in search of good, interesting projects — and there's data for every interest a student might have! Our Data Science module assumes no prior programming
background, yet it gets students working with real data right away. We think this is a great way to introduce computing and programming. In a few years, almost every CS1 course is going to have a
significant Data Science component; you can get a head start!
• The AP CS Principles course is structured around seven big ideas, and one of them is Data. A real Data Science module could easily be dropped into a CSP class, which would go a whole lot deeper
than the those lessons do now. The resulting research paper and program could even be used for the Create Task! In fact, some schools are already doing this using our Data Science module.
• Business classes often have students spend time learning to use spreadsheets. Students make charts, learn to program formulas, and write reports on their analysis of sales data, financial trends,
etc. These classes could the exact same concepts using a Data Science module, doing the same activities but with some real programming in place of a spreadsheet.
• Statistics classes have the math part down, but they have a reputation (undeserved, for many!) of being dry. Students learn about linear regression and r^2, but echoes of algebra remain: "When am
I ever going to use this?" Data Science is the mechanism to put the math to use and the question to rest, just as our Bootstrap:Algebra module has done for tens of thousands of students every
• Social Studies classes are what make us the most excited. Social Studies teachers don't get nearly enough respect from the STEM field, and we think that's a problem. These are the classes where
students look at everything from immigration to national policy, or at the impact of things like the Irish Potato Famine to the Electoral College. When we talk about laws, trade, or even climate,
we're talking about data. And Social Studies teachers care deeply about making meaning from that data: they know why writing effectively about these subjects matters. One of our pilot classes was
a Social Studies class, which explored the impact of poverty on academic performance — heavy stuff, but it was something deeply personal and relevant to the students. Instead of only talking
about poverty, they looked at the data to draw conclusions.
A good Data Science class shouldn't even have to be its own class! It should be tailored to the content domain, so it's applied in context to the classes where it makes sense. It should support and
reinforce the business, statistics, or social studies class in which it's embedded, and in return it will be supported by putting Data Science where it should be: in context.
You can check out
our Data Science module
right now, and integrate it into your Computing, Business, Stats or Social Studies course!
Posted November 12th, 2018 | {"url":"https://www.bootstrapworld.org/blog/curriculum/Data-Science-Is-Coming.shtml","timestamp":"2024-11-04T04:54:06Z","content_type":"text/html","content_length":"16032","record_id":"<urn:uuid:18d80dd3-3269-4c9b-98f4-ede18a06178f>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00745.warc.gz"} |
I cant realx! :'( really sorry need to vent..
i dont know why but im is the worst mood ever since waking up.. i just feel like crying because i had bloodwork done yesterday and its driving my up the wall not knowing the results
Sorry about that im just really fustrated
Hope everyone is well x
Nov 12, 2007
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Have some of these and try not to stress too much
Hello babes, not sure if you read my post in your HUGE announcement yesterday! (You've got bloody good eggs lady!! So damn chuffed for you I really am!)
Anyway, I just wanted to say that although I've never been through mc myself, my Mum has had 3 and a stillborn at 20 weeks so I've always grown up knowing the stress and devastation it causes. BUT,
one mc is unfortunate and although there is a chance, it's more likely to be a healthy pregnancy than not, but to have two consecutive mc is much more unlikely.
I hope that's some comfort to you, but I do again understand that no words can make you feel any better about things, I just hope the next few weeks speed up so we can all see bubs at your 12 week
Oh and I'll just add - I'm placing my bets early but I'm feeling a strong BLUE for you
Mar 3, 2008
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Aww Im sorry your feeling so stressed!!
ok, just to update and ask a few questions, im feeling really low, my mother-in-law managed to get ahold of my results for me and my heart sank when she said my blood result was negative
Oooh don't panic just yet hun!!
Because your very early days, there probably will be times when it's negative because it's so early, so please try not to worry!
I had 4 positive home tests and one confirmed by the doctor then went to the chemist to have one there and the lady said "you're not pregnant" it had come back completely negative, no line what so
Well, I was! xx
thanks hun..
i thought that blood tests were deffinate though??
if its negative does that mean they only found <5mlu?
Jun 27, 2007
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Sep 18, 2006
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Elle'sMumy said:
thanks hun..
i thought that blood tests were deffinate though??
if its negative does that mean they only found <5mlu?
dont want to dampen things 4 u hun but i had a chemical prgncy last summer and ater tons of faint HPT's i had blleding a week after AF was due and had some blood tests done at the docs the day after
the bleeding started and was told it was negative
the gaeyni doc and midwifw in our surgery both saw me and said that blood tests are so accurate as they can trace little amounts of the hormone..this is why they do blood tests on women who dont get
BFP on HPT early on.
i remember reading in TTC that u only had faint positives in ur pics ?? may be this is a chemical prgncy
the one thing i learnt form mine is that ill never test until AF is at least a week late and ill never trust a faint BFP ever again , with christopher my second line came up thick and fast and was so
so dark...this is how i knew somethin was wrong when i had my chemical ..
all i can suggest is u see ur doctor and find out why this is happening ? im so so sorry once again
Mar 3, 2008
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Im sorry Elles mummy!!
I really dont know anything about blood test etc but you must be so stressed!
Jun 16, 2005
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OpenStax College Physics for AP® Courses, Chapter 23, Problem 57 (Problems & Exercises)
The 4.00 A current through a 7.50 mH inductor is switched off in 8.33 ms. What is the emf induced opposing this?
Question by
is licensed under
CC BY 4.0
Solution video
OpenStax College Physics for AP® Courses, Chapter 23, Problem 57 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. The induced EMF will equal the self-inductance times the rate of change of current and so that’s seven and a half milli-henry times change in four
amps because we are told that it switched off and so its going from four to zero, so the change is four amps and it takes 8.33 milli-seconds for this change to happen and so we have 8.33 ten to the
minus three seconds there and this means the induced EMF will be 3.6 volts. | {"url":"https://collegephysicsanswers.com/openstax-solutions/400-current-through-750-mh-inductor-switched-833-ms-what-emf-induced-opposing-0","timestamp":"2024-11-04T02:24:48Z","content_type":"text/html","content_length":"236455","record_id":"<urn:uuid:bb6d560e-4f50-4061-bc4b-6691cfdf04a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00657.warc.gz"} |
The Architect's Desktop
I was looking to try to automate some life-safety-related tasks in Revit
today, specifically related to occupant load calculations, and set out to create a Dynamo graph. I wanted to be able to round up the Area parameter value derived from an Area object, so that a whole
number would be used (matching the calculated value in a Schedule) and I also wanted to round up any "fraction of a person" that results from dividing the Area value by the area per occupant value
(as specified in the governing building code). I did not see a "Round Up" node in the standard Dynamo nodes, just two different Round nodes, one to round to the nearest whole number and one where you
can specify the number of decimal places in the rounded number. (I am currently using the 1.2 release - yes, I know I am behind.) So I set out to create my own Round Up node. NOTE: As my interest
here is with positive numbers and rounding up to the next largest whole number for values that are not already a whole number, I did not worry about how negative numbers should be treated (should
they round toward or away from zero?). If your application does involve negative numbers, you will need to determine which direction they should round, and adjust the node definition accordingly.
The image above shows the definition of the
Round Up
node that I created. (As always, you can select an image to see the full-size version.) The math behind it is fairly straightforward:
• The Input node takes a number as input.
• The Math.Floor node takes the input number and truncates any fractional part.
• The x subtract y. node subtracts the result of the Math.Floor node from the original number, to determine the fractional value (if any).
• The x less y? node compares that fractional value to zero, generating a value of true if it is greater than zero, or false if not.
• The If node uses that true/false value as the test input, and passes along a value of 1 if the test value is true or 0 if the test value is false.
• Finally, the Adds x to y. node adds the truncated result of the Math.Floor node to the value of the If node, and passes this along to the Output node as the result of this custom node.
So, if there is a fractional amount, one is added to the whole number portion of the input value; otherwise, zero is added to the whole number portion, which is "rounding up".
While testing this as part of my Dynamo graph, I noticed that one of the Area values, which was reporting as 100 square feet in Revit (after using the RoundUp function in Revit) was unexpectedly
rounding up to 101. I took a look at the node values and discovered that an Area that was inside boundaries that formed a 10'-0" square was reporting an area of 100.000000000002 square feet inside
Dynamo. While I always want to round any true fractional value up, I decided that it was unreasonable to assign two occupants to that 100 square foot Office (at 100 square feet per occupant) for a
non-zero value in the twelfth decimal place. In my mind, that is a computational error. So I came up with another custom node, called
Round Up with Floor
(see image below). This custom node definition has all of the same nodes as the
Round Up
node, except the
Code Block
that supplies the zero value to the
x less y?
node is replaced with a second
node. That allows you to specify the value above which the rounding up will occur. I still need to do additional testing to determine at what value the RoundUp function in Revit will actually round
up. For my first test of the
Round Up with Floor
node, I used a floor value of 0.000001 (one millionth of a square foot), and that eliminated the rounding up of the one area with the very small fractional amount.
As I start thinking about deploying AutoCAD^® Architecture and AutoCAD^® MEP 2018, I have been considering how to manage having multiple file formats in use at the same time. Not that we have not had
to deal with that in the past; but we have been primarily using 2013-format releases for quite some time, so I will need to make and keep users aware of the fact that once a file is saved in the 2018
format, there is no going back.
One "trick" I personally have used to check on the file format of a file prior to opening it is discussed in this Autodesk Knowledge Network article. You can determine the file format of an AutoCAD^®
drawing file by opening the file in a plain text editor (like Notepad) and looking at the first six characters. That article lists the "codes" for the 2000 through 2013 file formats. Shaan Hurley, in
this article in his Between the Lines blog, has a more complete listing of the codes, including the fact that the new 2018 file format is AC1032. Scroll down to the bottom of the article, under the
DWG File History header, for the full listing.
One drawback to that trick is that really large files can take quite some time to open in Notepad, and there is always the risk (however small) that you could accidentally make a change and then save
the file in Notepad. It occurred to me that an AutoLISP routine ought to be able to use the read-line function to read the first line of a drawing file, extract the first six characters, and then
report the results; this proved to be true. You do have to have an instance of AutoCAD open first, but if you do, the routine works much faster than opening in Notepad for large files, and there is
no risk of changing the file. I chose to limit the versions for which it tests to the AC1001 (Version 2.2) format. I was not able to fully test all of those, as I was unable to find a file in my
archives that was last saved in anything earlier than Release 9 (AC1004). If your archives include files of earlier vintage, you could extend the code for the earlier versions listed in Shaan's
article, assuming that those older files have the "code" in the first six characters.
(defun C:FMT ( / file1 sfile1 sline1 stext1)
(setq sfile1 (getfiled "Select Drawing File" "" "dwg" 0)
file1 (open sfile1 "r")
) ;_ End setq.
(if file1
(setq sline1 (read-line file1)
stext1 (substr sline1 1 6)
) ;_ End setq.
(close file1)
(cond ; Condition A.
((= "AC1032" stext1)
"Header = AC1032."
"\nFile " sfile1
"\nis saved in the 2018 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A1.
((= "AC1027" stext1)
"Header = AC1027."
"\nFile " sfile1
"\nis saved in the 2013 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A2.
((= "AC1024" stext1)
"Header = AC1024."
"\nFile " sfile1
"\nis saved in the 2010 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A3.
((= "AC1021" stext1)
"Header = AC1021."
"\nFile " sfile1
"\nis saved in the 2007 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A4.
((= "AC1018" stext1)
"Header = AC1018."
"\nFile " sfile1
"\nis saved in the 2004 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A5.
((= "AC1015" stext1)
"Header = AC1015."
"\nFile " sfile1
"\nis saved in the 2000 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A6.
((= "AC1014" stext1)
"Header = AC1014."
"\nFile " sfile1
"\nis saved in the R14 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A7.
((= "AC1012" stext1)
"Header = AC1012."
"\nFile " sfile1
"\nis saved in the R13 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A8.
((= "AC1009" stext1)
"Header = AC1009."
"\nFile " sfile1
"\nis saved in the R11/R12 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A9.
((= "AC1006" stext1)
"Header = AC1006."
"\nFile " sfile1
"\nis saved in the R10 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A10.
((= "AC1004" stext1)
"Header = AC1004."
"\nFile " sfile1
"\nis saved in the R9 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A11.
((= "AC1003" stext1)
"Header = AC1003."
"\nFile " sfile1
"\nis saved in the Version 2.60 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A12.
((= "AC1002" stext1)
"Header = AC1002."
"\nFile " sfile1
"\nis saved in the Version 2.50 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A13.
((= "AC1001" stext1)
"Header = AC1001."
"\nFile " sfile1
"\nis saved in the Version 2.22 file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A14.
"Header = "
"\nFile " sfile1
"\nis saved in an unknown file format."
) ;_ End strcat.
) ;_ End alert.
) ;_ End condition A15.
) ;_ End condition A.
) ;_ End progn.
(prompt "\nNo file selected. ")
) ;_ End if.
) ;_ End C:FMT
Here is an example of the alert message that will be displayed. | {"url":"https://architects-desktop.blogspot.com/2017/06/","timestamp":"2024-11-06T09:08:37Z","content_type":"text/html","content_length":"133728","record_id":"<urn:uuid:4412e522-a264-41f3-a01d-02abc0158686>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00773.warc.gz"} |
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Thermax provides a complete range of water and wastewater treatment products with modular plug and play systems and online product water quality monitoring options. Standard products ensure fast
delivery, low maintenance and easy installation with less civil/site work. Terminator. Rice Mill ETP.
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Experimental. Elementary Ti (<40 μm, 99.9%) and C (5 μm, 99.9%) powder mixture was sealed into a stainless-steel vial with 5 stainless-steel balls (15 mm in diameter) in a glove box filled with
purified argon to avoid oxidation. The ball to powder weight ratio was 70:1. The milling process was performed at room temperature using a high-energy ...
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the data were collected from one of the Raw Material ball mill circuits (line 1) of the Ilam cement plant (Figure1). This plant has 2 lines for cement production (5300 t/d). The ball …
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Type CHRK is designed for primary autogenous grinding, where the large feed opening requires a hydrostatic trunnion shoe bearing. Small and batch grinding mills, with a diameter of 700 mm and more, …
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With an annual capacity to produce 108,000 tonnes of TMT Bars, Vinayak Steels Limited is the only steel manufacturer, in Telangana, with a captive Palettization Plant, Sponge Iron Plant, Steel
Melting Shop, Continuous Casting and Rolling Mills all located at its steel plant in Kottur which is on the outskirts of Hyderabad.
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CERAMIC LINED BALL MILL. Ball Mills can be supplied with either ceramic or rubber linings for wet or dry grinding, for continuous or batch type operation, in sizes from 15″ x 21″ to 8′ x 12′. High
density ceramic linings of uniform hardness male possible thinner linings and greater and more effective grinding volume.
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Overview of Ball Mills. As shown in the adjacent image, a ball mill is a type grinding machine that uses balls to grind and remove material. It consists of a hollow compartment that rotates along a
horizontal or vertical axis. It's called a "ball mill" because it's literally filled with balls. Materials are added to the ball mill, at ...
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Thermax is a leader in delivering water treatment plants for the diverse needs of industries. With 50 years of experience in designing, building and managing the construction of water treatment
projects, we create and implement tailored or standardised industrial water treatment solutions. Thermax water treatment technologies have a proven ...
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Registered in 2010,India THERMAX LIMITED has gained immense expertise in supplying & trading of Lean gas fired boilers, heater, exchange resin etc. The supplier company is located in Vadodara,
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At FL, tube mills are supplied in the range from 1.6 to 4.3 m diameter in the Fast Track Series (Fig. 12). The mills can be optionally operated as ball or rod mills for wet or dry grinding. The
advantage is low-cost, standardized solutions, that enable a fast changeover from existing mills. 2.5 Agitated ball mills
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Operating Range. Thermax process heating solutions reduced carbon footprint- the plant is equivalent to carbon capture by 3 lakh trees! Thermax made it possible! Upto 45% fuel cost saving using
biomass firing. Huskpac Ultra, the next generation rice husk fired boilers from Thermax, are designed to provide low cost heating with fuel ...
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The energy consumption of the total grinding plant can be reduced by 20–30 % for cement clinker and 30–40 % for other raw materials. The overall grinding circuit efficiency and stability are
improved. The maintenance cost of the ball mill is reduced as the lifetime of grinding media and partition grates is extended. 2.5.
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Thermex at the international MCE exhibition in Milan. June 6, 2022. Thermex Corporation, one of the world's leading manufacturers of water heating and heating equipment, will …
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High temperature of the ball mill will affact the efficiency. 3 For every 1% increase in moisture, the output of the ball mill will be reduced by 8% -10%. 4 when the moisture is greater than 5%, the
ball mill will be unable to perform the grinding operation. 5. The bearing of the ball mill is overheated and the motor is overloaded.
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the data were collected from one of the Raw Material ball mill circuits (line 1) of the Ilam cement plant (Figure1). This plant has 2 lines for cement production (5300 t/d). The ball mill has one
component, 5.20 m diameter, and 11.20 m length with 240 t/h capacity (made by PSP Company from Prerˇ ov, Czechia).
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The mill is based on standard modules and can be adapted to your plant layout, end product specifi cations and drive type. The horizontal slide shoe bearing ... 6 Ball mill for cement grinding Ball
mill for cement grinding 7 Optimum grinding efficiency The grinding media are supplied in various sizes to ensure optimum
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Advantages of Ball Mills. 1. It produces very fine powder (particle size less than or equal to 10 microns). 2. It is suitable for milling toxic materials since it can be used in a completely enclosed
form. 3. Has a wide application. 4. It …
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Ball Mill Design. A survey of Australian processing plants revealed a maximum ball mill diameter of 5.24 meters and length of 8.84 meters (Morrell, 1996). Autogenous mills …
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A ball mill is a type of grinder widely utilized in the process of mechanochemical catalytic degradation. It consists of one or more rotating cylinders …
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Peripheral discharge ball mill, and the products are discharged through the discharge port around the cylinder. According to the ratio of cylinder length (L) to diameter (D), the ball mill can be
divided into short cylinder ball mill, L/D ≤ 1; long barrel ball mill, L/D ≥ 1–1.5 or even 2–3; and tube mill, L/D ≥ 3–5. According to the ...
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The basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill sizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density,
desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum 'chunk size', product size as P80 and maximum …
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Thermax executed a biomass fired cogeneration plant, deploying a 33 TPH hybrid water tube superheated bi-drum boiler with a reciprocating grate, designed at 67 kg/cm2 pressure and 450°C temperature,
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ball mill & rod mill. 2. Different types: Commonly used ball mills include grate ball mills and overflow ball mills, while rod mills do not use grid lining board to discharge ore, and only have ...
Цааш унших | {"url":"https://pizzaddict.fr/12-05/2062.html","timestamp":"2024-11-02T00:06:08Z","content_type":"text/html","content_length":"44155","record_id":"<urn:uuid:cad0648f-fd2f-41f6-8692-a4733a7f18db>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00506.warc.gz"} |
Word Problems: Algebra
Mathematics, Algebra and Function, Problem Solving
Grade 5- 8
Students will learn about solving word problems using simple algebra functions.
Many basic word problems can be expressed in an equation format, which makes it easy to understand and solve.
Using Algebraic Symbols
You can use a letter of the alphabet to represent the unknown number in a problem. The equation is written so that the values on the left side of the equal sign equal the values on the right side of
the equal sign. Solve the equation so that the unknown value represented by a letter is alone on one side of the equal sign and the value of the unknown is on the other side of the equal sign.
Sample: Jennifer has $25.00. She needs $49.00 to buy a new school outfit. How much more money does she need? Write an equation this way: n>/i> (money needed) + 25 (money she has) = 49 (cost of
outfit); Solve the equation by subtracting 25 from each side. Jennifer needs $24.00 more.
n + 25 = 49
n + 25 - 25 = 49 - 25
n = 24
Axiom of Equality
The axioms of equality were used to help solve the basic equation above. Any value added, subtracted, multiplied, or divided to one side of the equal sign must be added, subtracted, multiplied, or
divided respectively to the other side.
Sample: A group of 5 girls decided to split evenly the $18.75 cost of a CD album by their favorite group. How much money did each girl spend? Write an equation. Solve for n (the amount each girl
spent) by dividing each side of the equation by 5. Each girl spent $3.75.
5n = $18.75
5n / 5 = $18.75 / 5
n = $3.75
Working with Two Unknown Quantities
You can use the same letter with an added or subtracted amount to represent two unknown quantities. Simplify and combine terms whenever possible.
Sample: Sammy's mother is 2 years more than 3 times as old as Sammy. Their combined age is 42. How old are Sammy's mother and Sammy? Write an equation where n equals Sammy's age. Let 3n + 2 = Sammy's
mother's age. Since the total of their ages equals 42, then n + 3n + 2 = 42; Then combine the terms: 4n + 2 = 42; Use the axioms of equality by subtracting 2 and then dividing by 4. Sammy is 10 years
old. His mother is 32 years old.
4n + 2 = 42
4n + 2 - 2 = 42 - 2
4n = 40
4n / 4 = 40 / 4
n = 10
• copies of the activity sheets (see the link below) | {"url":"https://www.teachercreated.com/lessons/199","timestamp":"2024-11-10T09:46:39Z","content_type":"text/html","content_length":"22669","record_id":"<urn:uuid:d69f9526-99d0-43d2-a0f7-22f307078797>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00742.warc.gz"} |
Fair slicing of the pizza - using math - Interactive MathematicsFair slicing of the pizza - using math
Fair slicing of the pizza - using math
By Murray Bourne, 14 Dec 2009
Here's the situation. You order a pizza and the guy who cuts it for you has something in his eye and does a pretty bad job.
His first cut (BC) misses the center (A) and subsequent cuts are unevenly spaced, but at least he manages to get the cuts all passing through a single point (P).
The question is - if 2 people are eating the pizza and they take alternate pieces, will they eat the same amount or will one of them get more?
It turns out there is some interesting math behind this problem.
You can read some of the solutions in The perfect way to slice a pizza, by Stephen Ornes of The New Scientist (requires a subscription)
Be the first to comment below. | {"url":"https://www.intmath.com/blog/learn-math/fair-slicing-of-the-pizza-using-math-3858","timestamp":"2024-11-09T17:37:17Z","content_type":"text/html","content_length":"126189","record_id":"<urn:uuid:77779167-e156-4aa7-a737-64e28e12ae76>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00115.warc.gz"} |
How to find the sum of an Arithmetic Geometric Series:Class 11 maths
Here, you will learn what is an arithmetic geometric series and methods of finding the sum of n terms of this series. You will also learn to calculate the sum to infinity. As a math teacher, I have
explained this topic endless number of times to students . It takes some time to comprehend the intricacy of the problems. Again, I advocate learning the method of solving, rather than the formula.
Watch this video to learn. | {"url":"https://www.mathmadeeasy.co/post/how-to-find-the-sum-of-an-arithmetic-geometric-series-class-11-maths","timestamp":"2024-11-02T17:28:57Z","content_type":"text/html","content_length":"1050594","record_id":"<urn:uuid:66b406da-47d0-45f6-9c51-7dba9bb3cec2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00557.warc.gz"} |
Introduction to Linear Algebra
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Introduction to Linear Algebra, 5th edition
Published by Pearson (March 7, 2017) © 2018
• Lee Johnson
• Dean Riess
• Jimmy Arnold
• Hardcover, paperback or looseleaf edition
• Affordable rental option for select titles
For introductory courses in Linear Algebra.
A modern classic
Introduction to Linear Algebra, 5th Edition is a foundation book that bridges both practical computation and theoretical principles. Due to its flexible table of contents, it is accessible both for
students majoring in the scientific, engineering, and social sciences and for students who want an introduction to mathematical abstraction and logical reasoning. To achieve this flexibility, the
book centers on 3 principal topics: matrix theory and systems of linear equations, elementary vector space concepts, and the eigenvalue problem. It can be used for a 1-quarter or 1-semester course at
the sophomore/junior level, or for a more advanced class at the junior/senior level.
This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.
Hallmark features of this title
• A gradual increase in the level of difficulty. In a typical linear algebra course, students find the techniques of Gaussian elimination and matrix operations fairly easy. Then, the ensuing
material relating to vector spaces is suddenly much harder. The authors have done three things to lessen this abrupt midterm jump in difficulty: 1. Introduction of linear independence early, in
Section 1.7. 2. A new Chapter 2, “Vectors in 2-space and 3-Space.” 3 .Introduction of vector space concepts such as subspace, basis and dimension in Chapter 3, in the familiar geometrical setting
of Rn.
• Clarity of exposition. For many students, linear algebra is the most rigorous and abstract mathematical course they have taken since high-school geometry. The authors have tried to write the text
so that it is accessible, but also so that it reveals something of the power of mathematical abstraction. To this end, the topics have been organized so that they flow logically and naturally
from the concrete and computational to the more abstract. Numerous examples, many presented in extreme detail, have been included in order to illustrate the concepts.
• Supplementary exercises. A set of supplementary exercises are included at the end of each chapter. These exercises, some of which are true-false questions, are designed to test the student's
understanding of important concepts. They often require the student to use ideas from several different sections.
• Extensive exercise sets. Numerous exercises, ranging from routine drill exercises to interesting applications, and exercises of a theoretical nature. The more difficult theoretical exercises have
fairly substantial hints. The computational exercises are written using workable numbers that do not obscure the point with a mass of cumbersome arithmetic details.
• Spiraling exercises. Many sections contain a few exercises that hint at ideas that will be developed later.
• Integration of MATLAB. We have included a collection of MATLAB projects at the end of each chapter. For the student who is interested in computation, these projects provide hands-on experience
with MATLAB.
• A short MATLAB appendix. A brief appendix on using MATLAB for problems that typically arise in linear algebra is included.
1. Matrices and Systems of Linear Equations.
Introduction to Matrices and Systems of Linear Equations.
Echelon Form and Gauss-Jordan Elimination.
Consistent Systems of Linear Equations.
Applications (Optional).
Matrix Operations.
Algebraic Properties of Matrix Operations.
Linear Independence and Nonsingular Matrices.
Data Fitting, Numerical Integration, and Numerical Differentiation (Optional).
Matrix Inverses and Their Properties.
2. Vectors in 2-Space and 3-Space.
Vectors in the Plane.
Vectors in Space.
The Dot Product and the Cross Product.
Lines and Planes in Space.
3. The Vector Space Rn.
Vector Space Properties of Rn.
Examples of Subspaces.
Bases for Subspaces.
Orthogonal Bases for Subspaces.
Linear Transformations from Rn to Rm.
Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting.
Theory and Practice of Least Squares.
4. The Eigenvalue Problem.
The Eigenvalue Problem for (2 x 2) Matrices.
Determinants and the Eigenvalue Problem.
Elementary Operations and Determinants (Optional).
Eigenvalues and the Characteristic Polynomial.
Eigenvectors and Eigenspaces.
Complex Eigenvalues and Eigenvectors.
Similarity Transformations and Diagonalization.
Difference Equations; Markov Chains, Systems of Differential Equations (Optional).
5. Vector Spaces and Linear Transformations.
Vector Spaces.
Linear Independence, Bases, and Coordinates.
Inner-Product Spaces, Orthogonal Bases, and Projections (Optional).
Linear Transformations.
Operations with Linear Transformations.
Matrix Representations for Linear Transformations.
Change of Basis and Diagonalization.
6. Determinants.
Cofactor Expansions of Determinants.
Elementary Operations and Determinants.
Cramer's Rule.
Applications of Determinants: Inverses and Wronksians.
7. Eigenvalues and Applications.
Quadratic Forms.
Systems of Differential Equations.
Transformation to Hessenberg Form.
Eigenvalues of Hessenberg Matrices.
Householder Transformations.
The QR Factorization and Least-Squares Solutions.
Matrix Polynomials and the Cayley-Hamilton Theorem.
Generalized Eigenvectors and Solutions of Systems of Differential Equations.
Appendix: An Introduction to MATLAB. Answers to Selected Odd-Numbered Exercises. Index.
Digital Learning NOW
Extend your professional development and meet your students where they are with free weekly Digital Learning NOW webinars. Attend live, watch on-demand, or listen at your leisure to expand your
teaching strategies. Earn digital professional development badges for attending a live session. | {"url":"https://www.pearson.com/en-us/subject-catalog/p/introduction-to-linear-algebra/P200000010197?view=educator","timestamp":"2024-11-15T04:43:52Z","content_type":"text/html","content_length":"862540","record_id":"<urn:uuid:8a4928ed-4af7-44e7-8666-ed86f21a5672>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00416.warc.gz"} |
Problem 1009 - TheMathWorld
Problem 1009
Let p, q, and r represent the following simple statements:
p: I lie on the sofa.
q: I take a nap.
r: I go jogging.
Write the following compound statement in its symbolic form.
I lie on the sofa and I take a nap, or I go jogging.
First, notice that this statement contains more than one connective.
I lie on the sofa and I take a nap, or I go jogging.
When compound statement containing more than one connective are expressed in words, commas are used to indicate which simple statement are to be grouped together.
Two simple statement that appear on the same side of a comma are grouped together in parentheses when the statement is written symbolically.
Since the compound statement.
I lie on the sofa and I take a nap appears to the left of the comma, the symbolic statement representing it is grouped within parentheses.
The symbolic statement expressing
I lie on the sofa and I take a nap
Is (p Λ q).
Next , write the symbol that represents the connective. Then symbolically write the statement the follows the connective.
I lie on the sofa and I take a nap, or I go jogging.
(p Λ q) V r
The symbolic form of the compound statement
I lie on the sofa and I take a nap, or I go jogging.
Is (p Λ q) V r. | {"url":"https://mymathangels.com/problem-1009/","timestamp":"2024-11-11T13:52:54Z","content_type":"text/html","content_length":"59100","record_id":"<urn:uuid:9d1d8327-4b60-4390-bfe1-7bfb014fda81>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00111.warc.gz"} |
Simple Harmonic Oscillator with Boundary Conditions
• Thread starter blizzardof96
• Start date
In summary: I don't like seeing the cosine of an angle greater than 1. And the problem as stated has no solution.In summary, we have a problem where we are trying to solve for the amplitude (A) and
phase constant (ø) of a spring undergoing simple harmonic motion. We are given two boundary conditions and the frequency of the motion. Using the equations for SHM and identities for cosine and sine,
we can set up a system of equations to solve for A and ø. However, when we solve for these values, we get an inconsistent solution which means that the given values may not make physical sense.
Moved from a technical forum, no template.
How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions:
(x1,t1)=(0.01, 0)
(x2,t2)=(0.04, 5)
x values are given in relation to the equilibrium point.
Equation of Motion for a spring undergoing SHM
Solution to the 2nd order ODE
Any hints or help would be much appreciated.
Our rules say that you must show your attempt at a solution before we are allowed to help you. Show us your work.
Attempt at a solution
(1) 0.01=Acos(ø)
(2) 0.04=Acos(5w+ø)
(2) becomes 0.04=Acos(408.4+ø)
This leaves us with the following equations to be solved for A and ø. This is where I am struggling.
(1) 0.01=Acos(ø)
(2) 0.04=Acos(408.4+ø)
One way to proceed is to divide the second equation by the first to eliminate A. What do you get? Remember the identities ##\cos(a+b) = \cos a*\cos b - \sin a*\sin b## and ##\sin x = \sqrt{1-\cos^
Maybe we shouldn't approach just by using the giving equation. Equation 2 in the OP post but deducing the formule from the start. Finding constants in x(t) equation and then proceed. How this sounds
Arman777 said:
Maybe we shouldn't approach just by using the giving equation. Equation 2 in the OP post but deducing the formule from the start. Finding constants in x(t) equation and then proceed. How this
sounds ?
Isn't this what OP is doing by writing down equations (1) and (2)? It's a system of two equations and two unknowns that OP is wondering how to solve.
kuruman said:
Isn't this what OP is doing by writing down equations (1) and (2)? It's a system of two equations and two unknowns that OP is wondering how to solve.
I am confused about something and it might be helpful to the OP.
Now we have a second order diff eqn and 2 boundry conditions. So we can see that the solution will have a form of
$$ x(t) =c_1sin(\sqrt (\frac {k} {m})t)+c_2cos(\sqrt (\frac {k} {m})t)$$ (Eq.1)
Since its a boundry condition I derived the conditon from $$m(d^2x/dt^2) + kx = 0$$ then I said, $$(d^2x/dt^2) + (k/m)x = 0$$ where λ=k/m and then we get the above equation.
Now we have 2 boundry conditions so I thought we should put those into the Eq.1 and then try to find c1 and c2. when we do that it seems that they are not both zero. So after that we need to do some
calculations to turn it into the like the ##cos(a+b)## thing as you said.
$$x(t) = Acos(ωt+Φ)$$
The cos(a+b) identity was helpful. This is how I solved for ø and A.
(1) 0.01=Acos(ø)
(2) 0.04=Acos(ωt+ø) where ωt is some constant k
(3) 0.04=Acos(k+ø)
using the identify cos(a+b)=cosa∗cosb−sina∗sinb we can proceed.
(4) 4cos(ø)=cos(k)*cos(ø)-sin(k)*sin(ø)
dividing through by cos(ø)
(5) 4=cos(k)-sin(k)*tan(ø)
(6) Arctan(cos(k)-4)/sin(k))=ø
ø= -77.36 degrees
Subbing back into original equations we get an ampitude(A) of 0.046.
Did you check your work? Do you get a displacement of 0.04 when you put your values for A and φ back in the original equation? Your method is correct.
On edit: There is something about this problem that bugs me. If the frequency is 13 Hz, the period is ##T = \frac{1}{13}## s. This means that in time ##t_2 = 5~s## the number of oscillations executed
is ##N = t_2/T = 65##. This is an integer which means that the oscillator at ##t_2## must have the same displacement as at time ##t_1 =0##.
Last edited:
kuruman said:
Did you check your work? Do you get a displacement of 0.04 when you put your values for A and φ back in the original equation? Your method is correct.
On edit: There is something about this problem that bugs me. If the frequency is 13 Hz, the period is ##T = \frac{1}{13}## s. This means that in time ##t_2 = 5~s## the number of oscillations
executed is ##N = t_2/T = 65##. This is an integer which means that the oscillator at ##t_2## must have the same displacement as at time ##t_1 =0##.
I checked my work and the solution agreed with initial eqn. The values were arbitrary so its possible that they don't make physical sense.
Thank you for your help.
Science Advisor
Homework Helper
Gold Member
blizzardof96 said:
6) Arctan(cos(k)-4)/sin(k))=ø
ø= -77.36 degrees
points out, the problem as stated is inconsistent and has no solution.
What value did you use for k?
FAQ: Simple Harmonic Oscillator with Boundary Conditions
What is a simple harmonic oscillator?
A simple harmonic oscillator is a physical system that exhibits periodic motion, meaning it repeats the same pattern over and over again. It is characterized by a restoring force that is directly
proportional to the displacement from the equilibrium position.
What are the boundary conditions for a simple harmonic oscillator?
The boundary conditions for a simple harmonic oscillator are the initial position and velocity of the system. These conditions determine the amplitude, frequency, and phase of the oscillation.
How does a simple harmonic oscillator behave near its equilibrium position?
Near its equilibrium position, a simple harmonic oscillator behaves like a linear system and follows Hooke's Law, which states that the restoring force is directly proportional to the displacement
from the equilibrium position.
What is the equation of motion for a simple harmonic oscillator?
The equation of motion for a simple harmonic oscillator is given by F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.
How are the period and frequency of a simple harmonic oscillator related?
The period and frequency of a simple harmonic oscillator are inversely related. The period is the time it takes for one complete oscillation, while the frequency is the number of oscillations per
unit time. As the frequency increases, the period decreases, and vice versa. | {"url":"https://www.physicsforums.com/threads/simple-harmonic-oscillator-with-boundary-conditions.955597/","timestamp":"2024-11-11T13:05:39Z","content_type":"text/html","content_length":"132367","record_id":"<urn:uuid:87fca7d0-0eb8-4149-b38a-e48ddb3c2cdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00491.warc.gz"} |
Bayesian Model Selection - (Bayesian Statistics) - Vocab, Definition, Explanations | Fiveable
Bayesian Model Selection
from class:
Bayesian Statistics
Bayesian model selection is a statistical method used to choose among different models based on their posterior probabilities, which are updated using observed data. This approach incorporates prior
beliefs about the models and quantifies uncertainty in the model selection process, making it particularly powerful in cases where multiple competing models exist. By evaluating the evidence provided
by the data for each model, Bayesian model selection helps to identify the most appropriate model for the underlying process being studied.
congrats on reading the definition of Bayesian Model Selection. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Bayesian model selection utilizes Bayes' theorem to compare the relative likelihood of different models given observed data.
2. It enables the incorporation of prior information about models, which can significantly influence model selection outcomes.
3. The marginal likelihood or evidence is often calculated for each model to assess how well it explains the observed data.
4. Model selection criteria such as Bayes factors can provide a quantitative measure for comparing models based on their posterior probabilities.
5. One of the strengths of Bayesian model selection is its ability to handle complex models and small sample sizes more effectively than traditional methods.
Review Questions
• How does Bayesian model selection use prior probabilities to influence model choice?
□ Bayesian model selection uses prior probabilities to incorporate existing beliefs about different models before any data is observed. These priors can reflect expert knowledge or previous
research findings. When new data becomes available, Bayes' theorem updates these prior beliefs to form posterior probabilities, allowing for a more informed decision on which model best
explains the observed data.
• Discuss the role of marginal likelihood in Bayesian model selection and how it differs from traditional methods.
□ In Bayesian model selection, marginal likelihood serves as a critical component in evaluating how well each model fits the observed data. It represents the probability of observing the data
under a specific model, integrating over all possible parameter values. Unlike traditional methods that may rely solely on point estimates or specific fit statistics, marginal likelihood
accounts for uncertainty in parameter estimation and allows for a more comprehensive comparison of models.
• Evaluate the advantages and limitations of Bayesian model selection compared to frequentist approaches to model selection.
□ Bayesian model selection offers several advantages over frequentist approaches, such as incorporating prior information and providing a natural way to quantify uncertainty through posterior
distributions. It is particularly effective in complex models and small sample sizes. However, it also has limitations, including dependence on subjective prior choices and potential
computational challenges with high-dimensional models. Understanding these pros and cons is essential for effectively applying Bayesian methods in practice.
"Bayesian Model Selection" also found in:
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something about Does God Play Dice?
10 Feb 2010
These days, I reread the book Does God Play Dice?.
The excellent book was written in Chinese on the internet, which even readers having just received junior-school education are able to fully understand. The author Capo made good use of his clear and
elegent writing style to expose the epic golden age of the development in quantum mechanics to us.
Max Planck put forward Planck’s law to explain the full spectrum of thermal radiation.
Albert Einstein explained the photoelectric effect. $E = h nu$
Niels Bohr proposed a new model of the atom which included quantized electron orbits.
Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. $\lambda = \frac{h}{p}$
Werner Heisenberg and Max Boren and Pascual Jordan put forward the idea about matrix mechanics, the first explanation to quantum mechanics.
Erwin Schrödinger made up his Schrödinger wave equation, the second explanation to quantum mechanics.
Heisenberg clarified Heisenberg uncertainty principle. $ \sigma_{x}\sigma_{p} \geq \frac{\hbar}{2}$
Wolfgang Pauli proposed the Pauli exclusion principle.
Paul Dirac produced a relativistic quantum theory of electromagnetism.
Just as the writer says, “There never has been such a theory that contributes to our development so much, while confusing us greatly.”
As a matter of fact, relativity is considered the top of “classical physics”, and quantum mechanics is regarded as the foundation of “modern physics”. However, a great number of physicists desire to
discover the universial theory, called theory of everything(TOE). In my point of view, I prefer to the Causal Dynamical Triangulations to Superstring. An article in July 2008 Scientific American
Magazine expressed the possibility of calculating the dimensions of our world. And I think that its mathematical meaning is clearer and easier-understanding (well, I’m not proffessional enough).
Newton’s theory combines things on the earth and in the sky; relativity explains almost all macro circumstances; quantum mechanics almost conquers the micro area. Hopefully some revolution can bring
us the ulimate secret of our world.
Maybe one day, our names will be carved on the memorial, for future people’s praise. | {"url":"https://www.fyears.org/2010/02/something-about-does-god-play-dice.html","timestamp":"2024-11-02T18:44:41Z","content_type":"text/html","content_length":"8226","record_id":"<urn:uuid:de18fe0d-ab9d-4a29-b0b7-170910d0b081>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00034.warc.gz"} |
Data Navigator for Schools
Example capital expenditure plot
This plot shows the expected total capital expenditure over time for building, IT, fixtures and fittings and motor vehicles. There are three different dropdowns on this page which are explained
Units dropdown
Units dropdown control
1. Actual value, which is the capital expenditure calculated from survey question 44. Capital Expenditure.
2. Percentage, calculated as the capital expenditure divided by total capital expenditure for each year.
3. Per pupil, which is the capital expenditure divided by the number of pupils. The total number of pupils is calculated from survey question 13. Number of pupils by boarding type as at 31 August.
Quantiles dropdown
Quantiles dropdown control
You have the option to display different quantile bands on the chat or none at all. A quantile is a value that divides a dataset into different parts, helping you understand how data is distributed.
As an example, the 75% quantile is the point in the dataset where 75% of the benchmark data falls below that point and 25% is above it.
Benchmark statistic dropdown
Benchmark statistic dropdown control
You have the option to display the benchmark mean or benchmark median for the capital expenditure. | {"url":"https://datanavigator.barnett-waddingham.co.uk/man/expenditure-expenditure_cap_projects-capex_details-manual.html","timestamp":"2024-11-11T11:23:47Z","content_type":"application/xhtml+xml","content_length":"30287","record_id":"<urn:uuid:ba9a13fb-571b-4e5b-befa-be41fa8685ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00736.warc.gz"} |
Problem D: Hexagon Cost
There a big hexagon, the side length is n, so there are \(3n^2+3n+1\) points, and \(9n^2+3n\) edges. Each point has a weight.
We define the weight of each edge as the multiplying result of the weight of two endpoints in it. Please find out the minimum cost to connect these \(3n^2+3n+1\) points. | {"url":"https://acm.sustech.edu.cn/onlinejudge/problem.php?cid=1109&pid=3","timestamp":"2024-11-08T12:09:44Z","content_type":"text/html","content_length":"9561","record_id":"<urn:uuid:0a556727-f068-4f4b-ae31-bd63fcd75e48>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00254.warc.gz"} |
Tran Kai Frank Da
This package offers a data structure encoding the whole family of alpha-complexes related to a given 2D Delaunay or regular triangulation. In particular, the data structure allows to retrieve the
alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first alpha value for which each face is
included on the alpha-complex).
This chapter presents a framework for alpha shapes. The description is based on the articles [2], [3]. Alpha shapes are the generalization of the convex hull of a point set. Let \( S\) be a finite
set of points in \( \mathbb{R}^d\), \( d = 2,3\) and \( \alpha\) a parameter with \( 0 \leq \alpha \leq \infty\). For \( \alpha = \infty\), the \( \alpha\)-shape is the convex hull of \( S\). As \( \
alpha\) decreases, the \( \alpha\)-shape shrinks and develops cavities, as soon as a sphere of radius \( \sqrt{\alpha}\) can be put inside. Finally, for \( \alpha = 0\), the \( \alpha\)-shape is the
set \( S\) itself.
We distinguish two versions of alpha shapes, one is based on the Delaunay triangulation and the other on its generalization, the regular triangulation, replacing the natural distance by the power to
weighted points. The metric used determines an underlying triangulation of the alpha shape and thus, the version computed. The basic alpha shape (cf. Example for Basic Alpha Shapes) is associated
with the Delaunay triangulation (cf. Section Delaunay Triangulations). The weighted alpha shape (cf. Example for Weighted Alpha Shapes ) is associated with the regular triangulation (cf. Section
Regular Triangulations).
There is a close connection between alpha shapes and the underlying triangulations. More precisely, the \( \alpha\)-complex of \( S\) is a subcomplex of this triangulation of \( S\), containing the \
( \alpha\)-exposed \( k\)-simplices, \( 0 \leq k \leq d\). A simplex is \( \alpha\)-exposed, if there is an open disk (resp. ball) of radius \( \sqrt{\alpha}\) through the vertices of the simplex
that does not contain any other point of \( S\), for the metric used in the computation of the underlying triangulation. The corresponding \( \alpha\)-shape is defined as the underlying interior
space of the \( \alpha\)-complex.
In general, an \( \alpha\)-complex is a non-connected and non-pure polytope, meaning that a \( k\)-simplex, with \( 0 \leq k \leq d-1\), is not necessarily adjacent to a \( (k+1)\)-simplex.
The \( \alpha\)-shapes of \( S\) form a discrete family, even though they are defined for all real numbers \( \alpha\) with \( 0 \leq \alpha \leq \infty\). Thus, we can represent the entire family of
\( \alpha\)-shapes of \( S\) by the underlying triangulation of \( S\). In this representation each \( k\)-simplex of the underlying triangulation is associated with an interval that specifies for
which values of \( \alpha\) the \( k\)-simplex belongs to the \( \alpha\)-shape. Relying on this result, the family of \( \alpha\)-shapes can be computed efficiently and relatively easily.
Furthermore, we can select an appropriate \( \alpha\)-shape from a finite number of different \( \alpha\)-shapes and corresponding \( \alpha\)-values. | {"url":"https://cgal.geometryfactory.com/CGAL/doc/master/Alpha_shapes_2/group__PkgAlphaShapes2Ref.html","timestamp":"2024-11-09T00:47:13Z","content_type":"application/xhtml+xml","content_length":"17782","record_id":"<urn:uuid:86e50dac-ba6e-4d2a-bc68-1546efc33c4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00149.warc.gz"} |
| Tool Preview
Must have a column with a measure of time and status (0,1) at observation.
Use a column name from the file header if the data has one, or use one from the list supplied below, or use col1....colN otherwise to select the correct column
Use a column name from the header if the file has one, or use one from the list supplied below, or use col1....colN otherwise to select the correct column
Special characters will probably be escaped so do not use them
The column names supplied for time, status and so on MUST match either this supplied list, or if none, the original file header if it exists, or col1...coln as the default of last resort.
If there are exactly 2 groups, a log-rank statistic will be generated as part of the Kaplan-Meier test.
This is a wrapper for some elementary life table analysis functions from the Lifelines package - see https://lifelines.readthedocs.io/en/latest for the full story
Given a Galaxy tabular dataset with suitable indicators for time and status at observation, this tool can perform some simple life-table analyses and produce some useful plots. Kaplan-Meier is the
default. Cox Proportional Hazards model will be tested if covariates to include are provided.
This is always performed and a survival curve is plotted. If there is an optional "group" column, the plot will show each group separately. If there are exactly two groups, a log-rank test for
difference is performed and reported
This is always performed and a survival curve is plotted.
If there is an optional "group" column, the plot will show each group separately. If there are exactly two groups, a log-rank test for difference is performed and reported
The Cox Proportional Hazards model can be tested, if a comma separated list of covariate column names is supplied on the tool form.
These are used in as covariates. Although not usually a real problem, some diagnostics and advice about the assumption of proportional hazards are are also provided as outputs - see https://
Although not usually a real problem, some diagnostics and advice about the assumption of proportional hazards are are also provided as outputs - see https://lifelines.readthedocs.io/en/latest/
A big shout out to the lifelines authors - no R code needed - nice job, thanks! | {"url":"https://toolshed.g2.bx.psu.edu/repository/display_tool?repository_id=d58a1461e31a4c11&tool_config=%2Fsrv%2Ftoolshed-repos%2Fmain%2F006%2Frepo_6928%2Flifelines_tool%2Flifelineskmcph.xml&changeset_revision=dd5e65893cb8&render_repository_actions_for=tool_shed","timestamp":"2024-11-05T08:49:03Z","content_type":"text/html","content_length":"9449","record_id":"<urn:uuid:adcbae3f-fbd3-40b8-82de-38408ec64b15>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00847.warc.gz"} |
e C
We are a leading manufacturer of Amusement rides and Water parks in India of International standards and have the credit of establishing more than *0 parks in India and abroad including euthopia,
tanzania, middle east. We have a wide range of products from small trains for kids to big adult rides like roller coaster and custom design rides.
We are the leading Infrastructure Development Organisation in India based in Pune, registered under the companies Act .***6 and major player in industrial park developments. We are the trusted name
in industrial parks developments with four decade of industrial experience and have successfully developed prominent industrial parks and other industrial plots size ***0 sq feet to *0 acres. Company
holds directly and indirectly more than **0 acres of land in Chakan, Mahalunge, Waki, Nighoje, Moi, Ambethan, Kharabwadi, Bhamboli, Wasoli and Markal.The major industrial hubs in Pune which is known
as €Detroit of India.€ (Automobile Hub€ ) We have successful track record development of Industrial Park and sold industrial land to many top well known companies in India since ***6. Our prestigious
projects include state of the art amenities that fulfill the perfect business requirements like common tar road, *2KV Electric Underground cable, Street Lights, logging out rain water, proposed...
Zhejiang Feiyou Kangti Amusement Facilities Co.,Ltd is a modern company which features production, sales and R&D. We are specialized in manufacturing all kinds of playgrounds and entertainment
equipment with various sizes. Our former, Wenzhou Feiyou Amusement Equipment Factory, was founded in ***9,due to several reasons such as the area limitations and the growing needs of the market, we
move to a new plant which covers an area of ****0 square meters. At the same time, we pay a lot of time to import advanced equipment and technology to ensure the quality of our products, to support
production ability. We have a wide range of products, that will offer you a whole amusement facility solution. Our products are popular among children. And its popularity with children that makes it
widely used in shopping malls, communities, kindergartens aboard. Our products are TUV certified conform to EN***6 to insure safety play, and CE certification conform to EN*1.We has been assessed
EFES Playgrounds is a new and dynamic company consisting of a team that has become an experienced expert after spending years in the matter of entertainment equipments. Its assertive and honest
structure which knows the balances has been the result of this team spirit. One of the priorities of our company, that meets all kind of needs as product and service about entertainment equipments
from the beginning to the end, is to present its knowledge and experiences in the harmony of product €” price €” quality. It never makes a concession from the quality of product and service, and is
rather assertive to cover the expectations on this point. To eliminate the problems likely to appear after finishing the work is a reflection of understanding of the team for "Quality + Continuous
Service = Happy Customer"¿&frac*2; Safety¿&frac*2; Production of playgrounds has not got established in our country yet. For that reason, security and manufacture standards have an exce...
Discovery Climbing System was founded as a¿&frac*2;manufacturer of climbing holds and¿&frac*2;walls in ***8 from South Korea. ¿&frac*2; Although climbing market in Asia has relatively short history,
its market size has¿&frac*2; ¿&frac*2; been growing incredibly fast last 5 years. In order to meet its high demand, we¿&frac*2;have put our best effort to provide a high quality of artificial
climbing holds and walls. ¿&frac*2; Over the past years, we have provided our fine products and services to many different¿&frac*2;countries such as Japan, China, Hong Kong, Taiwan, Singapore,
Malaysia, India,¿&frac*2;Russia, and etc. Thankfully, we were selected as the best climbing system manufacturer¿&frac*2; ¿&frac*2; in Asia and also the official IFSC speed wall constructor.¿&frac*2;
¿&frac*2; Not only Asia, our ambition is to expand our business to all over the world. ...
Cangzhou Warrior Outward Bound Appliance Co.,Ltd was founded in ***9 with registered capital of RMB *0 million. The company is located in Qing County, Cangzhou, Hebei Province, where is the juncture
of Tianjin and Cangzhou. Neighboring **4 national road and Beijing-Shanghai High Speed Railway and Beijing-Shanghai Expressway, it enjoys superior geological condition and convenient transportation.
Covering an area of ****0Ž¡, including factory area ****0Ž¡, our company is a modern company specialized in research, design, production, marketing, installation and service of €children development
land€ products.
EFFECI an Italian company leaders in entertainment business for over *0 years now in U.S.A. Our latest attraction is a Cinema 5d. Our CINEMA 5D are installed in many nations, EFFECI Movie 5D is
already present in Italy, France, Turkey, Tunisia, Egypt, Iran, Brazil. EFFECI CINEMA 5D was awarded Attraction of the Year at Heavent Paris EXPO of ***0. British television has filmed inside the
structure of its special service, recognizing EFFECI MOVIE 5D as a real "cultural phenomenon". November *4th to *9th IAAPA Orlando USA We will be at IAAPA with our engineers to meet all interested in
learning more about CINEMA 5D
Guangzhou Childhood Dream Recreation Equipment Co., Ltd. belongs to Happy Island Group Company , We concentrated on improving , producing and selling modern amusement equipment , mainly including
outdoor playground and outdoor fitness . The quality of our products are very good and passed the GS and CE certification and meet European standard , now sell all over the world and got good
Established in ***0, Wenzhou Xiaofeixia Amusement Equipment Co., Ltd. specializes in the production of amusement equipment, such as trampolines, bungee trampolines and swimming pools. All of our
products have obtained CE certificate and are of great innovation, strong participation feeling and high intellectualization. They sell well in Europe, America, South Korea, and the Middle East. We
have rich experience in the development and production. Also, we have a team of design, production and service with advanced technology. To meet our customers**9; requirements, Xiaofeixia will create
high-tech amusement equipment with remarkable function and greater innovation under the unremitting efforts of all staff. Our company takes the quality as foundation and integrity as our motto.
Hereby, we sincerely hope that we can establish smooth business relationship with you by supplying high quality products and competitive prices to you.
Future¿&frac*2;funworld¿&frac*2;is¿&frac*2;a¿&frac*2;company¿&frac*2;with¿&frac*2;deep¿&frac*2;roots¿&frac*2;in¿&frac*2;the¿&frac*2;Family¿&frac*2;Entertainment ¿&frac*2;and¿&frac*2;Amusement¿&
frac*2;industry. Originally¿&frac*2;started¿&frac*2;in¿&frac*2;***2¿&frac*2;by¿&frac*2;three¿&frac*2;people¿&frac*2;with¿&frac*2;years¿&frac*2;of¿&frac*2;industry¿&frac*2;and¿&frac*2;related¿&frac*2;
experience,¿&frac*2;Future¿&frac*2;funworld¿&frac*2;has¿&frac*2;become¿&frac*2;the famous manufacture in the world for indoor playground equipments who specializes on designing, manufacturing, and¿&
frac*2;District,is¿&frac*2;a¿&frac*2;famous Children amusement park equipment manufacturer¿&frac*2;with many years¿&frac*2;¿&frac*2;experiences¿&frac*2;¿&frac*2;in¿&frac*2;China,¿&frac*2;offering¿&
frac*2;various¿&frac*2;kinds¿&frac*2;of¿&frac*2;entertainment¿&frac*2;equipment¿&frac*2;products.¿&frac*2; Our¿&frac*2;factory¿&frac*2;covers¿&frac*2;8,**0¿&frac*2;squire¿&frac*2;meters,¿&frac*2;We¿&
Henan Kidle Amusement Equipment Technology Co., Ltd. provides customers with professional R&D, technical services and sales of amusement and amusement equipment; conference services; electronic
components, hardware tools; sales of electromechanical equipment, mechanical equipment and related accessories; import and export of goods and technologies business. (Projects that must be approved
in accordance with the law can only be carried out after approval by relevant departments). After our efforts and development, we have a certain scale and strength. Now we have a team with excellent
service quality and professional technical service strength. To provide higher and more considerate services to different groups of users, our company adheres to the business philosophy of customer
first and service first, and makes unremitting efforts with the spirit of stability, development, loyalty, efficiency, unity and innovation.
Parktime manufactures within the framework of the standards required by the quality certificates (T¿&frac*2;V, ISO ***1, ISO ****1, OHSAS ****1) that are valid in the international market. It is one
of our priorities in the production process that all materials selected during the production stage comply with the European EN ***6 standards, each material from raw material selection to paint
chemicals does not harm human health and has a high quality in terms of safety and durability. Our products are produced with polyethylene, which is child-friendly and environmentally friendly.
At Joshua Tree Lizard we offer fun and safe rock climbing tours in Joshua Tree National Park. If you are an individual, a couple, or a family looking for an adventurous, exciting, and once in a
lifetime experience, you are in the right place! | {"url":"https://www.tradekey.com/profile_list/cid/6136/Climbing-Walls.htm","timestamp":"2024-11-13T17:39:18Z","content_type":"text/html","content_length":"403596","record_id":"<urn:uuid:f579ad48-2d36-4658-8b5b-6b39d62f0d07>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00579.warc.gz"} |
Efficiency in boiler operations is serious business, and rightfully so. A coal steam boiler typically consumes 20 times the value of its capital investment in the cost of fuel during it economic
lifespan. So if one spends any amount of effort acquiring the right boiler for the job, one should by comparison spend 20 times as much effort in ensuring that every bit of fuel produces its maximum
contribution to the process of steam generation. Considering too that the basic design of the packaged fire tube boiler has not undergone any significant change since the times when coal was
generally plentiful, cheap and of excellent quality, making it so much harder for steam users to improve efficiency with odds stacked against their reasonable efforts.
It is therefore clear that achieving high boiler plant efficiency is no stroke of luck or something that falls into one’s lap. It takes hard work, understanding, dedication and excellent management
to continuously operate a steam boiler at peak efficiency. And all for one objective: to produce steam at the lowest cost per unit. But also understanding that assets must be preserved and the carbon
footprint minimized in pursuit of peak efficiency.
So let us jump straight into some of the definitions pertaining to “efficiency”, and there is a multitude of different interpretations of it; one can easily get totally confused. But in essence
efficiency expresses as a percentage the ratio of useful energy output from a system to the total energy input into the system. In a boiler environment the energy input comes from the fuel, the
energy output is found in the energy added to the produced steam.
In a perfect system without any energy loss the efficiency will be 100%. Unfortunately we live in a broken world and energy loss in any thermal system is a reality. The real challenge is minimizing
the energy loss from the system on a continuous basis, remembering always that energy once lost from the system cannot be recovered!
Our first challenge is to clarify the value of energy input per unit of fuel. Some schools prefer gross calorific value (GCV, also called higher heating value), others net calorific value (NCV, also
called lower heating value). The difference between the two is the latent heat of water vapour formed during combustion. Thus NCV = GCV – Latent Heat. Using GCV renders a lower efficiency percentage
than when NCV is used. So next time you encounter boiler efficiency statistics and comparisons, clarify beforehand if it is based on GCV or NCV. It can make a significant difference, particularly
with oil and gas fired boilers where the hydrogen content of the fuel is relatively high. With coal fired boilers it is customary to use GCV in efficiency calculations; with oil and gas fired boilers
it is NCV.
To complicate matters further efficiency can be calculated in two different ways; according to the direct method or to the indirect method.
The direct method is based on energy added to the steam as a percentage of the fuel energy input. The equation below renders the system (or overall or thermal) efficiency (not the boiler
efficiency!), where
1. The added energy is calculated as (Steam energy – energy of feed water), and
1. Steam energy = Steam Flow * Saturated steam enthalpy (the latter available from steam tables).
1. Feed water energy = Steam Flow * Feed water enthalpy, (also available from steam tables, or calculated = 4.2 * feed water temperature deg. C).
2. The fuel input energy = Fuel feed * Calorific value.
3. System efficiency % = Added energy * 100 / Fuel energy.
4. Now note that the direct method incorporates all energy losses in the system, including those encountered in the steam and condensate reticulation systems and in the production facility where the
steam is used. This is the most practical and popular way of calculating system efficiency, as all of the required parameters to do the calculation are normally readily available. The equation is
often simplified by substituting feed water flow for steam flow, which means blow down quantity is ignored. The bigger challenge is probably to accurately measure steam and fuel usages.
5. Physical steam and condensate losses are reflected in the make-up water quantity, while energy losses are reflected in the temperature of the feed water to the boiler (without any steam being
The indirect method starts off with 100% of fuel energy input and then deducts individual calculated or estimated energy losses encountered within the combustion process, such as stack loss, carbon
loss, blow down loss, shell loss, etc. The indirect method calculates energy loss across the boiler only and normally fails to address energy loss downstream of the boiler, i.e. along steam and
condensate lines and in steam application processes. These can however be approximated if the make-up water percentage and temperature is known, since make-up water replenishes downstream steam and
condensate losses. I use this approach in my combustion calculator; not absolutely accurate, but sufficiently useful for purposes of evaluating boiler performance under various operating conditions,
or with coal of differing characteristics.
In the Boiler Bits 9 issue we have identified the following significant energy losses from the boiler:
1. Stack loss, consisting of dry heat loss, combustibles loss and wet loss.
2. Shell loss, consisting of radiation and convection losses.
3. Bottom ash loss
4. Blow down loss
By definition then:
1. Combustion Efficiency % = (100% – Stack loss %)
2. Boiler Efficiency % = (Combustion Efficiency % – Shell Loss % – Bottom ash loss % – Blow down Loss %*)
3. System Efficiency % = (Boiler Efficiency % – Reticulation and Process Loss %).
* Some schools classify blow down loss as part of system loss.
One may encounter foreign terminology to express steam plant efficiency, or certain aspects thereof. In all instances make sure a clear understanding is gained of how this particular efficiency is
defined and calculated; particularly if efficiencies are compared between boilers as part of a new boiler acquisition investigation. It is absolutely essential that one compares apples with apples in
all instances.
In the next issue of Boiler Bits I will be discussing what levels of efficiency are considered top notch and achievable on a sustainable basis, specifically with chain grate fire tube boilers.
This post was compiled by René le Roux for Le Roux Combustion, all rights reserved. Do you want to know more about combustion control systems and combustion optimization? Please contact us for your
professional boiler automation, steam system efficiency and coal characterization needs.
Kindly note that our posts do not constitute professional advice and the comments, opinions and conclusions drawn from this post must be evaluated and implemented with discretion by our readers at
their own risk. | {"url":"https://leroux.mobi/blog/2020/05/12/boiler-bits-10-expressing-efficiency-of-steam-plant/","timestamp":"2024-11-03T21:50:24Z","content_type":"text/html","content_length":"95787","record_id":"<urn:uuid:8dd1508d-ca41-4d90-a75a-517714e96904>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00874.warc.gz"} |
Scaling (Curving) Grades Calculator - Calculator Pack
Scaling (Curving) Grades Calculator
Are you a student who has ever struggled with a tough grading system? You know you did well on an exam, but your grade just doesn't reflect it. Or maybe you're a teacher who wants to curve grades for
your class, but you're not sure how to do it. Look no further, because our Scaling (Curving) Grades Calculator is here to help! With our user-friendly calculator, you can easily input your grades and
see what they would be if a curve was applied. This allows teachers to adjust their grading system and students to figure out what their final grades will be. No more stressing over a test, wondering
if you'll fail or pass - our calculator will give you accurate results. Whether you're in college, high school, or even middle school, our Scaling (Curving) Grades Calculator can be a valuable tool
for both students and teachers. It's simple, efficient, and just what you need to make sure your grades accurately reflect your efforts. Give it a try and see for yourself!
Scaling (Curving) Grades Calculator
Calculate the scaled (curved) grades based on the original grades and the desired scale.
Scaling (Curving) Grades Calculator Results
Original Grades: 0
Desired Scale: 0
Scaling Method: 0
Curve Value: 0
Curved Grades: 0
Share results with your friends
How to Use the Scaling (Curving) Grades Calculator
The Scaling (Curving) Grades Calculator is designed to help educators, instructors, and students in various educational contexts. It enables them to apply scaling methods to the original grades,
ensuring fair grading practices and maintaining consistency. By utilizing this calculator, you can easily adjust grades to align with desired scales and curves, allowing for accurate assessment and
evaluation of performance.
Primary Applications
The Scaling (Curving) Grades Calculator finds applications in diverse educational scenarios, including:
• Academic Institutions: Scaling grades to align with standard grade distributions.
• Competitive Examinations: Applying curves to account for the difficulty of the test.
• Proficiency Tests: Normalizing scores across different versions of the test.
• Research Evaluation: Applying scaling methods to analyze and compare research metrics.
Instructions for Utilizing the Calculator
To effectively utilize the Scaling (Curving) Grades Calculator, follow these steps:
Input Fields
• Original Grades: Enter the original grades separated by commas or new lines. Provide the grades as numerical values.
• Desired Scale: Select the desired scale for the curved grades. Choose from options such as A (90-100), B (80-89), C (70-79), D (60-69), or F (0-59).
• Scaling Method: Select the preferred scaling method for the grade adjustment. Choose between linear, logarithmic, or exponential.
• Curve Value: Enter the curve value as a percentage. This value represents the amount by which the grades will be adjusted.
Output Fields and Interpretations
Upon submitting the form, the Scaling (Curving) Grades Calculator will provide the following output:
• Original Grades: Displays the original grades you entered.
• Desired Scale: Shows the desired scale selected for the curved grades.
• Scaling Method: Indicates the scaling method chosen for the grade adjustment.
• Curve Value: Displays the curve value you entered as a percentage.
• Curved Grades: Displays the adjusted grades based on the original grades, desired scale, scaling method, and curve value.
Scaling (Curving) Grades Calculator Formula
The Scaling (Curving) Grades Calculator does not follow a specific formula but rather employs various scaling methods to adjust the grades based on the desired scale and curve value. Here's an
overview of the process:
• Determine the minimum, maximum, and average grades from the original grades.
• Calculate a multiplier based on the desired scale and curve value.
• Apply the scaling method to each original grade, multiplying it by the multiplier.
• Round the curved grades to an appropriate precision.
Illustrative Example
Let's consider an example to illustrate the functionality of the Scaling (Curving) Grades Calculator.
Suppose you have the following original grades: 75, 80, 90, 65, and 85. You want to adjust these grades based on a desired scale of B (80-89) using the linear scaling method and a curve value of 5%.
After inputting the values and submitting the form, you would obtain the following result:
• Original Grades: 75, 80, 90, 65, 85
• Desired Scale: B
• Scaling Method: Linear
• Curve Value: 5%
• Curved Grades: 77.5, 82.5, 90.0, 72.5, 85.0
The calculator adjusts each grade proportionally, ensuring the desired scale is achieved while considering the specified curve value.
Illustrative Table Example
Consider a table example with multiple rows of data demonstrating the Scaling (Curving) Grades Calculator for different scenarios:
Original Grades Desired Scale Scaling Method Curve Value Curved Grades
85, 78, 92, 70 A Linear 10% 93.5, 86.8, 97.2, 79
60, 72, 87, 68 C Logarithmic 7.5% 58.61, 68.21, 84.94, 66.01
92, 88, 79, 85 B Exponential 15% 8976.83, 7655.55, 4301.54, 7522.11
The Scaling (Curving) Grades Calculator is an indispensable tool for educators and students alike. It streamlines the process of adjusting grades based on desired scales and curve values. By
utilizing this calculator, educational institutions can ensure fair and consistent grading practices, while students can gain a better understanding of their performance relative to desired
standards. Embrace the convenience and accuracy provided by the Scaling (Curving) Grades Calculator to enhance your grading processes and promote effective learning environments. | {"url":"https://calculatorpack.com/scaling-curving-grades-calculator/","timestamp":"2024-11-05T17:03:24Z","content_type":"text/html","content_length":"36856","record_id":"<urn:uuid:52d93469-02e0-4299-a885-d0ad0c081af2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00888.warc.gz"} |
Aggregation: Quantile
Aggregation: Quantile#
This notebook explains the b-ary tree technique for releasing quantiles. Examples in this notebook will a dataset sampled from \(Exponential(\lambda=20)\).
# privacy settings for all examples:
max_contributions = 1
epsilon = 1.
import numpy as np
data = np.random.exponential(20., size=1000)
bounds = 0., 100. # a best guess!
import seaborn as sns
sns.displot(data, kind="kde");
# true quantiles
true_quantiles = np.quantile(data, [0.25, 0.5, 0.75])
[5.342567934782284, 13.46163987459231, 27.466767771051295]
Any constructors that have not completed the proof-writing and vetting process may still be accessed if you opt-in to “contrib”. Please contact us if you are interested in proof-writing. Thank you!
import opendp.prelude as dp
Quantile via Histogram#
One approach for releasing quantiles is to estimate the cumulative distribution via a histogram query.
The basic procedure for estimating an \(\alpha\)-quantile is as follows:
1. bin the data, and count the number of elements in each bin privately
2. divide the counts by the total to get the probability density of landing in each bin
3. sum the densities while scanning from the left until the sum is at least \(\alpha\)
4. interpolate the bin edges of the terminal bin
quart_alphas = [0.25, 0.5, 0.75]
input_space = dp.vector_domain(dp.atom_domain(T=float)), dp.symmetric_distance()
def make_hist_quantiles(alphas, d_in, d_out, num_bins=500):
edges = np.linspace(*bounds, num=num_bins + 1)
bin_names = [str(i) for i in range(num_bins)]
def make_from_scale(scale):
return (
input_space >>
dp.t.then_find_bin(edges=edges) >> # bin the data
dp.t.then_index(bin_names, "0") >> # can be omitted. Just set TIA="usize", categories=list(range(num_bins)) on next line:
dp.t.then_count_by_categories(categories=bin_names, null_category=False) >>
dp.m.then_laplace(scale) >>
# we're really only interested in the function on this transformation- the domain and metric don't matter
dp.t.make_cast_default(dp.vector_domain(dp.atom_domain(T=int)), dp.symmetric_distance(), TOA=float) >>
dp.t.make_quantiles_from_counts(edges, alphas=alphas)
return dp.binary_search_chain(make_from_scale, d_in, d_out)
hist_quartiles_meas = make_hist_quantiles(quart_alphas, max_contributions, epsilon)
[5.33, 13.411111111111111, 29.875]
A drawback of using this algorithm is that it can be difficult to choose the number of bins.
If the number of bins is chosen to be very small, then the postprocessor will need to sum fewer instances of noise before reaching the bin of interest, resulting in a better bin selection. However,
the bin will be wider, so there will be greater error when interpolating the final answer.
If the number of bins is chosen to be very large, then the same holds in the other direction.
Estimating quantiles via the next algorithm can help make choosing the number of bins less sensitive.
Quantile via B-ary Tree#
A slightly more complicated algorithm that tends to provide better utility is to privatize a B-ary tree instead of a histogram. In this algorithm, the raw counts form the leaf nodes, and a complete
tree is constructed by recursively summing groups of size b. This results in a structure where each parent node is the sum of its b children. Noise is added to each node in the tree, and then a
postprocessor makes all nodes of the tree consistent with each other, and returns the leaf nodes.
In the histogram approach, the postprocessor would be influenced by a number of noise sources approximately \(O(n)\) in the number of scanned bins. After this modification, the postprocessor is
influenced by a number of noise sources approximately \(O(log_b(n))\) in the number of scanned bins, and with noise sources of similarly greater magnitude.
This modification introduces a new hyperparameter, the branching factor. choose_branching_factor provides a heuristic for the ideal branching factor, based on information (or a best guess) of the
dataset size.
b = dp.t.choose_branching_factor(size_guess=1_500)
We now make the following adjustments to the histogram algorithm:
• insert a stable (Lipschitz) transformation to construct a b-ary tree before the noise mechanism
• replace the cast postprocessor with a consistency postprocessor
def make_tree_quantiles(alphas, b, d_in, d_out, num_bins=500):
edges = np.linspace(*bounds, num=num_bins + 1)
bin_names = [str(i) for i in range(num_bins)]
def make_from_scale(scale):
return (
input_space >>
dp.t.then_find_bin(edges=edges) >> # bin the data
dp.t.then_index(bin_names, "0") >> # can be omitted. Just set TIA="usize", categories=list(range(num_bins)) on next line:
dp.t.then_count_by_categories(categories=bin_names, null_category=False) >>
dp.t.then_b_ary_tree(leaf_count=len(bin_names), branching_factor=b) >>
dp.m.then_laplace(scale) >>
dp.t.make_consistent_b_ary_tree(branching_factor=b) >> # postprocessing
dp.t.make_quantiles_from_counts(edges, alphas=alphas) # postprocessing
return dp.binary_search_chain(make_from_scale, d_in, d_out)
tree_quartiles_meas = make_tree_quantiles(quart_alphas, b, max_contributions, epsilon)
[5.0371403927139795, 13.146207218782838, 26.037387422664267]
As mentioned earlier, using the tree-based approach can help make the algorithm less sensitive to the number of bins:
def average_error(num_bins, num_trials):
hist_quantiles_meas = make_hist_quantiles(quart_alphas, max_contributions, epsilon, num_bins)
tree_quantiles_meas = make_tree_quantiles(quart_alphas, b, max_contributions, epsilon, num_bins)
def sample_error(meas):
return np.linalg.norm(true_quantiles - meas(data))
hist_err = np.mean([sample_error(hist_quantiles_meas) for _ in range(num_trials)])
tree_err = np.mean([sample_error(tree_quantiles_meas) for _ in range(num_trials)])
return num_bins, hist_err, tree_err
import pandas as pd
[average_error(nb, num_trials=25) for nb in [70, 100, 250, 500, 750, 1_000, 3_000]],
columns=["number of bins", "histogram error", "tree error"]
).plot(0); # type: ignore
Privately Estimating the Distribution#
Minor note: instead of postprocessing the noisy counts into quantiles, they can be left as counts, which can be used to visualize the distribution.
def make_distribution_counts(edges, scale):
bin_names = [str(i) for i in range(len(edges - 1))]
return (
input_space >>
dp.t.then_find_bin(edges=edges) >> # bin the data
dp.t.then_index(bin_names, "0") >> # can be omitted. Just set TIA="usize", categories=list(range(num_bins)) on next line:
dp.t.then_count_by_categories(categories=bin_names, null_category=False) >>
edges = np.linspace(*bounds, num=50)
counts = make_distribution_counts(edges, scale=1.)(data)
import matplotlib.pyplot as plt
plt.hist(range(len(edges)), edges, weights=counts, density=True)
plt.ylabel("noisy density"); | {"url":"https://docs.opendp.org/en/nightly/api/user-guide/transformations/aggregation-quantile.html","timestamp":"2024-11-11T04:35:41Z","content_type":"text/html","content_length":"59940","record_id":"<urn:uuid:88b9faf5-b29c-45e6-984a-3c103593dd71>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00478.warc.gz"} |
Linear Algebra/Topic: Cramer's Rule - Wikibooks, open books for an open world
We have introduced determinant functions algebraically by looking for a formula to decide whether a matrix is nonsingular. After that introduction we saw a geometric interpretation, that the
determinant function gives the size of the box with sides formed by the columns of the matrix. This Topic makes a connection between the two views.
First, a linear system
${\displaystyle {\begin{array}{*{2}{rc}r}x_{1}&+&2x_{2}&=&6\\3x_{1}&+&x_{2}&=&8\end{array}}}$
is equivalent to a linear relationship among vectors.
${\displaystyle x_{1}\cdot {\begin{pmatrix}1\\3\end{pmatrix}}+x_{2}\cdot {\begin{pmatrix}2\\1\end{pmatrix}}={\begin{pmatrix}6\\8\end{pmatrix}}}$
The picture below shows a parallelogram with sides formed from ${\displaystyle {\binom {1}{3}}}$ and ${\displaystyle {\binom {2}{1}}}$ nested inside a parallelogram with sides formed from ${\
displaystyle x_{1}{\binom {1}{3}}}$ and ${\displaystyle x_{2}{\binom {2}{1}}}$.
So even without determinants we can state the algebraic issue that opened this book, finding the solution of a linear system, in geometric terms: by what factors ${\displaystyle x_{1}}$ and ${\
displaystyle x_{2}}$ must we dilate the vectors to expand the small parallegram to fill the larger one?
However, by employing the geometric significance of determinants we can get something that is not just a restatement, but also gives us a new insight and sometimes allows us to compute answers
quickly. Compare the sizes of these shaded boxes.
The second is formed from ${\displaystyle x_{1}{\binom {1}{3}}}$ and ${\displaystyle {\binom {2}{1}}}$, and one of the properties of the size function— the determinant— is that its size is therefore
${\displaystyle x_{1}}$ times the size of the first box. Since the third box is formed from ${\displaystyle x_{1}{\binom {1}{3}}+x_{2}{\binom {2}{1}}={\binom {6}{8}}}$ and ${\displaystyle {\binom {2}
{1}}}$, and the determinant is unchanged by adding ${\displaystyle x_{2}}$ times the second column to the first column, the size of the third box equals that of the second. We have this.
${\displaystyle {\begin{vmatrix}6&2\\8&1\end{vmatrix}}={\begin{vmatrix}x_{1}\cdot 1&2\\x_{1}\cdot 3&1\end{vmatrix}}=x_{1}\cdot {\begin{vmatrix}1&2\\3&1\end{vmatrix}}}$
Solving gives the value of one of the variables.
${\displaystyle x_{1}={\frac {\begin{vmatrix}6&2\\8&1\end{vmatrix}}{\begin{vmatrix}1&2\\3&1\end{vmatrix}}}={\frac {-10}{-5}}=2}$
The theorem that generalizes this example, Cramer's Rule, is: if ${\displaystyle \left|A\right|eq 0}$ then the system ${\displaystyle A{\vec {x}}={\vec {b}}}$ has the unique solution ${\displaystyle
x_{i}=\left|B_{i}\right|/\left|A\right|}$ where the matrix ${\displaystyle B_{i}}$ is formed from ${\displaystyle A}$ by replacing column ${\displaystyle i}$ with the vector ${\displaystyle {\vec
{b}}}$. Problem 3 asks for a proof.
For instance, to solve this system for ${\displaystyle x_{2}}$
${\displaystyle {\begin{pmatrix}1&0&4\\2&1&-1\\1&0&1\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}2\\1\\-1\end{pmatrix}}}$
we do this computation.
${\displaystyle x_{2}={\frac {\begin{vmatrix}1&2&4\\2&1&-1\\1&-1&1\end{vmatrix}}{\begin{vmatrix}1&0&4\\2&1&-1\\1&0&1\end{vmatrix}}}={\frac {-18}{-3}}}$
Cramer's Rule allows us to solve many two equations/two unknowns systems by eye. It is also sometimes used for three equations/three unknowns systems. But computing large determinants takes a long
time, so solving large systems by Cramer's Rule is not practical.
Problem 1
Use Cramer's Rule to solve each for each of the variables.
1. ${\displaystyle {\begin{array}{*{2}{rc}r}x&-&y&=&4\\-x&+&2y&=&-7\end{array}}}$
2. ${\displaystyle {\begin{array}{*{2}{rc}r}-2x&+&y&=&-2\\x&-&2y&=&-2\end{array}}}$
Problem 2
Use Cramer's Rule to solve this system for ${\displaystyle z}$ .
${\displaystyle {\begin{array}{*{4}{rc}r}2x&+&y&+&z&=&1\\3x&&&+&z&=&4\\x&-&y&-&z&=&2\end{array}}}$
Problem 3
Prove Cramer's Rule.
Problem 4
Suppose that a linear system has as many equations as unknowns, that all of its coefficients and constants are integers, and that its matrix of coefficients has determinant ${\displaystyle 1}$ .
Prove that the entries in the solution are all integers. (Remark. This is often used to invent linear systems for exercises. If an instructor makes the linear system with this property then the
solution is not some disagreeable fraction.)
Problem 5
Use Cramer's Rule to give a formula for the solution of a two equations/two unknowns linear system.
Problem 6
Can Cramer's Rule tell the difference between a system with no solutions and one with infinitely many?
Problem 7
The first picture in this Topic (the one that doesn't use determinants) shows a unique solution case. Produce a similar picture for the case of infintely many solutions, and the case of no solutions. | {"url":"https://en.m.wikibooks.org/wiki/Linear_Algebra/Topic:_Cramer%27s_Rule","timestamp":"2024-11-03T16:07:50Z","content_type":"text/html","content_length":"83499","record_id":"<urn:uuid:4dcb0c84-0dbc-4fba-ac10-4543d7352c48>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00541.warc.gz"} |
How do you solve -6+ 8x = - 38? | HIX Tutor
How do you solve #-6+ 8x = - 38#?
Answer 1
#-6 + 8x = -38#
#8x = -38 + 6, " bringing constant term to R H S"#
#8x = -32 #
#x = -cancel(32)^color(red)(4) / cancel(8)^color(red)(1) = = -4#
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Answer 2
To solve the equation -6 + 8x = -38, you need to isolate the variable x. Start by adding 6 to both sides of the equation to get rid of the constant term on the left side. This gives you 8x = -32.
Then, divide both sides by 8 to solve for x. This results in x = -4.
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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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How important are accurate AI timelines for the optimal spending schedule on AI risk interventions? — EA Forum
I present an extension to my optimal timing of spending on AGI safety model for calculating the value of information of AGI timelines via informing one’s spending schedule.
I show, using my best guess of the model parameters, that for an AI risk funder uncertain between a ‘short timelines’ model and ‘medium timelines’ model:
• Updating from near certainty in medium timelines to short timelines (and following the new optimal spending strategy) leads to a 40% increase in utility.
• Updating from near certainty in short timelines to medium timelines (and following the new optimal spending strategy) leads to a 20% increase in utility.
The gains are greater when considering a model of the community’s capacity, rather than capital.
I also show that small changes in one’s credence in short or medium timelines has relatively little impact on one’s optimal spending schedule, especially when one starts out with roughly equal
credence in each^[1].
You can enter your own parameters - such as AGI timelines, discount rate and diminishing returns to spending - here.
In an appendix I apply a basic model to consider the opportunity cost of timelines work. This model does not assume novel research is done.
The setup
Suppose you have two ‘models’ of AGI timelines and , with credence in and in ^[2]. You use your mixture distribution for AGI timelines to calculate the optimal spending schedule for AI risk
You could do some thinking and come to some credence in and in . How much better is the optimal spending schedule as a result of to the optimal spending schedule as a result of , both supposing ?
Writing for optimal spending schedule according to and for utility of supposing , I compare to (the former, by definition of optimality is greater or equal to the latter).
In the model, utility is the discount adjusted probability we ‘succeed’ with making AGI go well. The maximum utility is where are the AGI timelines and is the discount rate.
Example results for short and medium timelines
I take and as the following, and other parameters as described here. (See the appendix for some basic statistics on the mixtures of and .
: ‘Short’ 2027 median timelines : Metaculus community timelines (2039 median)
I compute the results for both the ‘main’ model and the ‘alternate’ model described in the previous post.
• The main model gives the optimal spending rate (in $ per year) for every time step on (1) research and (2) influence (being able to get AI risk reduction ideas implemented).
□ Supposing there was only one funder of AI risk interventions, they should follow this spending strategy.
• The alternate model gives the optimal ‘crunch’ rate at each time. At each time point we can either invest in our own capacity (be it through training ,hiring, some types of research, investing)
or ‘crunch’ - spend capacity to produce work directly beneficial to reducing AI risk.
□ This model gives more abstract results and is more applicable for individuals and teams.
I first show two examples of the optimal spending schedule for each of the models: one that is optimal supposing , short timelines, and one that is optimal supposing medium timelines, .
Main model
The optimal spending schedule assuming , , (left) and the optimal spending schedule assuming , , (right), both assuming medium difficulty.
and : going from certainty in to certainty in and following the new optimal spending schedule leads to 20% subjective increase in utility.
and : going from certainty in to certainty in and following the new optimal spending schedule leads to 35% subjective increase in utility
Alternate model
The optimal spending schedule assuming , (left) and the optimal spending schedule assuming , (right), both assuming medium difficulty.
and : going from certainty in to certainty in and following the new optimal spending schedule leads to 18% subjective increase in utility.
and : going from certainty in to certainty in and following the new optimal spending schedule leads to 45% subjective increase in utility
Results & analysis
Disclaimer: these results are based on my guesses of the model parameters. Further, the models are of course not without limitations. I expect the results to be directionally equal, but lower, for
the robust spend-save model. Overall, I’m less confident in these results than the previous spending results.
For each pairing (main capital spending model, alternate community capacity spending model) x AGI success is (easy, medium, hard) I show three graphs. The leftmost plot shows the % increase in
discount-adjusted probability of success (utility). The central plot shows this % increase after first factoring out the utilons we get for free - our probability of success if we contributed
nothing. The rightmost plot shows the absolute increase in utility.
Main model, easy^[3]
Main model, medium^[4]
Main model, hard^[5]
Alternate model, easy^[6]
Alternate model, medium^[7]
Alternate model, hard^[8]
Both the main and alternate model show that:
• The greatest gains are achievable in the hard case where we are highly confident in short timelines but mistaken. The easy case has the lowest gains: this is likely because we can do well
regardless of our spending schedule.
• There are greater gains to be had when one is very confident in short timelines than when one is very confident in medium to long timelines,
• The maximum increases in utility are comparable to the gain in utility achieved from the approximate gain the community can make from moving from its current spending strategy to the optimal
spending strategy.
• There is little to no gain in utility when one’s credence changes in by less than 10% (this is the bottom-left to top-right diagonal band of purple).
• The main model has greater difference between strategies than the alternate model (compare the examples above) which I believe leads to the rectangular-like areas in the plots: small increases in
can push greatly towards early spending.
• The alternate model sees greater utility gains from accurate timelines than the main model. This is likely due to:
□ slightly greater marginal returns to ‘crunching’ than there are to spending capital in the main model
□ greater opportunity cost of spending early (since there are higher returns to saving).
In practise
Naturally, those with low credal resilience in their AGI timelines have better returns to work on timelines.
The greatest potential gains of timelines work is for people already highly convinced of short timelines. This is particularly true if they are sacrificing gains in capacity now in order to ‘crunch’,
or spending at a high rate now (sacrificing a greater amount of capital later). However, it seems that:
1. many crunch-like activities may also be building capacity (e.g. building skills necessary for crunchtime), especially because it is relatively neglected in the community
2. that the current community spending rate is much lower than the optimal schedule implied even by medium timelines, and so marginally pushing for greater spending is supported regardless of one's
credence in short timelines.
In the other direction, the value of information moving from high confidence in medium timelines to high confidence in short timelines are likely to be higher than the results suggest because the
current spending rate and crunching rate are too low. That is, if you become convinced of short timelines, your marginal spending/crunching has lower diminishing returns (because, by your lights, the
rest of the community is at a suboptimally low spending/crunching rate).
Acknowledgements: thanks to Daniel Kokotajlo for the idea, comments and suggestions. Thanks to Tom Barnes for comments. All remaining errors are my own.
Statistics about the mixture of A and B
Potentially useful for calculating your own .
25th percentile Median 75th percentile
1 () 2025 2027 2030
0.75 2025 2028 2034
0.5 2026 2030 2041
0.25 2027 2033 2051
0 () 2030 2039 2060
Toy model
This model is highly flawed but potentially illustrative. Please take it with a massive grain of salt!
Suppose someone deliberates for ^[9] The person does not regret having done this if
Taking ^[10]
Note The white areas show . I hide these because I don’t condition the expected time remaining on years having passed.
The asymmetry in the plot due to the fact that in the case one in fact has longer than they’d guessed, their impact multiplier applies for more years (this could make sense if there are things you
can do down that grow over time, even if in 2042 both the you-who-did-timelines-work-in-2022 and the you who didn’t have similar AGI timelines).
To use these results, one must have a prior over how their credence in A will change over the duration (in expectation it must not change at all though!). Someone with low credal resilience will have
a 'wider' distribution than someone with higher credal resilience. One could simplify this step further by supposing discrete credences. For example, one could have 1/4 probability of staying at
their current credence in A of 0.6, 1/4 credence in moving to 0.2 and 1/2 credence in moving to 0.8.
The toy model could be greatly improved. For example, one could model the situation as spending non-time resources, in which the above limitation does not occur. Further, one could allow for parallel
work on timelines.
1. ^^
For example, moving from 40-60% credence in short to medium, or vice versa, has very little gains to the optimal spending schedule.
2. ^^
For example, A is ‘scale is all you need to AGI’ and B is ‘we need more difficult insights’. Or ‘’ is your independent impression, which you could become more confident in, and ‘’ is deference
towards others (having accounted for information cascades etc)
3. ^^
25% probability of AGI going well if it arrived this year, and slope parameter
4. ^^
10% probability of AGI going well if it arrived this year, and slope parameter
5. ^^
4% probability of AGI going well if it arrived this year, and and slope parameter
6. ^^
25% probability of AGI going well if it arrived after a year and the community had been crunching at rate 1 unit of capacity per year (we start with one unit of capacity) and the slope parameter
7. ^^
10% probability of AGI going well if it arrived after a year and the community had been crunching at rate 1 unit of capacity per year (we start with one unit of capacity) and the slope parameter
8. ^^
5% probability of AGI going well if it arrived after a year and the community had been crunching at rate 1 unit of capacity per year (we start with one unit of capacity) and the slope parameter
9. ^^
Assuming is sufficiently small such that conditioning on no AGI in that time makes little difference.
10. ^^
This roughly approximates the results above
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Suchergebnis: Katalogdaten im Herbstsemester 2022
Materialwissenschaft Master
Nummer Titel Typ ECTS Umfang Dozierende
327-0505-00L Surfaces, Interfaces and their Applications I W 3 KP 2V + 1U N. Spencer, M. P. Heuberger, L. Isa
Kurzbeschreibung After being introduced to the physical/chemical principles and importance of surfaces and interfaces, the student is introduced to the most important techniques that can be used to
characterize surfaces. Later, liquid interfaces are treated, followed by an introduction to the fields of tribology (friction, lubrication, and wear) and corrosion.
Lernziel To gain an understanding of the physical and chemical principles, as well as the tools and applications of surface science, and to be able to choose appropriate surface-analytical
approaches for solving problems.
Introduction to Surface Science
Physical Structure of Surfaces
Surface Forces (static and dynamic)
Adsorbates on Surfaces
Surface Thermodynamics and Kinetics
Inhalt The Solid-Liquid Interface
Electron Spectroscopy
Vibrational Spectroscopy on Surfaces
Scanning Probe Microscopy
Introduction to Tribology
Introduction to Corrosion Science
Skript Script Download:
Script Download:
Literatur https://moodle-app2.let.ethz.ch/course/view.php?id=17455
Book: "Surface Analysis--The Principal Techniques", Ed. J.C. Vickerman, Wiley, ISBN 0-471-97292
General undergraduate chemistry
Voraussetzungen including basic chemical kinetics and thermodynamics
/ Besonderes
General undergraduate physics
including basic theory of diffraction and basic knowledge of crystal structures
Fachspezifische Kompetenzen Konzepte und Theorien geprüft
Verfahren und Technologien geprüft
Methodenspezifische Kompetenzen Analytische Kompetenzen geprüft
Kompetenzen Entscheidungsfindung geprüft
Problemlösung geprüft
Persönliche Kompetenzen Kreatives Denken geprüft
Kritisches Denken geprüft
327-1201-00L Transport Phenomena I W Dr 5 KP 4G J. Vermant
Kurzbeschreibung Phenomenological approach to "Transport Phenomena" based on balance equations supplemented by thermodynamic considerations to formulate the undetermined fluxes in the local species
mass, momentum, and energy balance equations; Solutions of a few selected problems relevant to materials science and engineering both analytical and using numerical methods.
The teaching goals of this course are on five different levels:
(1) Deep understanding of fundamentals: local balance equations, constitutive equations for fluxes, entropy balance, interfaces, idea of dimensionless numbers and scaling, ...
Lernziel (2) Ability to use the fundamental concepts in applications
(3) Insight into the role of boundary conditions (mainly part 2)
(4) Knowledge of a number of applications.
(5) Flavor of numerical techniques: finite elements and finite differences.
Part 1 Approach to Transport Phenomena
Equilibrium Thermodynamics
Balance Equations
Inhalt Forces and Fluxes
1. Measuring Transport Coefficients
2. Fluid mechanics
3. combined heat and flow
Skript The course is based on the book D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018) and the book by W. M. Deen, Analysis of
Transport Phenomena (Oxford University Press, 1998)
1. D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018)
2. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd Ed. (Wiley, 2001)
Literatur 3. L.G. Leal, Advanced Transport Phenomena (Oxford University Press, 2011)
4. W. M. Deen, Analysis of Transport Phenomena (Oxford University Press, 1998)
5. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287
Complex numbers. Vector analysis (integrability; Gauss' divergence theorem). Laplace and Fourier transforms. Ordinary differential equations (basic ideas). Linear algebra (matrices;
Voraussetzungen functions of matrices; eigenvectors and eigenvalues; eigenfunctions). Probability theory (Gaussian distributions; Poisson distributions; averages; moments; variances; random
/ Besonderes variables). Numerical mathematics (integration). Equilibrium thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms). Maxwell equations.
Programming and simulation techniques (Matlab, Monte Carlo simulations).
Fachspezifische Kompetenzen Konzepte und Theorien geprüft
Kompetenzen Verfahren und Technologien geprüft
Methodenspezifische Kompetenzen Problemlösung geprüft
327-1202-00L Solid State Physics and Chemistry of Materials I W Dr 5 KP 4G N. Spaldin
Kurzbeschreibung In this course we study how the properties of solids are determined from the chemistry and arrangement of the constituent atoms, with a focus on materials that are not well described
by conventional band theories because their behavior is governed by strong quantum-mechanical interactions.
Electronic properties and band theory description of conventional solids
Lernziel Electron-lattice coupling and its consequences in functional materials
Electron-spin/orbit coupling and its consequences in functional materials
Structure/property relationships in strongly-correlated materials
In this course we study how the properties of solids are determined from the chemistry and arrangement of the constituent atoms, with a focus on materials that are not well described
by conventional band theories because their behavior is governed by strong quantum-mechanical interactions. We begin with a review of the successes of band theory in describing many
properties of metals, semiconductors and insulators, and we practise building up band structures from atoms and describing the resulting properties. Then we explore classes of
Inhalt systems in which the coupling between the electrons and the lattice is so strong that it drives structural distortions such as Peierls instabilities, Jahn-Teller distortions, and
ferroelectric transitions. Next, we move on to strong couplings between electronic charge and spin- and/or orbital- angular momentum, yielding materials with novel magnetic
properties. We end with examples of the complete breakdown of single-particle band theory in so-called strongly correlated materials, which comprise for example heavy-fermion
materials, frustrated magnets, materials with unusual metal-insulator transitions and the high-temperature superconductors.
Skript An electronic script for the course is provided in Moodle.
Literatur Hand-outs with additional reading will be made available during the course and posted on the moodle page accessible through MyStudies
all of:
Statistical Thermodynamics (327-0315-00)
Voraussetzungen Quantenmechanik für Materialwissenschaftler/innen (327-0316-00)
/ Besonderes Festkörpertheorie für Materialwissenschaftler/innen (327-0416-00)
Electronic, Optical and Magnetic Properties of Materials (327-0512-00)
or equivalent classes from another institution
327-1203-00L Complex Materials I: Synthesis & Assembly W Dr 5 KP 4G M. Niederberger, A. Lauria
Kurzbeschreibung Introduction to materials synthesis concepts based on the assembly of differently shaped objects of varying chemical nature and length scales
Lernziel The aim is a) to learn how to design and create objects as building blocks with a particular composition, size and shape, b) to understand the chemistry that allows for the creation
of such hard and soft objects, and c) to master the concepts to assemble these objects into materials over several length scales.
The course is divided into two parts: I) synthesis of 0-, 1-, 2-, and 3-dimensional building blocks with a length scale from nm to µm, and II) assembly of these building blocks into
1-, 2- and 3-dimensional structures over several length scales up to cm.
Inhalt In part I, various methodologies for the synthesis of the building blocks will be discussed, including Turkevich and Brust-Schiffrin-method for gold nanoparticles, hot-injection for
semiconducting quantum dots, aqueous and nonaqueous sol-gel chemistry for metal oxides, or gas-and liquid-phase routes to carbon nanostructures.
Part II is focused on self- and directed assembly methods that can be used to create higher order architectures from those building blocks connecting the microscopic with the
macroscopic world. Examples include photonic crystals, nanocrystal solids, colloidal molecules, mesocrystals or particle-based foams and aerogels.
Literatur References to original articles and reviews for further reading will be provided on the lecture notes.
Voraussetzungen 1) Materialsynthese II (327-0412-00)
/ Besonderes 2) Kristallographie (327-0104-00L), in particular structure of crystalline solids
3) Materials Characterization II (327-0413-00)
327-1204-00L Materials at Work I W Dr 4 KP 4S R. Spolenak, E. Dufresne, R. Koopmans
This course attempts to prepare the student for a job as a materials engineer in industry. The gap between fundamental materials science and the materials engineering of products
Kurzbeschreibung should be bridged. The focus lies on the practical application of fundamental knowledge allowing the students to experience application related materials concepts with a strong
emphasis on case-study mediated learning.
Teaching goals:
to learn how materials are selected for a specific application
to understand how materials around us are produced and manufactured
Lernziel to understand the value chain from raw material to application
to be exposed to state of the art technologies for processing, joining and shaping
to be exposed to industry related materials issues and the corresponding language (terminology) and skills
to create an impression of how a job in industry "works", to improve the perception of the demands of a job in industry
This course is designed as a two semester class and the topics reflect the contents covered in both semesters.
Lectures and case studies encompass the following topics:
Strategic Materials (where do raw materials come from, who owns them, who owns the IP and can they be substituted)
Materials Selection (what is the optimal material (class) for a specific application)
Materials systems (subdivisions include all classical materials classes)
Inhalt Processing
Joining (assembly)
Materials and process scaling (from nm to m and vice versa, from mg to tons)
Sustainable materials manufacturing (cradle to cradle) Recycling (Energy recovery)
After a general part of materials selection, critical materials and materials and design four parts consisting of polymers, metals, ceramics and coatings will be addressed.
In the fall semester the focus is on the general part, polymers and alloy case studies in metals. The course is accompanied by hands-on analysis projects on everyday materials.
Manufacturing, Engineering & Technology
Literatur Serope Kalpakjian, Steven Schmid
ISBN: 978-0131489653
Voraussetzungen Profound knowledge in Physical Metallurgy and Polymer Basics and Polymer Technology required (These subjects are covered at the Bachelor Level by the following lectures: Metalle 1,
/ Besonderes 2; Polymere 1,2)
327-1207-00L Engineering with Soft Materials W Dr 5 KP 4G J. Vermant, L. Isa
In this course the engineering with soft materials is discussed. First, scaling principles to design structural and functional properties are introduced a. Second, the
Kurzbeschreibung characterisation techniques to interrogate the structure property relations are introduced, which include rheology, advanced optical microscopies, static and dynamic scattering and
techniques for liquid interfaces.
Lernziel The learning goals of the course are to introduce the students to soft matter and its technological applications, to see how the structure property relations depend on fundamental
formulation properties and processing steps. Students should also be able to select a measurement technique to evaluate the properties.
Skript slides with text notes accompanying each slide are presented. | {"url":"https://www.vorlesungen.ethz.ch/Vorlesungsverzeichnis/sucheLehrangebot.view?seite=1&semkez=2022W&ansicht=2&lang=de&abschnittId=100097","timestamp":"2024-11-06T14:56:29Z","content_type":"text/html","content_length":"29541","record_id":"<urn:uuid:b96976a6-2bd5-46bd-886d-9a3496fbee08>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00275.warc.gz"} |
Heads up for a hot new root seeking algorithm!!
01-20-2017, 09:27 AM
(This post was last modified: 01-20-2017 09:38 AM by Namir.)
Post: #19
Namir Posts: 1,107
Senior Member Joined: Dec 2013
RE: Heads up for a hot new root seeking algorithm!!
If you download the ZIP file from my web site, you get the report and an Excel file that has worksheets for the dozen test functions. What I observed is that in the majority of the cases, the methods
of Halley and Ostrowski give the same results or don't differ by more than one iteration (both require three function calls per iteration). The results DO SHOW a consistent improvement over Newton's
method. You can say that the methods of Halley and Ostrowski have a higher convergence rate than that of Newton. My new algorithm (which uses the Ostrowski's approach to enhance Halley) shows several
cases where either the number of iterations or both the number of iterations AND number of function calls is less than that of Halley and Ostrowski. Since I developed the new algorithm using an
empirical/heuristic approach I did not yield a long set of mathematical derivations that indicate the convergence rate. It may well be at least one order more than that of Ostrowski's method. It's
hard to measure ... compared to, say determining the order of array sorting methods where you can widely vary the array size and calculate the number of array element comparisons and number of
element swaps. See my article about
enhancing the CombSort method
where I was able to calculate the sort order of this method in Tables 3 and 4 of the article.
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Journal Papers
1. Chitgarha, F., Ommi, F., Farshchi, M., “Assessment of optimal reaction progress variable characteristics for partially premixed flames,” Combustion Theory and Modelling, https://doi.org/10.1080/
13647830.2022.2070549, Published online 19 May 2022.
2. Poormahmood, A., Salehi, M.M., Farshchi, M., "A methodology for modeling the interaction between turbulence and non-linearity of the equation of state," Physics of Fluids 34 (1), 015106, 2022.
3. Dehghan-Nezhad, S., Fahim, M., Farshchi, M. “Experimental Study of Continuous H2/Air Rotating Detonations,” Combustion Science and Technology, 194 (3), 449-463, 2022.
4. Rezayat, S., Farshchi, M., Berrocal, E., "High-speed imaging database of water jet disintegration Part II: Temporal analysis of the primary breakup," International Journal of Multiphase Flow 145,
103807, 2021.
5. Rezayat, S., Farshchi, M., "Effects of Radial–Tangential Discharge Channels on a Rotary Atomizer Performance," AIAA Journal 59 (6), 2262-2281, 2021.
6. Zeinivand, H., Farshchi, M. “Numerical study of the pseudo-boiling phenomenon in the transcritical liquid oxygen/gaseous hydrogen flame,” Proceedings of the Institution of Mechanical Engineers,
Part G: Journal of Aerospace Engineering, 235(8), pp. 893-911, 2021.
7. Zeinivand, H., Rezaei, H., Farshchi, M., "Influence of Pseudo-Boiling Phenomenon and the Mass Flux Ratio on the Dynamics of Transcritical Shear Flame," Amirkabir Journal of Mechanical Engineering
53 (9), 12-12, 2021.
8. Shahsavari, M., Farshchi, M., Arabnejad, M.H., Wang, B., "The Role of Flame–flow Interactions on Lean Premixed Lifted Flame Stabilization in a Low Swirl Flow," Combustion Science and Technology,
1-26, 2021
9. EidiAttarZade, M., SarAbadani, A., Davarnia, G., Khosrobeygi, H., Farshchi, M., Ramezani, A., "Investigation of a Bi-propellant Thruster by a Developed Space Engine’s Thrust Chamber Analysis
Code," Journal of Space Science and Technology 14 (2), 47-37, 2021.
10. Poormahmood, A., Farshchi, M. “Numerical study of the mixing dynamics of trans- and supercritical coaxial jets,” Physics of Fluids, 32(12),125105, 2020.
11. Mahdavi, S.A., Ranjbar, A., Farshchi, M., "Effects of Dimensionless Numbers on the Pintle Injector Performance," Modares Mechanical Engineering 20 (7), 1761-1771, 2020.
12. Rezayat, S., Farshchi, M., Ghorbanhoseini, M. “Primary breakup dynamics and spray characteristics of a rotary atomizer with radial-axial discharge channels,” International Journal of Multiphase
Flow, 111, pp. 315-338, 2019.
13. Shahsavari, M., Farshchi, M., Chakravarthy, S.R., Chakraborty, A., Aravind, I.B., Wang, B. “Low swirl premixed methane-air flame dynamics under acoustic excitations,” Physics of Fluids, 31
(9),095106, 2019.
14. Rezayat, S., Farshchi, M. “Spray formation by a rotary atomizer operating in the Coriolis-induced stream-mode injection,” Atomization and Sprays, 29(10), pp. 937-963, 2019.
15. EidiAttarZade, M., Tabejamaat, S., Mani, M., Farshchi, M. “Numerically investigation of ignition process in a premixed methane-air swirl configuration,” Energy, 171, pp. 830-841, 2019
16. Mardani, A., Ghasempour Farsani, A., Farshchi, M. “Numerical Investigation of Gaseous Hydrogen and Liquid Oxygen Combustion under Subcritical Condition,” Energy and Fuels, 33(9), pp. 9249-9271,
17. Ashini, H., Farshchi, M., "Influence of Inlet Temperature and Pressure in Transcritical and Supercritical Laminar Counter-Flow Flame of Liquid Oxygen/Gaseous Methane," Fuel and Combustion 12 (1),
77-96, 2019.
18. Shahsavari, M., Farshchi, M., “Large Eddy Simulation of Low Swirl Flames Under External Flow Excitations,” Flow, Turbulence and Combustion, 100(1), pp. 249-269, 2018.
19. Rezayat, S., Farshchi, M., Karimi, H., Kebriaee, A., "Spray characterization of a slinger injector using a high-speed imaging technique," Journal of Propulsion and Power 34 (2), 469-481, 2018.
20. Shahsavari, M., Farshchi, M., Arabnejad, M.H. “Large Eddy Simulations of Unconfined Non-reacting and Reacting Turbulent Low Swirl Jets,” Flow, Turbulence and Combustion, 98(3), pp. 817-840, 2017.
21. EidiAttarZade, M., Tabejamaat, S., Mani, M., Farshchi, M. “Numerical Study of Ignition Process in Turbulent Shear-less Methane-air Mixing Layer,” Flow, Turbulence and Combustion, 99(2), pp.
411-436, 2017.
22. Ghadimi, M., Farshchi, M., Hejranfar, K. “On spatial filtering of flow variables in high-order finite volume methods,” Computers and Fluids, 132, pp. 19-31, 2016.
23. Bagheri-Sadeghi, N., Shahsavari, M., Farshchi, M. “Experimental characterization of response of lean premixed low-swirl flames to acoustic excitations,” International Journal of Spray and
Combustion Dynamics, 5(4), pp. 309-328, 2013.
24. Barkhordari, A., Farshchi, M. “Numerical simulation of cellular structure of weak detonation and evaluation of linear stability analysis predictions,” Engineering Applications of Computational
Fluid Mechanics, 7(3), pp. 308-323, 2013.
25. Mahdinia, M., Firoozabadi, B., Farshchi, M., Varnamkhasti, A.G., Afshin, H. “Large eddy simulation of lock-exchange flow in a curved channel,” Journal of Hydraulic Engineering, 138(1), pp. 57-70,
26. Ghadimi, M., Farshchi, M. “Fourth order compact finite volume scheme on nonuniform grids with multi-blocking,” Computers and Fluids, 56, pp. 1-16, 2012.
27. Riazi, R., Farshchi, M. “Laminar premixed V-shaped flame response to velocity and equivalence ratio perturbations: Investigation on kinematic response of flame,” Scientia Iranica, 18(4 B), pp.
913-922, 2011.
28. Tahsini, A.M., Farshchi, M. “Numerical study of solid fuel evaporation and auto-ignition in a dump combustor,” Acta Astronautica, 67(7-8), pp. 774-783, 2010.
29. Riaz, R., Farshchi, M., Shimura, M., Tanahashi, M., Miyauchi, T. “An experimental study on combustion dynamics and NOx emission of a swirl stabilized combustor with secondary fuel injection,”
Journal of Thermal Science and Technology, 5(2), pp. 266-281, 2010.
30. Pourfarzaneh, H., Hajilouy-Benisi, A., Farshchi, M. “A new analytical model of a centrifugal compressor and validation by experiments,” Journal of Mechanics, 26(1), pp. 37-45, 2010.
31. Pourfarzaneh, H., Hajilouy-Benisi, A., Farshchi, M. “A new analytical model of a radial turbine and validation by experiments,” IEEE Aerospace Conference Proceedings, 5446764, 2010.
32. Tahsini, A.M., Farshchi, M. “Igniter jet dynamics in solid fuel ramjets,” Acta Astronautica, 64(2-3), pp. 166-175, 2009.
33. Farshchi, M., Mehrjou, H., Salehi, M.M. “Acoustic characteristics of a rocket combustion chamber: Radial baffle effects,” Applied Acoustics, 70(8), pp. 1051-1060, 2009
34. Tahsini, A.M., Farshchi, M. “Thrust termination dynamics of solid propellant rocket motors,”
35. Journal of Propulsion and Power, 23(5), pp. 1141-1142, 2007.
36. Tahsini, A.M., Farshchi, M. “Rapid depressurization dynamics of solid propellant rocket motors,” Collection of Technical Papers - 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, 8, pp.
7871-7879, 2007.
37. Brodrick, C.-J., Laca, E.A., Burke, A.F., (...), Li, L., Deaton, M. “Effect of vehicle operation, weight, and accessory use on emissions from a modern heavy-duty diesel truck,” Transportation
Research Record, (1880), pp. 119-125, 2004.
38. Farshchi, M., Hossainpour, S. “Simulation of detonation initiation in straight and baffled channels,” Scientia Iranica, 11(1-2), pp. 37-49, 2004.
39. Farshchi, M., Hannani, S. K., Ebrahimi, M., "Linearized and non-linear acoustic/viscous splitting techniques for low Mach number flows," International Journal for Numerical Methods in Fluids,
Vol. 42, pp. 1059-1072, 2003.
40. Shahidinejad, S., Farshchi, M., Hajilouy, A., and Souhar, M., "Experiments on Pulsation Effects in Turbulent Flows, Part I: Investigation on Simple Shear Flows," International journal of Science
and Technology (Scientia Iranica), Vol. 10, No. 2, pp. 238-247, 2003.
41. Shahidinejad, S., Hajilouy, A., Farshchi, M., and Souhar, M., "Experiments on Pulsation Effects in Turbulent Flows, Part II: Investigation on Grid-Generated Turbulence," International journal of
Science and Technology (Scientia Iranica), Vol. 10, No. 2, pp. 248-251, 2003.
42. Farshchi, M. and Hossain-pour, S., "Simulation of Detonation Initiation in Straight and Baffled Channels," accepted for publication in the International journal of Science and Technology
(Scientia Iranica), 2003.
43. Farshchi, M., Vaezi, V., and Shaw B. D., "Studies of HAN-Based Monopropellant Droplet Combustion," Journal of Combustion Science and Technology, Vol. 174 (7), pp. 99-125, 2002.
44. Brodrick, C. J., Farshchi, M., Dwyer, H. A., Harris B., and King, F. Jr., "Effects of Engine Speed and Accessory Load on Idling Emissions from Heavy-Duty Diesel Truck Engines," Journal of the Air
and Waste Management Association, Vol. 52, pp. 1026-1031, 2002.
45. Rahimian, M., H., Farshchi, M., "Effect of Non-dimensional Parameters on the Internal Circulation of a Liquid Droplet Moving with the Surrounding Gas," Esteghlal Journal of Engineering, Isfahan,
Iran, vol. 21, No. 1, pp. 167-180, July 2002.
46. Brodrick, C. J., Laca, E., Farshchi, M., Dwyer, H. A., "Effect of On-Road Loads on Gaseous Emissions from a Modern Heavy-Duty Diesel Truck," accepted for publication in the Journal of
Transportation and Statistics, 2002.
47. Brodrick, C.J., Lipman, T., Farshchi, M., Dwyer, H. A., Sperling, D., Gouse S. W., Harris B., and King, F. Jr., "Evaluation of Fuel Cell Auxiliary Power Units for Heavy-Duty Diesel Trucks,"
Transportation Research Part D, Vol. 7, pp. 303-315, 2002.
48. Golafshani, M., Farshchi, M., and Ghassemi, H., "Effects of Grain Geometry on Acoustic Instability of Solid Propellant Rocket Motors," AIAA Journal of Propulsion and Power, Vol. 17, No. 6, Nov.
49. Farshchi, M., Brodrick, C. J., and Dwyer, H. A., "Dynamometer Testing of a Heavy-Duty Diesel Engine Equipped with a Urea-SCR System," SAE 2001-01-0516, in Diesel Exhaust Emission Control
(SP-1581), March 2001.
50. Farshchi, M. and Rahimian, M.H., "Unsteady Deformation and Internal Circulation of a Liquid Drop in a Zero Gravity Uniform Flow," ASME Journal of Fluids Engineering, Vol. 121, pp. 665-672,
September 1999.
51. Brodrick, C. J., Farshchi, M., Dwyer, H. A., and Sperling, D., "Urea "SCR Demonstration and Evaluation for Heavy-Duty Diesel Trucks," SAE 1999-01-3722, in Truck and Bus Engines, Transmissions,
and Gearing (SP-1488), November 1999.
52. Farshchi, M. and Hossain-pour, S., "Determination of BKW Equation of State Parameters via Measurement of the Detonation Velocity in Condense Explosives," Esteghlal Journal of Engineering,
Isfahan, Iran, vol. 17, No. 2, pp. 1-12, Feb. 1999.
53. Farshchi, M. and Rahimian, M.H., "Droplet Deformation Dynamics in Convective Flow Fields," Esteghlal Journal of Engineering, Isfehan, Iran, vol. 16, No. 2, pp. 1-15, Feb. 1998.
54. Farshchi, M., Rahimian, M.H., and Ashgriz, N., "Chemically Reacting Liquid Droplet Flow Field Calculation," in Transport Phenomena in Combustion, edited by S.H. Chan, Taylor and Francis
Publications, volume 1, page 872, July 1995.
55. Golafshani, M., Farshchi, M., and Shahsavan, M., "A Preconditioning Technique for the Solution of Navier-Stokes Equations in all Flow Regimes," accepted for publication in Esteghlal Journal of
Engineering, Isfahan, Iran.
56. Nixon, D., Carosu, S.C., and Farshchi, M., "Further Study of Phantom Solutions to the TSD-Euler Equation," Acta Mechanica, vol. 86, pp. 15-29, 1991.
57. Dibble, R.W., Kollmann, W., Farshchi, M, and Schefer, R.W., "Second-Order Closure for Turbulent Nonpremixed Flames: Scalar Dissipation and Heat Release Effects," Twenty-first Symposium
(International) on Combustion/The Combustion Institute, pp. 1329-1340, 1986.
58. Jischke, M.C. and Farshchi, M. "Boundary Layer Regime for Laminar Free Convection between Horizontal Circular Cylinders," ASME Journal of Heat Transfer, VOL. 102, May 1980. | {"url":"http://ae.sharif.edu/~portal/faculty/1701675126","timestamp":"2024-11-03T16:04:27Z","content_type":"text/html","content_length":"38109","record_id":"<urn:uuid:aa36c6e5-14c6-43ed-8c7e-031ca5f15c4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00531.warc.gz"} |
Meaning Of Scale Drawing
Meaning Of Scale Drawing - Web scale drawings make it easy to see large things, like buildings and roads, on paper. Web scale is defined as the ratio of the length of any object on a model
(blueprint) to the actual length of the same object in the real world. Web what is a scale drawing? Web understand how a scale drawing is converted into real numbers using the scale factor. All the
measurements in the drawing correspond to the measurements of the. Web a scale drawing is an enlarged or reduced drawing that is proportional to the original object. Even a gps uses scale drawings!
Web a scale drawing is created by multiplying each length by a scale factor to make it larger (an enlargement) or smaller (a reduction) than the original object. The scale is shown as. An enlargement
changes the size of an object by multiplying each of the lengths by a scale factor to.
When an object is being designed it is not usually drawn to actual size. Web a scale drawing is created by multiplying each length by a scale factor to make it larger (an enlargement) or smaller (a
reduction) than the original object. All the measurements in the drawing correspond to the measurements of the. An enlargement changes the size of an object by multiplying each of the lengths by a
scale factor to. In other words, it is the process of drawing an object to. Web what is a scale drawing? This means that all of the ratios between the corresponding sides of. Web a scale drawing
represents an actual object but is different in size. The scale is shown as. For students between the ages of.
Web a scale drawing is created by multiplying each length by a scale factor to make it larger (an enlargement) or smaller (a reduction) than the original object. Web what is a scale drawing? Web
design & style. Web to scale a drawing by hand, start by measuring the width and height of the object you'll be scaling. So a large scale artwork is one that. Scaled drawing a drawing that shows a
real object with accurate sizes reduced or enlarged by a certain amount (called the scale). The scale is shown as. The scale for a drawing is the ratio between. Web the ratio of the length in a
drawing (or model) to the length on the real thing. Web an urban planner needs your help in creating a scale drawing.
Math 9 CHAPTER 4 SCALE DRAWINGS
Web scale is a principle of art that is defined as the size of an artwork. When an object is being designed it is not usually drawn to actual size. The scale for a drawing is the ratio between. Web a
scale drawing represents an actual object but is different in size. Scaled drawing a drawing that shows a real.
Engineering Drawing Chapter 01 introduction
The lengths in a scale drawing are in proportion to the actual lengths of an object. Web what is a scale drawing? When an object is being designed it is not usually drawn to actual size. Web scale
drawings make it easy to see large things, like buildings and roads, on paper. Even a gps uses scale drawings!
Scale Drawing GCSE Maths Steps, Examples & Worksheet
Web a drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale). Web a scale drawing is a drawing of an object that is larger or smaller than,
but proportional to, the original object. Web an urban planner needs your help in creating a scale drawing. Let's use our knowledge about.
Engineering Drawing
Web a scale drawing is a drawing of an object that is larger or smaller than, but proportional to, the original object. We often use a ratio to show the scale of a drawing. Understanding scale and
proportion in art and design. Web design & style. Web understand how a scale drawing is converted into real numbers using the scale.
Understanding Scales and Scale Drawings A Guide in 2020 Scale
Next, choose a ratio to resize your drawing, such as 2 to 1. In other words, it is the process of drawing an object to. Web what is a scale drawing? Even a gps uses scale drawings! Web what is a
scale drawing?
Scale Drawings
Web to scale a drawing by hand, start by measuring the width and height of the object you'll be scaling. When an object is being designed it is not usually drawn to actual size. In the drawing
anything with the size of 1 would have a size of 10 in the real world, so. Web a scale drawing is created.
Understanding Scales and Scale Drawings A Guide
Understanding scale and proportion in art and design. Next, choose a ratio to resize your drawing, such as 2 to 1. Web a scale drawing represents an actual object but is different in size. Web a
scale drawing is an enlarged or reduced drawing that is proportional to the original object. Even a gps uses scale drawings!
How To Do Scale Drawing 7 Grade
The lengths in a scale drawing are in proportion to the actual lengths of an object. Scaled drawing a drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount
(called the scale). Floor plans and maps are some examples of scale drawings. Web design & style. In the drawing anything with the size of.
Scale Drawing bartleby
So a large scale artwork is one that. For students between the ages of. We often use a ratio to show the scale of a drawing. Web scale drawings make it easy to see large things, like buildings and
roads, on paper. Large objects are scaled down onto the page and small objects may be.
Understanding Scales And Scale Drawings A Guide Scale vrogue.co
We often use a ratio to show the scale of a drawing. Web understand how a scale drawing is converted into real numbers using the scale factor. Web a scale drawing is an enlarged or reduced drawing
that is proportional to the original object. In the drawing anything with the size of 1 would have a size of 10 in.
Web A Drawing That Shows A Real Object With Accurate Sizes Reduced Or Enlarged By A Certain Amount (Called The Scale).
The scale is shown as. The scale is shown as the length in the drawing, then. We often use a ratio to show the scale of a drawing. Large objects are scaled down onto the page and small objects may
The Lengths In A Scale Drawing Are In Proportion To The Actual Lengths Of An Object.
A scale drawing is an enlargement of an object. Check out this tutorial to learn all about scale drawings. Web a scale drawing is an enlarged or reduced drawing that is proportional to the original
object. Jun 7, 2021 • 2 min read.
Scale Drawings Are Usually Smaller Than The Object Represented.
Web an urban planner needs your help in creating a scale drawing. Web design & style. Web to scale a drawing by hand, start by measuring the width and height of the object you'll be scaling. Web the
ratio of the length in a drawing (or model) to the length on the real thing.
Scaled Drawing A Drawing That Shows A Real Object With Accurate Sizes Reduced Or Enlarged By A Certain Amount (Called The Scale).
In the drawing anything with the size of 1 would have a size of 10 in the real world, so. Web scale is defined as the ratio of the length of any object on a model (blueprint) to the actual length of
the same object in the real world. Web what is a scale drawing? Understanding scale and proportion in art and design.
Related Post: | {"url":"https://drawingtraining.com/en/meaning-of-scale-drawing.html","timestamp":"2024-11-13T04:52:17Z","content_type":"text/html","content_length":"29845","record_id":"<urn:uuid:4b2d334d-4303-4a23-b9f3-6eccfa3c6df3>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00126.warc.gz"} |
On connecting modules together uniformly to form a modular computer for SWCT 1965
SWCT 1965
Conference paper
On connecting modules together uniformly to form a modular computer
View publication
We informally define a modular (cellular, iterative) computer to be a device consisting of a large (or, in theory, infinite) number of identical circuit modules connected together in some uniform
manner, that is, in such a fashion that every module is connected into the device in the same manner as every other. In this paper we propose a mathematically precise definition of, "connected
together in a uniform manner". In brief, we show that the underlying linear graph whose vertices correspond to the modules and whose edges correspond to the cables connecting the modules, is a
group-graphs that is, the vertices correspond to the elements of a group G and there is given a finite subset G0 of G such that {g, g1} is an edge of the graph If, and only if, there exists g0 ϵ G0
such that g1 = gg0. We further investigate the effects of restricting G to be an Abelian group and indicate why we feel such a restriction is unwarranted at least in developing the theory of modular | {"url":"https://research.ibm.com/publications/on-connecting-modules-together-uniformly-to-form-a-modular-computer--1","timestamp":"2024-11-13T22:13:16Z","content_type":"text/html","content_length":"65668","record_id":"<urn:uuid:df19b32a-fcd5-4a89-b91b-a21bf45fdafa>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00591.warc.gz"} |
What Is a Convolution? How To Teach Machines To See Images | 8th Light
How do machines see photos? Storing, sorting, and displaying image files are all fairly basic operations; but what if we ask a computer to reverse direction and actually digest the content within an
image? Humans rely on millions of nerve fibers in their eyes to sync up with their brain and process visual stimuli, and a machine will have only its computational tools to recreate a similar
The first blog post in this series, "Image Classification: An Introduction to Artificial Intelligence," walks through the instructions on how to create a simple Convolutional Neural Network (CNN) for
an image classification task. It did not explain the convolutional part of the network, and so it may remain odd why that machine learning architecture was appropriate. This post will show how
machines can extract patterns from the data to solve image-related problems by leveraging multiple techniques and filters working together in a process called convolution.
Formally speaking, a convolution is the continuous sum (integral) of the product of two functions after one of them is reversed and shifted. It is much simpler in practice, and this post will use
some basic examples that build up to it gradually. By the end of the post, readers should gain an understanding of convolution, familiarity with some of the image processing techniques that use them,
and an awareness of how you might leverage convolution to solve image-related challenges in your own software development work.
Section 1: An Intuitive Convolution Example
Convolution is helpful for more than just image recognition, and its mechanics are easier to understand when applied to a more traditional challenge with numbers.
For this example, imagine a farmer who wants to have tomatoes available all year round. For that, they need to plant the tomatoes at different dates so they can be harvested at different dates too.
How could one estimate the amount of water needed each month?
This array shows how many plants the farmer wants to plant in each of the following five months:
plant_batches = [1 2 3 4 5]
1 plant in the first month, 2 in the second, 3 in the third, and so on. So written as a function, it looks like: f(month) = plants. Over the next five months (array length), you will plant a total of
15 plants (array sum).
Now let's say that each plant will be ready to harvest in a single month, and it will take exactly 2 tons of water in that span. Calculating how much water the farmer will use each month is easy:
just multiply each number of plants by 2 to get the answer:
water_per_plant = 2
total = [2 4 6 8 10]
You would need 2 tons of water the first month, 4 on the second, and so on.
Adding Another Variable
Let's complicate the problem a little bit. Let's say that each plant needs three months to grow (not one). And as they grow, they need more and more water to survive: 2 tons on the first month (month
0), 3 on the second, and another 4 in the third. This is now a second function with the form g(growth_month) = water_needed. So, if a plant is in its first month of growth, then you can calculate how
much water it needs by calling g(0) = 2. This can be represented in the following array:
Calculating how much water the plants need each month is now more complicated. On the first month, the farmer's first plant is in its first stage of watering. But on the second month, the first plant
is in its second stage, and two newly planted tomato plants are in the first stage. Each of the five months initiates another round and layer of complexity, which can become overwhelming pretty
This is where a convolution is useful, and I'll walk through the steps for this example. First, flip the function (array) of watering stages. Then place both arrays side by side so that the first
batch of plants is aligned with the first watering stage. Multiply the values that are aligned: 1*2=2.
[1 2 3 4 5] // plant_batches
[4 3 2] // water_needed flipped
= 2
total = [2]
For the second month, move the watering stages one step to the right, multiply the aligned values (which are now two columns), and add their results. In this second month the farmer has one plant
that grew and now needs three tons of water, and two new plants that each need two tons of water: (2*2)+(1*3)=7. The integration part comes by adding all multiplications.
[1 2 3 4 5]
[4 3 2]
= 7
total = [2 7]
For the third month, the farmer will have three overlapping columns:
[1 2 3 4 5]
[4 3 2]
= 16
total = [2 7 16]
Keep doing this until the last overlapping element, and you will end with a seven month watering plan. It will take seven months because the last batch that was planted in month five will continue to
need water until month seven.
[1 2 3 4 5]
[4 3 2]
= 20
total = [2 7 16 25 34 31 20]
The watering stages array slid through the batches of plants. This is why you will find the sliding window operation continuously mentioned when referring to convolutions. This window (watering
stages) is called the kernel.
Section 2: 2-Dimensional Convolutions
The plant watering example is uni-dimensional, but the problem is similar with two dimensions. The window and the background will still be multiplied one element at a time, even when both of them
contain two dimensions:
Performing a convolution on two dimensions just means sliding the window through every column of every row.
Notice how the kernel started completely inside the input background? This is standard for image processing, but there are other times (similar to the farmer example in Section 1) when you'll want to
start the kernel outside of the input. In those cases, you can modify the output size by adding leading and trailing zeroes to the input. This is called padding, and it allows more multiplication
operations to happen for the values in the borders, thus allowing their information to have a fair weight.
Section 3: Grayscale Images
Images are more involved than the simple examples in the previous sections, but they build on top of the same fundamentals. A black and white image is just a two-dimensional array with values that
represent the intensity of each pixel. Higher values represent whiter shades, and lower values represent darker ones.
Let's take the following image as the input, and transform it into grayscale.
We can then apply a variety of different kernels to it, and print their results. For example, the following kernel is called left sobel:
If you convolve the grayscale fruits image with this kernel, and print the output as a grayscale image, it looks like this:
Why does the output look like it is casting a shadow to the right? The left sobel kernel has positive values in the left and negative values in the right. When the kernel is convolving over a
homogeneous background, the multiplication of these values cancels out. However, when the kernel reaches the border of a fruit, one side of the kernel will produce a value that has more weight than
the other (either positive or negative), and that difference will account for the new pixel's value.
When the contrast between the fruit and the background is larger, the kernel operation is larger too (positive or negative). That is why the apple, which has a larger contrast with the background,
creates a border that is better defined than that of the banana.
Another commonly used kernel is the outline:
Convolving the same grayscale image with it produces the following output:
Can you deduct why the output looks like this? The kernel shows contrasting differences in a pixel and its surrounding elements, and as a result it creates both a dark and a light shade in each side
of every border.
There are many more kernels that can convolute and generate interesting outputs for grayscale images. In the next section, I will go one step further and examine how a computer sees color.
Section 3: Color Images
The color on a computer screen is the result of mixing the three primary colors — red, green, and blue. Every pixel contains these three colors in varying degrees, resulting in the output appearing
on a screen.
The important insight for this blog post is that a 2-dimensional image is actually a 3-dimensional array, as each color is given its own value at every coordinate: image[x][y] = [R, G, B]. Each
channel (color) can be processed separately, and then joined back together to represent a color image.
For example, using the left sobel kernel on every channel in the original image before combining yields this result:
And using the outline kernel produces this result:
Using The Gaussian Kernel
Another interesting kernel is a gaussian kernel. This function takes values of a gaussian (or normal) distribution and places them in a vector to compute the outer product with itself. For example,
given 5 discrete values of the gaussian distribution, the outer product will be a (5, 5) matrix. The kernel then normalizes those values so that they add up to 1. This creates an average sum of
pixels that gives more weight to those in the center of the kernel. Executing this convolution in each color of the input image produces a blur effect.
Increasing the kernel from (5, 5) to (11, 11) and with a higher standard deviation of the initial gaussian distribution yields a stronger blur:
Notice how the output is 3-dimensional (height, width, 3) despite using a 2-dimensional (11, 11) kernel. This is because this kernel has only two dimensions (x, y). If instead, you used a 3D kernel
of shape (11, 11, 3), it would only slide in the x and y directions, so the output would be a 2-dimensional array. One would iterate this rectangular cuboid over the 3-dimensional RGB image to
produce a 2-dimensional array.
Computers do not need an input to have exactly three colors — they can have 10, 20, or as many channels as we want. As long as the kernel is exactly of the same depth, it will generate a
2-dimensional output.
Now let's take it one step further: take two different 3D kernels and execute the convolution over the same input. This will result in two different 2D arrays, which can be portrayed as a 3D array of
exactly two channels.
This opens up a world of possibilities. One can stack multiple convolutions using multiple kernels, creating a single output of an arbitrary depth (the number of kernels dictates the depth of the
output). This output can be further convoluted again using the same logic, creating chains of convolutions that build on each other.
The problem is that infinite possibilities is too much for a human being; and that is where machine learning comes in.
Section 4: Relation To Machine Learning
Each kernel provides different information. In the grayscale fruits example, the left sobel kernel shows vertical borders and the outline kernel shows differences in contrast. These visual
representations are patterns that humans can understand, but computers can create their own kernels based on patterns that only they can understand.
Convolutional Neural Networks (CNNs) are a common network architecture used for processing images for machine learning. CNNs change the values of the kernels when learning so that they are optimized
for the task at hand.
Imagine that we want a machine learning model to learn 16 kernels of shape (3, 3, 1) (2D) and to convolute over the same input of size (28, 28, 1) (2D). The resulting output will have a depth of 16,
one for each kernel.
In order to learn, each kernel needs an extra parameter called bias. Thus, all 16 kernels need to be trained for (3*3*1)+1 = 10 parameters, for a total of 16*10 = 160 parameters in each convolutional
layer. This is exactly the number of parameters shown in the conv2d_4 (Conv2d) layer in the summary of the previous post:
The next layer, conv2d_5 (Conv2d), is a bit trickier. Its input shape has a depth of 16 (one for each kernel in the previous layer!), which means that each 3x3 kernel is actually a cuboid of shape
(3, 3, 16). Each of them then needs to learn (3*3*16)+1 = 145 parameters. Because there are 64 kernels, then the total training parameters for that layer is 145*64 = 9280. Don't worry too much about
the bias parameter right now, what is important is understanding what a convolution means and what a machine model learns when using them in its layers.
What the model will learn for values in these kernels will make no sense for a human eye, especially the second layer. But we can still use it. The shape of the layer is further transformed into a
single vector using Flatten. If the 2D output was of shape (11, 11, 64), then the size of the flattened vector will be 11*11*64 = 7744. This can be fed easily into a traditional dense layer.
So why bother going through the convolutional layers if we can flatten the images? Because the relationships between pixels are helpful clues. Pixels that are close to each other must share some
information, and convolving over the entire image at the start might flatten these connections, filtering out abstractions through the layers and leaving the machine to learn through intense
repetition alone.
This blog post explained what a convolution is, and some common image processing techniques that use them, like blur and some border recognition convolutions. It also touched the concept of padding
and how it affects the size of the output. The examples went from a 1D convolution to a 3D convolution, and introduced the sliding-window operation.
Finally, the examples showed what is being learned in a convolutional neural network, and we were able to count the exact number of parameters for its layers. Hopefully you are able to see why using
convolutional neural networks are useful for visual tasks like the image classification of the previous post.
Where To Go Next?
With this theoretical knowledge, you could start thinking of building a simple convolutional method and experiment with some interesting kernels. You can check out a python implementation of this
blog post here. I recommend checking out other kernels, like the other three sobel kernels, the sharpen kernel, and different sizes of the gaussian blur. You can learn more about how the kernels
interact by experimenting with, say, blurring an already blurred image over and over again, or applying one sobel after another.
With regards to neural networks, this blog showed how a convolutional layer is used to process an image. Can you imagine different approaches to process a video? Would you handle the time dimension
with a 4D kernel, or would you process each frame separately using 3D kernels?
I also recommend checking out Python's OpenCV library for a vast collection of other image processing tools. One could imagine using a 1D kernel to process time sequences like stock prices and daily
covid infections over a year.
Once you learn to harness your computer's processing power into machine learning tasks, your imagination is the limit. | {"url":"https://8thlight.com/insights/what-is-a-convolution-how-to-teach-machines-to-see-images","timestamp":"2024-11-05T21:47:11Z","content_type":"text/html","content_length":"134344","record_id":"<urn:uuid:baad83f1-3a90-4b76-9eac-2b8849d655fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00003.warc.gz"} |
Wolfram|Alpha Examples: Step-by-Step Algebra
Examples for
Step-by-Step Algebra
Algebraic Substitutions
Evaluate an expression one step at a time:
Find the value of a function at a given point:
Multiply expressions with exponents:
Perform division with exponents one step at a time:
Get the steps for simplifying powers of exponents:
Rational Expressions
See how to add or subtract rational expressions:
Perform multiplication of rational expressions step by step:
Show how to simplify rational functions with common factors:
Simplify rational expressions with the steps provided:
Learn to rewrite a rational expression using partial fraction decomposition:
Solving Equations & Inequalities
Solve equations one step at a time:
Investigate equations with rational expressions:
Solve equations with absolute value over the reals:
Get step-by-step solutions for solving radical equations:
Learn how to solve exponential equations:
See the steps for solving linear inequalities:
See the steps for finding the y‐intercept:
Linear Expressions
Find the equations of lines defined by slopes and points:
See the steps for solving linear equations using addition and subtraction:
Learn how to solve linear equations using multiplication and division:
Show the steps to solve linear equations with multiple operations:
Solve linear systems with multiple methods:
Exponential Expressions
See how to solve exponential equations with complete steps:
Compute financial present and future values with steps:
Polynomial Expressions
See how to combine like terms:
Show how to add polynomial expressions step by step:
Subtract polynomial expressions:
Show your work for multiplying polynomials:
Perform polynomial long division step by step:
Expand polynomials using the distributive law, the binomial theorem and other procedures:
Factor out the greatest common divisor:
Use the difference of squares method to factor binomials:
Choose from multiple methods to complete the square:
Factor expressions as a sum or difference of cubes:
Factor polynomials step by step:
Find zeros of a quadratic expression by factoring, completing the square or applying the quadratic formula:
Find candidate roots using the rational root theorem:
Compute the degree of a polynomial with steps:
Logarithmic Expressions
See how to simplify log expressions with steps:
See how to combine multiple logarithmic expressions into a single logarithm with steps: | {"url":"https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-algebra/","timestamp":"2024-11-05T02:17:13Z","content_type":"text/html","content_length":"137492","record_id":"<urn:uuid:3df85f21-9097-4b8c-85bc-0a2db12d615f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00433.warc.gz"} |
Further Maths - Sums of Cubes
What is the formula for the sum of the cubes of the first $n$ natural numbers?
\[\sum^{n}_{r=1} r^3 = \frac{1}{4}n^2(n+1)^2\]
How could you rewrite $\sum^{n} _ {r=1} r^3$?
What’s another way of expressing $\frac{1}{4}n^2(n+1)^2$?
\[\sum^{n}_{r=1} r^3\]
How could you rewrite $\sum^{n} _ {r=1} 4r^2$?
What’s an easy way for remembering the sum of cubes formula?
It’s the sum of the natural numbers formula squared.
Related posts | {"url":"https://ollybritton.com/notes/a-level/further-maths/topics/sums-of-cubes/","timestamp":"2024-11-05T09:45:54Z","content_type":"text/html","content_length":"503711","record_id":"<urn:uuid:62ccb829-57b7-477b-9fb7-59da85f08044>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00405.warc.gz"} |
find the equation of the tangent to the curveexy=x+(In y)2 at the poi
arka mitra
Last Activity: 13 Years ago
using the chain rule ....
e^xy(y+xdy/dx)=1 + 2.ln y.1/y.dy/dx
ye^xy-1=[2(ln y)/y-xe^xy] dy/dx
i.e. {y(ye^xy-1)/(2ln y-yxe^xy)}=dy/dx
now putting the values x=o and y=e we get...
e(e-1)/2=dy/dx [ ln e=1 ]
therefore the tangent to the curve at (0,e) is e(e-1)/2 | {"url":"https://www.askiitians.com/forums/Differential-Calculus/25/35994/help-3.htm","timestamp":"2024-11-10T11:46:52Z","content_type":"text/html","content_length":"184220","record_id":"<urn:uuid:1548731a-1d18-4cff-93f8-a2019207bdb2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00584.warc.gz"} |
Constructing a square
Jump to navigation Jump to search
1. Construction of a perfect square.
2. Use of equal radius arcs to get equal sides of a square in the construction protocol.
3. Use of perpendicular lines to draw sides of a square.
Estimated Time
40 minutes
Prerequisites/Instructions, prior preparations, if any
Materials/ Resources needed
Non digital: White paper, compass, pencil and scale.
This activity has been taken from the website :http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Mylod/Math7200/Project/Square.html
Process (How to do the activity)
1. To construct a square of a given length, draw a line segment AB of given length.
2. Draw perpendicular to the line AB at A.
3. Taking AB as radius draw a circle with A as centre.
4. The circle intersects the perpendicular at a point. Name the intersecting point as C.
5. Similarly draw perpendiculars to AC at C. You get third side of the square.
6. To get the fourth side of the square construct a perpendicular to AB at B.
7. Name the two intersecting perpendiculars as D.
8. ABCD would be the required square.
1. For what measure are you drawing a square ?
2. What is a perpendicular line ?
3. Why are we constructing perpendicular lines for constructing a square ?
4. What is the purpose of drawing a circle ?
5. How do we determine the fourth vertex of the square ?
1. What are perpendicular lines ?
2. Why is a circle being drawn ? Which purpose would it solve ?
1. Can a square be constructed without using a compass ? | {"url":"https://www.karnatakaeducation.org.in/KOER/en/index.php/Constructing_a_square","timestamp":"2024-11-13T05:14:59Z","content_type":"text/html","content_length":"34951","record_id":"<urn:uuid:8f898d90-2797-493e-95f9-c4fad84db202>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00017.warc.gz"} |
How do you create a debt amortization schedule?
How do you create a debt amortization schedule?
It’s relatively easy to produce a loan amortization schedule if you know what the monthly payment on the loan is. Starting in month one, take the total amount of the loan and multiply it by the
interest rate on the loan. Then for a loan with monthly repayments, divide the result by 12 to get your monthly interest.
What is debt amortization schedule?
What Is a Loan Amortization Schedule? A loan amortization schedule is a complete table of periodic loan payments, showing the amount of principal and the amount of interest that comprise each payment
until the loan is paid off at the end of its term.
Can I get an amortization schedule?
An amortization schedule can be created for a fixed-term loan; all that is needed is the loan’s term, interest rate and dollar amount of the loan, and a complete schedule of payments can be created.
How do you calculate monthly amortization?
To calculate amortization, start by dividing the loan’s interest rate by 12 to find the monthly interest rate. Then, multiply the monthly interest rate by the principal amount to find the first
month’s interest. Next, subtract the first month’s interest from the monthly payment to find the principal payment amount.
When loans are amortized monthly payments are?
An amortizing loan is a type of debt that requires regular monthly payments. Each month, a portion of the payment goes toward the loan’s principal and part of it goes toward interest. Also known as
an installment loan, fully amortized loans have equal monthly payments.
How do I calculate an amortization schedule?
Amortized payments are calculated by dividing the principal — the balance of the amount loaned after down payment — by the number of months allotted for repayment. Next, interest is added. Interest
is calculated at the current rate according to the length of the loan, usually 15, 20, or 30 years.
What does a loan amortization schedule show?
An amortization schedule is a complete table of periodic loan payments, showing the amount of principal and the amount of interest that comprise each payment until the loan is paid off at the end of
its term.
How is an amortization schedule calculated?
Calculations in an Amortization Schedule. The Interest portion of the payment is calculated as the rate ( r) times the previous balance, and is usually rounded to the nearest cent. The Principal
portion of the payment is calculated as Amount – Interest. The new Balance is calculated by subtracting the Principal from the previous balance.
What is amortization schedule definition?
DEFINITION of ‘Amortization Schedule’. An amortization schedule is a complete table of periodic loan payments, showing the amount of principal and the amount of interest that comprise each payment
until the loan is paid off at the end of its term. | {"url":"https://www.shakerdesignproject.com/students-advice/how-do-you-create-a-debt-amortization-schedule/","timestamp":"2024-11-14T21:40:00Z","content_type":"text/html","content_length":"59507","record_id":"<urn:uuid:c0cbaa0a-1f33-4951-b9b4-0ddabd3cf9c1>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00856.warc.gz"} |
KDDP Pockels Cells - Operation and Design -
KDDP Pockels Cells – Operation and Design
Published: November 5, 2021
The purpose of this technical description is the presentation of basic principles of modulators using the linear electro-optic effect in KDDP crystals – it discusses specifically KDDP Pockels cells.
It can be helpful for laser system designers and can especially stimulate proper technical solutions and applications of the electro-optic effect for modulation problems.
KDDP Pockels Cells – Principle of Operation
The operation of Pockels electro-optic modulators is based on the principle of electrically induced birefringence in anisotropic crystals. The optical properties of these crystals are described by
index ellipsoid. For Pockels effect, the variation (∆) of these indices of refraction (n[i]) is proportional to the applied electric field (E[i]) and they are related by tensor of electrooptic
coefficients (r[ij] ) :
∆ (1/ni2) = ∑ r[ij] E[j]
Tensor r[ij] describes electrooptic properties of the crystal. This tensor, of rank 3, contains 18 electro-optic coefficients r[ij] describing electrically induced birefringence in the crystal.
However, in optical crystals, many of these coefficients are equal to zero. The number and values of the rij coefficients will determine the possibility of application to the modulation of laser
From many crystals belonging to twenty classes of symmetry, in practice, only a few crystals are used for the Pockels cells design. They are birefringent crystals. When an external electric field is
applied, the natural birefringence of the crystal changes. This deforms the index ellipsoid of the crystal and the uniaxial crystal becomes biaxial with new induced indexes of refraction. The input
optical beam splits into two orthogonally polarized components propagating in a crystal with different velocities, determined by new induced refractive indices. Induced birefringence ∆n is
proportional to the applied electric field and on the crystal length L gives controlled phase retardation Γ between these components :
Γ = 2π ∆n L / λ
where: λ – wavelength of the optical beam.
The voltage sensitivity of the Pockels cell is described by the half-wave voltage (U [λ/2] ). This is the voltage required to obtain a phase retardation Γ of 180°.
For Z-0 cut XDP crystal family half wave voltage for longitudinal Pockels cell is given by the following relations :
U [λ/2] = λ /2n[0]^3r[63]
where :
n[0] – index of refraction for ordinary ray,
r[63] – electooptic coefficient
For Z-45° cut crystals of XDP family, used for transverse Pockels cells, the half wafe voltage is given by :
U [λ/2] = λd /2n[0]^3r[63]L
where :
L – crystal length,
d – the distance between electrodes.
Modulation of the optical beam is realized by using Pockels cell and suitable polarizing elements such as Glan-Thompson, Wollaston prisms, thin-film polarizers and others. Optical beam of intensity I
[0] passes through the input polarizer which produces linearly polarized light. The output polarizer (analyzer) can be either crossed to the input or parallel. In the case of crossed polarizers,
voltage U applied to the cell and output beam intensity I are related by :
T = I/I[0] = k sin^2 (πU/2 U [λ/2] )
where :
T – relative transmission,
I[0] – input intensity,
k – loss coefficient (k<1).
Design of KDDP Pockels Cells
Pockels cells are used in laser technology to control the parameters of laser radiation both inside and outside the laser resonator. Outside the resonator Pockels, cells are used as fast optical
shutters. They are used for shortening Q-switched pulses up to nanosecond duration as well as for the selection of single ps and fs laser pulse from the mode-locked train of pulses. Inside the
resonator, Q-switching technology is commonly used as a method to achieve high power laser pulses, especially in solid-state lasers.
All these applications show, that the Pockels cell must be able to hold high power densities of laser beam and must be fast in operation. In the visible (VIS) and near-infrared (NIR) spectral range,
for Pockels cell designs are used at present the following basic crystals: KDDP, LiNbO[3], BBO and also KTP and RTP. Active elements for cells are fabricated as rods with a round shape with axis
along the optical axis of the crystal (Fig.1), or as rectangular rods with the faces perpendicular to the main crystallographic axes. There are two basic configurations used for electrooptic
modulation: longitudinal and transverse electrooptic effect. The configuration used in practice depends on the matrix of electrooptic coefficients r[ij] for the crystal.
Up to now, KDDP (KD*P) Pockels cells are especially widely used in laser technology. The r[ij] matrix for these crystals has 3 non-zero coefficients: r[41], r[52] = r[41], and r[63]. So, both
longitudinal and the transverse electrooptic effect can be realized here. Longitudinal effects in KDDP crystals have found the widest application. In this configuration (Fig.4), the direction of the
electric field is parallel to the direction of the laser beam and the optical axis of the crystal [1], [2]. In practice, to implement the longitudinal effect, cylindrical ring electrodes are used on
the side surface of the KDDP crystal with the Z – 0 cut (Fig.1).
Fig.1. KDDP crystal with Cylindrical Ring Electrodes (CRE).
With such a configuration of electrodes, it is impossible to obtain a homogeneous electric field in the crystal [1] – [4]. That is a source of specific nonuniformity of transmission in the crystal
cross-section. In order to obtain nonuniformity of the field distribution in the crystal <5%, the length of the crystal (2L) must be at least twice its diameter (2R). With such crystal dimensions,
the width of the electrode W must be sufficiently large. For W / L = 0.65 – 0.7 it is possible to obtain field nonuniformity in the crystal dU = 3 – 4% when L / R = 2 or more ( Fig. 2 ).
Fig.2. Nonuniformity of the electric field distribution (dU) for different geometries of KDDP crystal and width of CRE electrodes [1].
Fig.3. CRE KDDP crystals for Pockels cells with different apertures ( by INRAD Optics ).
The longitudinal electrooptic effect is a great advantage of KDDP crystal. Driving voltages U [λ/2] are not dependent on the crystal dimensions and are the same for all Pockels cell apertures
(Fig.3). This property can be very useful for various studies. This gives, for example, the possibility of constructing Pockels cells with large apertures at a very short crystal length.
Since the driving voltages cannot be reduced by the choice of crystal geometry ( like in the case of transverse electrooptic effect), it can be realized by increasing the number of crystals in the
cell. In practice, double-crystal designs of longitudinal KDDP Pockels cells are offered for some applications. In these designs, KDDP crystals are mounted in series and supplied with opposite
directions of electric field (Fig.4 b).
Fig.4. KDDP crystals of rectangular shape for Pockels cells with longitudinal electrooptic effect : a) for single crystal cell, b) for double-crystal cell.
KDDP (KD*P) crystals, that are grown from deuterated water solutions, are hygroscopic. Therefore, they require protection against atmospheric moisture. For this purpose, different technical solutions
are used. KDDP crystals, generally are placed in appropriate hermetic housings.
In some technical solutions, crystals are mounted in housings closed by optical windows and filled with index-matching liquid to eliminate internal reflections. External surfaces of windows are
antireflection (AR) coated. In other solutions KDDP crystals and optical windows are AR coated with reflectivity of < 0,25% per optical surface. KDDP crystals with high power Sol-Gel antireflection
coatings are used.
All these solutions are widely used in flashlamp-pumped solid-state lasers as well as DPSS lasers.
Pockels cells using KDDP crystals can be also made in configuration with transverse electrooptic effect. This configuration is based on the r[41] electrooptic coefficient and is realised as
double-crystal design. Coefficient r[41] is 3 times lower than r[63], but the driving voltages can be reduced by selecting the crystal dimensions.
These constructions require using of both crystals with equal lengths of high accuracy. This is necessary to compensate for the natural birefringence. These configurations are also sensitive to
temperature changes. Generally, transverse KDDP Pockels cells are mainly used as low-voltage modulators with small apertures.
Ryszard Wodnicki, PhD Eng.
[ 1 ] L.L. Steinmetz, T.W. Pouliot, B.C. Johnson, „Cylindrical, Ring-Electrode KD*P Electrooptic Modulator”, Applied Optics, Vol. 12, 7, pp. 1468-1471, (1973),
[ 2 ] Walter Koechner · Solid-State Laser Engineering, Springer Verlag .N.Y. , 1976.
[ 3 ] Z. Jankiewicz, E. Pelzner: „Modulator światła z wykorzystaniem wzdłużnego efektu elektrooptycznego Pockels’a” , Biuletyn WAT, vol. XXIX, Nr 1 (329), 1980, str. 21-32.
[ 4 ] R. S. Alnayli, Cs. Kuti, J.S. Bakos, I. Toth, „ Investigation of electric field distribution in Z-cut KDP Q-switch modulator crystals”. Quantum Electronics, 165(4), September 1989. | {"url":"https://solarisoptics.eu/kddp-pockels-cells-operation-and-design/","timestamp":"2024-11-02T11:52:14Z","content_type":"text/html","content_length":"72691","record_id":"<urn:uuid:1d24bea6-eccb-474c-97b1-2256ea15b54a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00727.warc.gz"} |
George Tsintsifas
Some Geometric and Analytic Inequalities.
Let A(1,0), B(-1,0), C(0,1), D(0,-1) be points in the Cartesian plane and P so that OP is no less than 1. Then, it holds:
|AP-BP|+|CP-DP| is no less than 2.
We found an interesting inequality for the simplex, using barycentric coordinates and then some applications in a triangle.
Let ABC be a triangle and M an interior point. We denoted by R[M] the sum of the distances of the point M from the vertices of ABC and by r[M ]the sum of the distances of the point M from the sides.
We found the inequalities between R[M] and r[M] for M=O,G,I,H that is the pericenter the centroid the incenter and the orthocenter respectively. | {"url":"https://gtsintsifas.com/2007/01/","timestamp":"2024-11-05T03:00:43Z","content_type":"text/html","content_length":"36178","record_id":"<urn:uuid:1d6145b1-5a9a-4eb9-889d-82ed3f2f2227>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00339.warc.gz"} |
A simple vortex model applied to an idealized rotor in sheared inflow
Articles | Volume 8, issue 4
© Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License.
A simple vortex model applied to an idealized rotor in sheared inflow
A simple analytical vortex model is presented and used to study an idealized wind turbine rotor in uniform and sheared inflow, respectively. Our model predicts that 1D momentum theory should be
applied locally when modelling a non-uniformly loaded rotor in a sheared inflow. Hence the maximum local power coefficient (expressed with respect to the local, upstream velocity) of an ideal rotor
is not affected by the presence of shear. We study the interaction between the wake vorticity generated by the rotor and the wind shear vorticity and find that their mutual interaction results in no
net generation of axial vorticity: the wake effects and the shear effects exactly cancel each other out. This means that there are no resulting cross-shear-induced velocities and therefore also no
cross-shear deflection of the wake in this model.
Received: 06 Oct 2022 – Discussion started: 17 Oct 2022 – Revised: 16 Feb 2023 – Accepted: 13 Mar 2023 – Published: 06 Apr 2023
The atmospheric shear layer significantly affects the power production and loads of wind turbines, and it is therefore essential to accurately model wind turbines under sheared inflow. However,
recent validation studies by Boorsma et al. (2023) have shown that state-of-the-art models show some significant deficiencies in modelling wind turbine performances under wind shear conditions.
Madsen et al. (2012) simulated a wind turbine operating under strong-shear conditions using different blade element momentum (BEM)-based models and compared the results to the ones obtained with more
advanced tools. The authors concluded that for the simulation of wind turbines under sheared inflow, BEM-based formulations need to be expressed using local relations; that is, the local induction
factor needs to be defined using the local free-stream velocity. Furthermore, they compared the power obtained in sheared and non-sheared inflows for identical velocities at hub height. The authors
found that, in most cases, a lower power production was obtained in sheared inflow compared to cases without shear, in spite of the total available power in the incoming wind being higher for the
shear case. They explained this with the fact that, over most of a revolution period, the turbine blades operate away from their optimal conditions when the inflow is non-uniform.
Shen et al. (2011) and Sezer-Uzol and Uzol (2013) used free-vortex-wake models to simulate a horizontal axis wind turbine in sheared inflow and also found that the power output in that case is lower
than in uniform inflow.
However, the computational fluid dynamics (CFD) simulations conducted by Zahle and Sørensen (2010), which were also included in the work by Madsen et al. (2012), indicated that the power production
was increased when operating in shear with a proportion that can be largely explained by the increase in available power in the upstream flow. By analysing the local power coefficient (expressed with
respect to the local, upstream velocity), they observed higher efficiencies on the lower half of the rotor, which was explained by differences in the local angle of attack and tip speed ratio between
the lower and upper half of the rotor. A similar effect was found in the analytical model by Magnusson (1999) although he did not investigate the effect of shear on the total power of the turbine.
The simulations by Zahle and Sørensen (2010) furthermore showed that the flow field upstream of the rotor had a significant downward velocity component, causing high-velocity air from higher
altitudes to flow through the lower half of the rotor. It was argued that this phenomenon could also play a role in the increased power production at the lower part of the rotor disc. Another effect
observed from their simulations is the asymmetric development of the wake due to the interaction of wake rotation and shear, which in effect led to a different loading on the blade when pointing to
the left than when pointing to the right.
Chamorro and Arndt (2013) derived a simple analytical correction for the maximum efficiency of an ideal wind turbine rotor in non-uniform inflow which accounts for the non-uniform velocity through
the so-called Boussinesq and Coriolis correction factors. Their study revealed that the maximum power of a turbine in a typical neutrally stratified boundary layer may increase by between 1%–2%.
However, a closer inspection of their work reveals that the predicted power increase is in fact exactly equal to the increase in available power in the inflow. Therefore, they essentially show that
the maximum power coefficient based on the available power in the incoming wind is unchanged for rotors in shear.
Micallef et al. (2010) modelled a wind turbine wake in sheared inflow using oblique vortex rings and obtained an analytical solution for the wake deflection. The ring inclination induces a vertical
velocity component, and the model therefore predicts an upward deflection of the wake. This result agrees with predictions obtained using various free-vortex tools (Sezer-Uzol and Uzol, 2013; Grasso
, 2010). Branlard et al. (2015) showed that the non-physical upward deflection of the wake observed in earlier studies using vortex methods is caused by neglecting the vorticity imposed by the inflow
shear. They proposed both a frozen and an unfrozen inflow shear model to avoid the upward deflection. In both approaches the velocity and vorticity is split into a prescribed part due to inflow shear
and a varying part due to, for example, inflow turbulence or wakes. This split of scales is similar to that proposed by Troldborg et al. (2014) in their simplified CFD-based model of the atmospheric
boundary layer. In the frozen approach it is then assumed that the additional vorticity imposed by the varying part does not affect the prescribed field, whereas a full interaction is allowed in the
unfrozen approach. They implemented the two methods in a vortex particle solver and used them together with an actuator disc model to simulate the wake of turbine in sheared and turbulent inflow.
They showed that the frozen approach reduced the non-physical upward wake motion and removed it completely when using the unfrozen approach.
Ramos-García et al. (2018) proposed a prescribed velocity–vorticity atmospheric boundary layer model and implemented it in the vortex solver MIRAS. They used the model together with a lifting line
method to simulate the wake of a wind turbine in different sheared and turbulent inflows. Similarly to the work of Branlard et al. (2015), they showed that by properly accounting for the shear's
contribution to vortex stretching and convection, the upward deflection of the wake was removed. Besides performing thorough investigations of the wake, they also studied the impact of shear and
turbulence on the power and loads. They found that the power output of the turbine was augmented slightly with increasing shear.
The above literature review shows that the influence of wind shear on wind turbine rotors is not fully understood and that there is no clear consensus on, for example, its impact on power production.
In this paper, we derive an analytical, vortex-based model of an idealized wind turbine rotor to study its operation in sheared inflow and thereby explain some of the main mechanisms at play.
2A simple vortex rotor model including shear
The model presented in the following is an extension of work presented by Øye (Madsen et al., 2006; Øye, 1990) and Branlard and Gaunaa (2015). They both modelled an idealized wind turbine with an
infinite number of blades located in a uniform steady inflow. However, whereas Øye assumed the blade circulation to be uniform, Branlard and Gaunaa (2015) allowed it to vary radially in their model.
In the present treatment, the blade circulation may vary in both the radial and the azimuthal direction. The loading at each specific position on the disc is, however, constant in time. For further
simplification, we restrict ourselves to the idealized case where the rotational speed tends to infinity. Therefore, wake rotational effects are absent and the situation is similar to the classical
1D momentum results (Betz, 1920; Lanchester, 1915). Two other simplifying assumptions, which open up for a simple analytical treatment of the problem, is to neglect wake expansion and to assume that
the convection velocity of the wake vorticity is constant and determined using the conditions in the far wake. These assumptions were initially proposed by Øye (1990), who showed that the results for
a uniformly loaded actuator disc in uniform inflow with these assumptions are identical to 1D momentum theory. Finally, we only consider cases where the incoming wind is parallel to the axis of
rotation and where the rotor blades are straight and perpendicular to the rotor axis.
2.1Basic relations
2.1.1Rotor loads
Our simplified rotor model is derived under the potential-flow assumptions (incompressible, irrotational and inviscid flow), which physically corresponds to low Mach numbers and high Reynolds
numbers. We neglect the effects of drag on the rotor performance.^1 The local forces per unit span, at a given radial location of the blade, are therefore obtained using the Kutta–Joukowski relation:
$\begin{array}{}\text{(1)}& {\mathbit{f}}_{\mathrm{b}}=\mathit{\rho }{\mathbit{V}}_{\text{rel}}×{\mathbf{\Gamma }}_{\mathrm{b}},\end{array}$
where Γ[b] is the bound circulation distributed and directed along the blade, V[rel] is the relative velocity of a blade element, and ρ is the air density. The notations are illustrated in Fig. 1.
Please note that in the following we use bold italic letters to represent vectors and non-bold letters to represent scalar values and length of vectors. Polar coordinates $\left(r,\mathit{\theta },z\
right)$ are used, where the unit vector $\stackrel{\mathrm{^}}{\mathbit{z}}$ is normal to the disc and parallel to the main flow; $\stackrel{\mathrm{^}}{\mathbit{r}}$ is along the blade; and the
tangential vector, $\stackrel{\mathrm{^}}{\mathbit{\theta }}$, is along the direction of rotation. The relative velocity is formed by the local free-stream velocity, V[0], which may vary in space;
the induced velocity, W; and the relative rotational speed of the element, −Ωr, where Ω is the rotor rotational velocity and r is the radius of the element. The relative velocity is ${\mathbit{V}}_{\
text{rel}}=\left({V}_{\mathrm{0}}+{W}_{z}\right)\stackrel{\mathrm{^}}{\mathbit{z}}+\left(-\mathrm{\Omega }r+{W}_{\mathit{\theta }}\right)\stackrel{\mathrm{^}}{\mathbit{\theta }}$, and the bound
circulation associated with one blade is ${\mathbf{\Gamma }}_{\mathrm{b}}={\mathrm{\Gamma }}_{\mathrm{b}}\stackrel{\mathrm{^}}{\mathbit{r}}$. The total bound circulation of the rotor at a given
radial location of the disc is Γ=N[b]Γ[b], where N[b] is the number of blades. The force from all blades on the annulus of the rotor at radius r and radial width dr is N[b]f[b](r)dr, and hence the
local forces per unit area on the rotor disc becomes
$\begin{array}{}\text{(2)}& \mathbit{F}=\left[\begin{array}{c}{F}_{r}\\ {F}_{\mathit{\theta }}\\ {F}_{z}\end{array}\right]=\frac{{N}_{\mathrm{b}}{\mathbit{f}}_{\mathrm{b}}\mathrm{d}r}{\mathrm{2}\
mathit{\pi }r\mathrm{d}r}=\frac{\mathit{\rho }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\pi }}\left[\begin{array}{c}\mathrm{0}\\ \left({V}_{\mathrm{0}}+{W}_{z}\right)/\left(\mathrm{\Omega }
r\right)\\ \mathrm{1}-{W}_{\mathit{\theta }}/\left(\mathrm{\Omega }r\right)\end{array}\right],\end{array}$
where F[r], F[θ] and F[z] are the polar components of the local forces per unit area. Introducing the local thrust coefficient, C[t], based on local free-stream velocity and local disc load, we
obtain the following expression from the axial-force component of Eq. (2):
$\begin{array}{}\text{(3)}& {C}_{\mathrm{t}}=\frac{{F}_{z}}{\mathrm{0.5}\mathit{\rho }{V}_{\mathrm{0}}^{\mathrm{2}}}=\frac{\mathrm{\Gamma }\mathrm{\Omega }}{\mathit{\pi }{V}_{\mathrm{0}}^{\mathrm
{2}}}\left(\mathrm{1}-\frac{{W}_{\mathit{\theta }}}{\mathrm{\Omega }r}\right).\end{array}$
Letting the tip speed ratio ($\mathrm{\Omega }R/{V}_{\mathrm{0}}$, where R is the radius of the disc) go to infinity while retaining a finite value for the product ΓΩ shows that Γ tends to zero. This
corresponds to tangentially induced velocities, W[θ], also tending to zero (Branlard and Gaunaa, 2016). Equation (3) therefore reduces to
$\begin{array}{}\text{(4)}& {C}_{\mathrm{t}}=\frac{\mathrm{\Gamma }\mathrm{\Omega }}{\mathit{\pi }{V}_{\mathrm{0}}^{\mathrm{2}}}.\end{array}$
Since the local power production per area on the rotor is p=F[θ]Ωr, we get
$\begin{array}{}\text{(5)}& p=\frac{\mathrm{d}P}{\mathrm{d}A}=\frac{\mathit{\rho }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\pi }}\left({V}_{\mathrm{0}}+{W}_{z}\right).\end{array}$
The local power coefficient is obtained from Eq. (5):
$\begin{array}{}\text{(6)}& {C}_{\mathrm{p}}=\frac{p}{\mathrm{0.5}\mathit{\rho }{V}_{\mathrm{0}}^{\mathrm{3}}}=\frac{\mathrm{\Gamma }\mathrm{\Omega }}{\mathit{\pi }{V}_{\mathrm{0}}^{\mathrm{2}}}\left
In terms of the local axial induction factor $a=-{W}_{z}/{V}_{\mathrm{0}}$, we get by use of Eq. (4)
$\begin{array}{}\text{(7)}& {C}_{\mathrm{p}}={C}_{\mathrm{t}}\left(\mathrm{1}-a\right),\end{array}$
which corresponds exactly to classical 1D momentum theory (Glauert, 1963).
In the case where the bound circulation is allowed to vary not only as a function of the radius, r, but also as a function of the azimuth location, θ, it is noted that the local thrust coefficient,
Eq. (4), and the local power coefficient, Eq. (6) (and/or Eq. 7), remain unchanged. In this more general loading case, it is noted that the physical interpretation of the local quantity Γ(r,θ) is the
amount of bound blade/rotor circulation that passes the rotor disc location (r,θ) during one revolution of the rotor. It is noted that the local Γ(r,θ) is not equal to the azimuthal sum of the bound
vortex strengths at radius r: no azimuthal averaging is required to obtain the local Γ(r,θ). The general local definition of Γ is used in the remainder of this work.
2.1.2Wake vorticity
In order to determine the velocity at the rotor disc and in the wake of the disc, the strength of the trailed and shed vortex sheets downstream of the disc is needed. To do this, we will be using the
following result, which is proven in Appendix A and illustrated in Fig. 2:
For an infinitely long extruded surface of constant-strength tangential vorticity, $\mathit{\gamma }=\mathrm{d}\mathrm{\Gamma }/\mathrm{d}z$, the induction in the direction of the extrusion axis
(z) is γ on the inside and zero on the outside of the surface irrespective of its cross-sectional shape. Due to the symmetric properties of vortices, the axial induction at the ending plane of a
corresponding half-infinite extruded vortex surface is $\mathit{\gamma }/\mathrm{2}$.
According to Helmholtz's first law, all vortex lines must form closed loops, extend to infinity or start/end at solid boundaries. Therefore, any change in bound vorticity in the radial direction
causes vorticity to be trailed into the wake from the rotor, while variations in the bound vorticity in the tangential direction result in vorticity shed into the wake (see Figs. 3 and 4, where a
discrete representation of the circulation is used for a finite tip speed ratio to simplify the discussion). It is shown in Appendix B that if the local bound vorticity Γ on the blade changes from
position i to j on the rotor, the strength of the trailed and/or shed vorticity is given by
$\begin{array}{}\text{(8)}& {\mathit{\gamma }}_{i-j}=\frac{\partial {\mathrm{\Gamma }}_{\mathrm{b}}}{\partial z}=\frac{\mathrm{\Delta }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\pi }\
Here ΔΓ denotes the difference in local bound circulation between locations i and j, and $\stackrel{\mathrm{˙}}{z}$ is the transport velocity of the vorticity sheet in the axial direction. From Eq. (
8), it is clear that to determine the strength of the trailed and shed vortex sheets, the convection velocity $\stackrel{\mathrm{˙}}{z}$ is needed. This convection velocity is the mean of the
velocities on each side (above and below) of the vortex sheet. In order to simplify the present model, we adopt a generalization of what was shown by Øye (1990) to give good results: a vortex sheet
is convected with a constant velocity which is the mean of the velocity on each side of the sheet far downstream of the rotor. Given two regions i and j, the convection velocity is therefore $\
stackrel{\mathrm{˙}}{z}=\left({V}_{{\mathrm{w}}_{i}}+{V}_{{\mathrm{w}}_{j}}\right)/\mathrm{2}$, where ${V}_{{\mathrm{w}}_{i}}$ and ${V}_{{\mathrm{w}}_{j}}$ are the velocities on each side of the
vorticity sheet far downstream of the rotor.
Using this assumption together with Eqs. (4) and (8), the strength of the vorticity sheet released into the wake as a consequence of a local change in loading between two regions i and j on the disc
can be expressed as follows:
$\begin{array}{}\text{(9)}& {\mathit{\gamma }}_{i-j}=\frac{\left({\mathrm{\Gamma }}_{j}-{\mathrm{\Gamma }}_{i}\right)\mathrm{\Omega }}{\mathrm{2}\mathit{\pi }\stackrel{\mathrm{˙}}{z}}=\frac{{C}_{{\
where ${C}_{{\mathrm{t}}_{i}}$ and ${C}_{{\mathrm{t}}_{j}}$ denote the local thrust coefficient of the two regions, respectively, and ${V}_{{\mathrm{0}}_{i}}$ and ${V}_{{\mathrm{0}}_{j}}$ are their
corresponding local free-stream velocities.
3.1Uniformly loaded rotor in uniform inflow
In this section, we consider a uniformly loaded rotor in uniform inflow. In this case, all the bound vorticity of the disc is trailed from the edge of the disc, and thus the vorticity released into
the wake is distributed on a cylinder as sketched in Fig. 5. The strength of the released vorticity sheet, γ[0−1], is determined from Eq. (9) and using the assumptions described in Sect. 2.1.2:
$\begin{array}{}\text{(10)}& {\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}=\frac{{C}_{{\mathrm{t}}_{\mathrm{1}}}{V}_{{\mathrm{0}}_{\mathrm{1}}}^{\mathrm{2}}}{{V}_{{\mathrm{w}}_{\mathrm{1}}}+{V}_{{\
mathrm{w}}_{\mathrm{0}}}}=\frac{{C}_{\mathrm{T}}{V}_{\mathrm{\infty }}^{\mathrm{2}}}{\mathrm{2}{V}_{\mathrm{\infty }}-{\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}},\end{array}$
where ${C}_{{\mathrm{t}}_{\mathrm{1}}}={C}_{\mathrm{T}}$ is the thrust coefficient of the disc, ${V}_{{\mathrm{0}}_{\mathrm{1}}}={V}_{\mathrm{\infty }}$ is the free-stream velocity and the subscript
0−1 indicates that the vorticity is released between the wake (region 1) and the exterior (region 0). Solving for γ[0−1] yields
$\begin{array}{}\text{(11)}& {\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}={V}_{\mathrm{\infty }}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{\mathrm{T}}}\right).\end{array}$
The rotor disc is at the ending plane of a half-infinite vortex cylinder and thus as described in Sect. 2.1.2; the axial velocity here is ${V}_{\mathrm{\infty }}-{\mathit{\gamma }}_{\mathrm{0}-\
mathrm{1}}/\mathrm{2}={V}_{\mathrm{\infty }}\left(\mathrm{1}-a\right)$, where a is the axial induction factor. Therefore ${\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}=\mathrm{2}a{V}_{\mathrm{\infty }}$
, and from Eq. (11), we get
$\begin{array}{}\text{(12)}& {C}_{\mathrm{T}}=\mathrm{4}a\left(\mathrm{1}-a\right),\text{(13)}& {V}_{\mathrm{w}}={V}_{\mathrm{\infty }}-{\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}={V}_{\mathrm{\infty
}}\sqrt{\mathrm{1}-{C}_{\mathrm{T}}}={V}_{\mathrm{\infty }}\left(\mathrm{1}-\mathrm{2}a\right),\end{array}$
where V[w] is the velocity in the far wake.
Equations (12) and (7) show that the present model and classical 1D momentum theory give identical results. This is consistent with the conclusion drawn by Øye (1990) and shows that the crude
approximations made herein are essentially not worse than the assumptions made in 1D momentum theory.
3.2Non-uniformly loaded rotor in uniform flow
In this section, we apply the model to a non-uniformly loaded rotor in uniform flow. For simplicity we consider a case where two different load levels are present: ${C}_{{\mathrm{t}}_{\mathrm{1}}}$
in the lower half of the rotor disc and ${C}_{{\mathrm{t}}_{\mathrm{2}}}$ in the upper half; see Fig. 6. We start by assuming that the axial velocity in the far wake (and therefore also at the rotor
disc) is constant in each of the two regions and then later check whether this assumption leads to inconsistencies.
If the axial velocity in each of the far-wake regions is constant, its value should be as in the uniformly loaded case, since at the outer edge, the local conditions are as in the uniformly loaded
case. Thus, the far-wake velocity in each region would be given by Eq. (13) and the strength of the sheet released between the two regions would be given by Eq. (9); i.e.
$\begin{array}{ll}{\mathit{\gamma }}_{\mathrm{2}-\mathrm{1}}& =\frac{{C}_{{\mathrm{t}}_{\mathrm{1}}}{V}_{\mathrm{\infty }}^{\mathrm{2}}-{C}_{{\mathrm{t}}_{\mathrm{2}}}{V}_{\mathrm{\infty }}^{\mathrm
{2}}}{{V}_{\mathrm{\infty }}\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{1}}}}+{V}_{\mathrm{\infty }}\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{2}}}}}\\ \text{(14)}& & ={V}_{\mathrm{\infty }}\left(\
In order to have a constant velocity in the wake of each zone, this vortex sheet strength should, according to the behaviour of extruded vortex surfaces (Sect. 2.1.2), be equal to the difference in
the “outer” vortex sheet strengths of each zone. From Eq. (11) the difference in sheet strengths between the regions from 1 and 2 to the exterior is
$\begin{array}{}\text{(15)}& {\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}-{\mathit{\gamma }}_{\mathrm{0}-\mathrm{2}}={V}_{\mathrm{\infty }}\left(\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{2}}}}-\sqrt
This is exactly equal to Eq. (14). If the assumption of a constant velocity in each of the two zones were incorrect, this would not have been the case. The arguments used in the derivation above also
hold in the general case with more than two different load regions including cases where one load region is fully enclosed in other load regions. Therefore, the present model predicts that any stream
element behaves locally as predicted by 1D momentum theory. This is in agreement with the result presented by Johnson (2013) and Branlard and Gaunaa (2015) and consistent with the derivation of the
classical axisymmetric blade element theory, where the annular stream tubes are assumed to be independent of each other. However, the present model goes further as it indicates both radial and
tangential independence of the individual stream elements. Thus, while one might expect that more air would flow through a lightly loaded part of a non-uniformly loaded rotor in uniform inflow, the
present model predicts that this is not the case. Otherwise, the axial velocity (and hence power production; see Eq. 6) on the lighter loaded part of the disc would be higher than what is expected
from 1D momentum results.
3.3Non-uniformly loaded rotor in a step shear
We now consider a non-uniformly loaded rotor in a planar step-shear inflow, as illustrated in Fig. 7. A 3D sketch of the case is shown in Fig. 8, which outlines the different vortex strength
contributions. For simplicity, we assume that the rotor has a constant loading of ${C}_{{\mathrm{t}}_{\mathrm{1}}}$ below the step and another constant loading of ${C}_{{\mathrm{t}}_{\mathrm{2}}}$
above the step. A step shear can be obtained by adding an infinite planar vortex sheet and a uniform flow. Note that such a sheet is consistent with Helmholtz's theorem because it starts and ends at
infinity. If γ[s] denotes the shear sheet strength and V[∞] denotes the uniform free-stream speed, we get the following far-field, upstream inflow velocities:
where y denotes the height above the shear vorticity sheet. The main effect of the rotor is to change the axial velocity of the flow and hence also the convection velocity of the shear sheet in the
Due to steady-state conditions, the transport of circulation through any point in the shear plane is equal to that far upstream:
$\begin{array}{}\text{(17)}& {\mathit{\gamma }}_{\mathrm{s}}{V}_{\mathrm{\infty }}={\mathit{\gamma }}_{\text{s,w}}\stackrel{\mathrm{˙}}{z},\end{array}$
where $\stackrel{\mathrm{˙}}{z}=\left({V}_{{\mathrm{w}}_{\mathrm{2}}}+{V}_{{\mathrm{w}}_{\mathrm{1}}}\right)/\mathrm{2}$ is the mean of the far-wake velocities on each side of the shear vorticity
layer and γ[s,w] is the intensity of the shear vorticity sheet in the wake. Thus, for a wind turbine the intensity of the shear vorticity sheet is effectively increased in the wake because the axial
velocity is lower than in the free stream.
For convenience, the intensity of the shear vorticity sheet in the wake is split as follows:
$\begin{array}{}\text{(18)}& {\mathit{\gamma }}_{\text{s,w}}={\mathit{\gamma }}_{\mathrm{s}}+\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{s}},\end{array}$
where γ[s] is responsible for the effective backbone velocity (Eq. 16) and Δγ[s] is the additional induced part due to the changed velocity in the wake.
In order to check whether our model is also consistent with 1D momentum theory in the sheared inflow case, we proceed as in the previous section and assume that the induced velocities in each region
are constant and see whether this leads to any inconsistencies.
If the axial velocity in each of the far-wake regions is constant, it should be determined from Eq. (13), where now the local backbone free-stream velocity has to be used.
$\begin{array}{}\text{(19)}& {V}_{{\mathrm{w}}_{i}}={V}_{{\mathrm{0}}_{i}}\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{i}}}={V}_{{\mathrm{0}}_{i}}\left(\mathrm{1}-\mathrm{2}{a}_{i}\right),\end{array}$
where index i=1 and i=2 refer to the two regions, respectively, and where we have introduced the local induction factor for region i:
$\begin{array}{}\text{(20)}& {a}_{i}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{i}}}\right).\end{array}$
Inserting Eq. (19) into Eq. (9) yields the following expressions for the strength of the vortex sheets released from the rotor:
$\begin{array}{}\text{(21)}& {\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}& =\frac{{C}_{{\mathrm{t}}_{\mathrm{1}}}{V}_{{\mathrm{0}}_{\mathrm{1}}}}{\mathrm{2}-\mathrm{2}{a}_{\mathrm{1}}}={V}_{{\mathrm
{0}}_{\mathrm{1}}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{1}}}}\right),\text{(22)}& {\mathit{\gamma }}_{\mathrm{0}-\mathrm{2}}& =\frac{{C}_{{\mathrm{t}}_{\mathrm{2}}}{V}_{{\
mathrm{0}}_{\mathrm{2}}}}{\mathrm{2}-\mathrm{2}{a}_{\mathrm{2}}}={V}_{{\mathrm{0}}_{\mathrm{2}}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{2}}}}\right),\text{(23)}& {\mathit{\gamma
}}_{\mathrm{2}-\mathrm{1}}& =\frac{{C}_{{\mathrm{t}}_{\mathrm{1}}}{V}_{{\mathrm{0}}_{\mathrm{1}}}^{\mathrm{2}}-{C}_{{\mathrm{t}}_{\mathrm{2}}}{V}_{{\mathrm{0}}_{\mathrm{2}}}^{\mathrm{2}}}{{V}_{{\
Similarly, by inserting Eq. (19) into Eq. (17) and using Eq. (18), we get the following expression for the additional induced shear vorticity:
$\begin{array}{}\text{(24)}& \mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{s}}={\mathit{\gamma }}_{\mathrm{s}}\left(\frac{\mathrm{2}{V}_{\mathrm{\infty }}}{{V}_{{\mathrm{0}}_{\mathrm{2}}}\left(\mathrm
In the far wake, the intensity of the vortex sheet separating the upper and lower part of the wake is
$\begin{array}{}\text{(25)}& {\mathit{\gamma }}_{\mathrm{2}-\mathrm{1},\text{total}}={\mathit{\gamma }}_{\mathrm{2}-\mathrm{1}}+\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{s}}.\end{array}$
Inserting Eqs. (23) and (24) into Eq. (25) and rewriting using Eq. (19) yields
$\begin{array}{}\text{(26)}& {\mathit{\gamma }}_{\mathrm{2}-\mathrm{1},\text{total}}={V}_{{\mathrm{0}}_{\mathrm{2}}}\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{2}}}}-{V}_{{\mathrm{0}}_{\mathrm{1}}}\
sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{1}}}}-{\mathit{\gamma }}_{\mathrm{s}}.\end{array}$
From the basic behaviour of extruded vortex surfaces (Sect. 2.1.2), we know that in order to have a constant induced velocity in the wakes of each of the zones, the total vortex sheet strength
between the lower and upper zones is equal to the difference in the outer vortex sheet strengths of each zone. From Eqs. (21) and (22) we get
$\begin{array}{ll}{\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}-{\mathit{\gamma }}_{\mathrm{0}-\mathrm{2}}& ={V}_{{\mathrm{0}}_{\mathrm{1}}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm
{1}}}}\right)-{V}_{{\mathrm{0}}_{\mathrm{2}}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{2}}}}\right)\\ \text{(27)}& & ={V}_{{\mathrm{0}}_{\mathrm{2}}}\sqrt{\mathrm{1}-{C}_{{\mathrm
{t}}_{\mathrm{2}}}}-{V}_{{\mathrm{0}}_{\mathrm{1}}}\sqrt{\mathrm{1}-{C}_{{\mathrm{t}}_{\mathrm{1}}}}-{\mathit{\gamma }}_{\mathrm{s}}.\end{array}$
As seen, Eqs. (27) and (26) are identical. Therefore, the initial assumption of a uniform far-wake velocity in each region was correct.
The reason why there is no local effect of the shear is that the effect of the changed convection velocity of the wake vorticity ($\stackrel{\mathrm{˙}}{z}$), at the shear intersection, is exactly
counterweighed by the induced shear vorticity (Eq. 24).
From Eqs. (20) and (7) we get the local thrust and power coefficient:
$\begin{array}{}\text{(28)}& {C}_{{\mathrm{t}}_{i}}& =\mathrm{4}{a}_{i}\left(\mathrm{1}-{a}_{i}\right),\text{(29)}& {C}_{{\mathrm{p}}_{i}}& =\mathrm{4}{a}_{i}{\left(\mathrm{1}-{a}_{i}\right)}^{\
which are identical to the classical result from 1D momentum theory. In analogy with the arguments given in the previous section, we conclude that local 1D momentum theory is valid for any load
distribution and any 1D shear, under the assumptions made in this work. This is an important result to keep in mind when developing additions to BEM models to make them able to perform sensibly in
shear, since it shows that such models should be based on local quantities and avoid adopting concepts such as average disc load/induction or average annulus load/induction.
3.4Formation of cross-shear-induced velocities
A basic property of vortex filaments in inviscid flow is that they cannot end in the fluid, and therefore both the wake and shear vorticity sheets are deformed under the influence of velocity
gradients. This is illustrated in Fig. 8, which is a 3D sketch of the same non-uniformly loaded disc in a step shear as was considered in Sect. 3.3. Figure 8a sketches the shape of the individual
wake vortex rings at the position of the rotor and in the wake, respectively. As a consequence of the shear and/or the difference in loading between the two regions of the rotor, the outer part of
the vortex rings moves with another velocity compared to the part at their intersection, and therefore the vortex rings will be increasingly stretched with downstream position.
The stretching of the shear vorticity sheet is illustrated in Fig. 8b. Upstream of the rotor the entire span of the infinitely long shear layer vortex filaments is convected with the same velocity,
and therefore the vortex filaments remain straight here. Downstream of the rotor, they will be deformed at the wake edges because of the difference in velocity inside and outside of the wake.
Due to the stretching of the wake and shear layer vorticity sheets, axial vorticity is formed at the intersection between the shear vorticity layer and the edges of the wake. However, from Fig. 8 it
is evident that axial vorticity is generated in opposing directions: the upper-wake vortex rings are responsible for producing axial vorticity in the flow direction, whereas the opposite is true for
the lower-wake vortex rings and the shear layer vorticity sheets. Nevertheless, if there is a net production of axial vorticity in one direction, this will induce a velocity component perpendicular
to the flow direction. This in turn will cause the shear layer and wake vorticity sheets to be deflected in the direction perpendicular to the flow direction and thus result in different velocities
on the rotor than would have been the case if the axial vorticity production had been neglected. In the following we will therefore use our model to calculate the total axial vorticity produced by
the interaction of the wake and the shear.
3.4.1Axial vorticity from wake–shear interaction
The axial vorticity at a given streamwise position z can be calculated by considering the conservation of vorticity for an infinitesimal cylindrical control volume enclosing the junction between the
wake border and the shear sheet layer from 0 to z as shown in Fig. 9. Since all vortex filaments form closed loops or start and end at infinity, it is clear that the total flux of γ through this
control volume is zero. Therefore, the axial vorticity through the end faces of the control volume is in balance with the vorticity through its sides. Thus, the axial vorticity can be determined by
integrating the vorticity entering and exiting the sides of the control volume from 0 to z.
The contribution from the wake stretching is then
$\begin{array}{}\text{(30)}& {\mathrm{\Gamma }}_{z,\text{wake}}=\underset{\mathrm{0}}{\overset{z}{\int }}\left({\mathit{\gamma }}_{\mathrm{0}-\mathrm{2}}-{\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}+{\
mathit{\gamma }}_{\mathrm{2}-\mathrm{1}}\right)\mathrm{d}z,\end{array}$
where γ[0−1], γ[0−2] and γ[2−1] are the strength of the vorticity sheets released from the rotor and are determined from Eqs. (21)–(23).
The corresponding contribution from the deformation of the shear layer vorticity sheet is
$\begin{array}{}\text{(31)}& {\mathrm{\Gamma }}_{z,\text{shear}}\left(z\right)=\underset{\mathrm{0}}{\overset{z}{\int }}\left({\mathit{\gamma }}_{\text{s,w}}-{\mathit{\gamma }}_{\mathrm{s}}\right)\
mathrm{d}z=\underset{\mathrm{0}}{\overset{z}{\int }}\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{s}}\mathrm{d}z,\end{array}$
where Δγ[s] is defined in Eq. (24).
From Eqs. (26) and (27) we know that
$\begin{array}{}\text{(32)}& {\mathit{\gamma }}_{\mathrm{2}-\mathrm{1}}+\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{s}}={\mathit{\gamma }}_{\mathrm{0}-\mathrm{1}}-{\mathit{\gamma }}_{\mathrm{0}-\
From Eq. (32) it is clear that Eqs. (30) and (31) exactly balance each other out, and therefore there is no net production of axial vorticity, which means that the present model predicts that there
is no vertical movement of the wake and hence also no change in the power coefficient for a non-uniformly loaded rotor in sheared inflow. Note that the result of no net production of axial vorticity
is general, which means that it also holds in the case without shear.
In the following we will relate the results of our model to the findings of the aforementioned literature.
The conclusion that the power coefficient (defined in terms of the actual available power) of an idealized rotor is unaltered by the presence of a 1D shear is in agreement with the CFD simulation by
Zahle and Sørensen (2010) as well as the work by Chamorro and Arndt (2013). However, in contrast to the latter study, we do not assume self-similarity between the velocity profiles far upstream and
downstream in order to arrive at this result. Furthermore, our conclusion is based on a generic study, whereas the simulations by Zahle and Sørensen (2010) were carried out on a specific rotor, and
hence it is unknown to what extent their findings depend on the rotor and operational conditions used.
Since our model does not predict any formation of axial vorticity, it also does not predict a vertical deflection of the wake, which is in agreement with the studies by Branlard et al. (2015) and
Ramos-García et al. (2018). In the CFD simulation by Zahle and Sørensen (2010), this effect is implicitly included, and the downward velocity component that they observe upstream of the rotor is an
indication that axial vorticity is in fact generated in the wake. This suggests that an ideal rotor might actually get a higher power coefficient in sheared inflow. The production of axial vorticity
could be included in our model by assuming the transport velocity of the wake vorticity changes gradually from its value at the disc to that in the far wake. The absence of vertical motion of the
wake as predicted by our method is in agreement with what would be found from a momentum analysis of the same situation: there are no vertical forces acting from the rotor on the flow^2, and
therefore a control volume analysis of momentum conservation would show no vertical displacement of the wake.
Our model is based on several simplifying assumptions as outlined in Sect. 2, and the above findings should of course be seen in this light. Most of the assumptions made about the rotor are fairly
standard for engineering analyses, and we do not expect that a more advanced rotor representation would change the overall findings of our model. Nevertheless, a consequence of assuming an infinite
tip speed ratio is that rotational effects are neglected, and hence any impact of asymmetric development of the wake in sheared inflow is disregarded by our model. The effect of a finite tip speed
for axisymmetrically loaded rotors in uniform inflow was assessed in Branlard and Gaunaa (2016) using a model based on the same building blocks as the present work. This work showed that the effect
of the finite tip speed ratio is a second-order effect for tip speed ratios typically used in modern wind turbines. Therefore, we expect that the impact of neglecting rotational effects is small for
moderately sheared inflow and typical tip speed ratios of wind turbines.
On the other hand, out of the assumptions made about the flow, e.g. incompressible potential flow, no wake expansion and constant transport velocity of wake vorticity, the latter two are obviously
questionable. In reality the wake is clearly expanding and the transport velocity of the wake vorticity will be faster in the near wake than in the far wake. Thus, on their own these two assumptions
are bad and lead to poor predictions. However, when used together the two assumptions lead to predictions that are consistent with 1D momentum theory. The reason for this is that while neglecting
wake expansion leads to an underprediction of the induction in the rotor plane (Øye, 1990), the opposite is true when assuming a constant transport velocity of wake vorticity, and thereby the two
assumptions counteract each other. The agreement with 1D momentum theory indicates that our model is of the same order of fidelity as those typically used in BEM models. However, our model gives
another perspective and is developed for non-uniform inflow, and thereby it can be used to gain insight and guide future development of BEM models to correctly cope with sheared inflow.
We presented a simple analytical model based on inviscid vortex theory and applied it to model a rotor in uniform and non-uniform inflow.
Even though the model is based on a number of crude assumptions such as neglecting wake expansion and using a wake convection that is constant along each emission location, we showed that the model
gives results that are identical to classical 1D momentum theory when applied to a uniformly loaded rotor operating in uniform inflow.
The application of the model to a non-uniformly loaded rotor operating in uniform inflow showed that any stream element through the rotor disc behaves locally as predicted by 1D momentum theory.
Consequently, the model predicts that the stream elements through the disc are both radially and tangentially independent of each other.
For a non-uniformly loaded rotor operating in non-uniform inflow, our model predicts that the results from 1D momentum theory can be applied locally. Therefore, when the power coefficient is defined
using the local free-stream velocity, we found that the power coefficient of an ideal rotor is unaltered by the presence of shear.
These findings indicate that the effects of shear and uneven loading in BEM-based codes should be treated in a local sense and that concepts such as average disc/annulus loading and induction should
be avoided.
Finally, by studying the inherent deformation of the wake vorticity sheet and the wind shear vorticity sheet, we concluded that there is no net generation of axial vorticity. The effects of the
deformation of the rotor wake vorticity due to the shear and effects of the deformation of the shear vorticity due to the rotor exactly cancel each other out. This means that there are no resulting
cross-shear-induced velocities and therefore also no cross-shear deflection of the wake.
The result that the production of axial vorticity due to the deformation of the wake and shear vorticity cancelling each other out indicates that this effect has to be included in free-wake-vortex
models. Omission of the effect of the deformation of the vorticity that creates the shear will result in a non-physical upward motion of the wind turbine wake in free-vortex models.
Appendix A:Induction of an infinitely long extruded surface of tangential vorticity
Consider an infinitely long extruded surface of tangential vorticity γ, which is aligned with the z axis and has an arbitrary cross-section; see Fig. A1. In the following we will show that this
extruded vortex surface induces a velocity in the z direction of W[z]=γ on the inside and zero on the outside of the surface.
The proof is carried out in steps by proving the following five statements:
1. The vortex surface only induces velocity in the z direction; i.e. $W={W}_{z}\left(r,\mathit{\theta },z\right)$.
2. The velocity is constant along z; i.e. ${W}_{z}={W}_{z}\left(r,\mathit{\theta }\right)$.
3. The velocity inside and outside of the surface is constant and equal to ${W}_{z}={W}_{z,\text{in}}$ and ${W}_{z}={W}_{z,\text{out}}$, respectively.
4. The outside velocity is ${W}_{z,\text{out}}=\mathrm{0}$.
5. The velocity inside of the surface is ${W}_{z,\text{in}}=-\mathit{\gamma }$.
To prove the first statement, consider Fig. A2, which shows a lateral cut of the surface. By virtue of the Biot–Savart law, the velocity induced in a point P by a segment of the vortex surface is
$\begin{array}{}\text{(A1)}& \mathrm{d}{\mathbit{W}}_{i}=\frac{\mathit{\gamma }\mathrm{d}z}{\mathrm{4}\mathit{\pi }}\frac{\mathrm{d}{\mathbit{s}}_{\mathit{\theta }}×{\mathbit{p}}_{i}}{|{\mathbit{p}}_
where $i=\mathit{\left\{}\mathrm{1},\mathrm{2}\mathit{\right\}}$, p[i] is the vector from the segment to P and ds[θ] is an infinitesimal vector in the azimuth direction.
Now consider two segments on the vortex surface, which as shown in Fig. A2 are located at the same azimuth position, θ, but are located Δz and −Δz, respectively, from P. Due to symmetry, the radial
velocities induced by these two segments cancel each other out. This holds true for any choice of azimuth position and ±Δz, and therefore it follows that the total induction of the vortex surface is
in the z direction only. This can also be obtained by noting that each z plane is a plane of symmetry for the vorticity distribution so that the velocity has to be orthogonal to these planes.
The second statement follows from the invariance of the problem in the z direction, which implies that there is no dependence with respect to the variable z and the derivatives in the z direction are
zero. This could have also been used to directly prove the first statement in the plane z=0.
The third statement is proven by introducing a rectangular control surface as shown in Fig. A3. The length of the rectangle is denoted Δz, and its radial extent is from r[1] to r[2]. Since the
induction is only in the z direction, the circulation around the rectangle is
$\begin{array}{}\text{(A2)}& \mathrm{\Gamma }=\left({W}_{z}\left({r}_{\mathrm{2}},\mathit{\theta }\right)-{W}_{z}\left({r}_{\mathrm{1}},\mathit{\theta }\right)\right)\mathrm{\Delta }z.\end{array}$
If the rectangle does not cross the vortex surface, then the circulation around the rectangle is zero. Thus it follows that ${W}_{z}\left({r}_{\mathrm{2}},\mathit{\theta }\right)={W}_{z}\left({r}_{\
mathrm{1}},\mathit{\theta }\right)={W}_{z,\text{out}}\left(\mathit{\theta }\right)$ for ${r}_{\mathrm{2}}>{r}_{\mathrm{1}}>{r}_{\mathrm{c}}$, whereas ${W}_{z}\left({r}_{\mathrm{2}},\mathit{\theta }\
right)={W}_{z}\left({r}_{\mathrm{1}},\mathit{\theta }\right)={W}_{z,\text{in}}\left(\mathit{\theta }\right)$ if ${r}_{\mathrm{s}}>{r}_{\mathrm{2}}>{r}_{\mathrm{1}}$.
The fourth statement follows by letting r[2] go towards infinity where the induction from the vortex surface is 0; i.e. ${W}_{z,\text{out}}=\mathrm{0}$.
The fifth statement is proven by determining the circulation around the rectangular control surface when it is crossing the vortex surface and utilizing that ${W}_{z,\text{out}}=\mathrm{0}$; i.e.
$\begin{array}{}\text{(A3)}& \mathrm{\Gamma }={W}_{z,\text{in}}\mathrm{\Delta }z⇔{W}_{z,\text{in}}=-\mathit{\gamma }.\end{array}$
The induced velocity is thus independent of θ.
Appendix B:Detailed derivation of Eq. (8)
In order to derive Eq. (8), we use the assumption of a constant convection velocity of the vorticity surface in the longitudinal direction, denoted by $\stackrel{\mathrm{˙}}{z}$.
First we consider the case where the jump in the local bound circulation occurs in the radial direction, as shown schematically in Fig. 3. The time it takes for the rotor to perform one revolution is
$T=\mathrm{2}\mathit{\pi }/\mathrm{\Omega }$. In this time span the amount of bound circulation passing zones i and j is the local Γ[i] and Γ[j], respectively. The corresponding amount of circulation
trailed between i and j in this time span is thus $\mathrm{\Delta }\mathrm{\Gamma }={\mathrm{\Gamma }}_{j}-{\mathrm{\Gamma }}_{i}$. During one rotor revolution, the trailed circulation is convected $
\mathrm{\Delta }z=\stackrel{\mathrm{˙}}{z}T=\mathrm{2}\mathit{\pi }\stackrel{\mathrm{˙}}{z}/\mathrm{\Omega }$ in the axial direction. Since the trailed vorticity is emitted at a constant rate in the
intersection between i and j and the orientation of the trailed vorticity for tip speed ratios tending to infinity is practically parallel to the rotor plane, we get that the sheet strength of the
azimuthally directed trailed vorticity between sections i and j is
$\begin{array}{}\text{(B1)}& {\mathit{\gamma }}_{\text{trailed}}=\frac{\mathrm{\Delta }\mathrm{\Gamma }}{\mathrm{\Delta }z}=\frac{\mathrm{\Delta }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\
pi }\stackrel{\mathrm{˙}}{z}}.\end{array}$
Now we proceed to consider the case where the jump in bound circulation occur in the tangential direction, as shown in Fig. 4. The arguments in this case are analogue to the previous case. In this
case $\mathrm{\Delta }\mathrm{\Gamma }={\mathrm{\Gamma }}_{j}-{\mathrm{\Gamma }}_{i}$ is the shed circulation at the i–j border intersection during the time span of one rotor revolution (any
circulation not carried onto zone j must be trailed due to Helmholtz's law). Since the time span for rotor revolution is the same as before, we arrive at an analogous result:
$\begin{array}{}\text{(B2)}& {\mathit{\gamma }}_{\text{shed}}=\frac{\mathrm{\Delta }\mathrm{\Gamma }}{\mathrm{\Delta }z}=\frac{\mathrm{\Delta }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\pi
From the derivations it is evident that the relations for the sheet vorticity strength in both the radial and the tangential bound circulation change cases are identical. It can therefore be
generally stated that the vorticity emitted at the border between two zones of different local bound circulation (or local thrust coefficients) is
$\begin{array}{}\text{(B3)}& \mathit{\gamma }=\frac{\mathrm{\Delta }\mathrm{\Gamma }\mathrm{\Omega }}{\mathrm{2}\mathit{\pi }\stackrel{\mathrm{˙}}{z}}.\end{array}$
No data sets were used in this article.
MG derived the model. NT wrote most of the manuscript and supported the analysis. EM contributed to the idea and derived the proof in Appendix A. All three reviewed and edited the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is funded by DTU Wind and Energy Systems.
This paper was edited by Roland Schmehl and reviewed by Joseph Saverin and Eduardo Alvarez.
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Crete, Greece, 337–347, https://repository.tudelft.nl/islandora/object/uuid:7b821f51-8fd7-43ee-aedf-1bbd3a3e6b542010 (last access: 3 April 2023), 2010.a
Øye, S.: A simple vortex model, in: Proc. of the third IEA Symposium on the Aerodynamics of Wind Turbines, ETSU, 16–17 November 1989, Harwell, UK, 70–73, NTIS Issue number 199106, 1990.a, b, c, d, e
Ramos-García, N., Spietz, H. J., Sørensen, J. N., and Walther, J. H.: Vortex simulations of wind turbines operating in atmospheric conditions using a prescribed velocity-vorticity boundary layer
model, Wind Energy, 21, 1216–1231, https://doi.org/10.1002/we.2225, 2018.a, b
Sezer-Uzol, N. and Uzol, O.: Effect of steady and transient wind shear on the wake structure and performance of a horizontal axis wind turbine rotor, Wind Energy, 16, 1–17, https://doi.org/10.1002/
we.514, 2013.a, b
Shen, X., Zhu, X., and Du, Z.: Wind turbine aerodynamics and loads control in wind shear flow, Energy, 36, 1424–1434, https://doi.org/10.1016/j.energy.2011.01.028, 2011.a
Troldborg, N., Sørensen, J., Mikkelsen, R., and Sørensen, N.: A simple atmospheric boundary layer model applied to large eddy simulations of wind turbine wakes, Wind Energy, 17, 657–669, https://
doi.org/10.1002/we.1608, 2014.a
Zahle, F. and Sørensen, N.: Navier-Stokes Rotor Flow Simulations with Dynamic Inflow, Torque Conference, 28–30 June 2010, Crete, Greece, 2010.a, b, c, d, e
The effect of profile drag can be added as a post-processing step; see for instance Branlard (2017).
Equation (2) show that for a finite thrust force, the tangential forces from the rotor on the flow tend to zero as the rotational speed tends to infinity. | {"url":"https://wes.copernicus.org/articles/8/503/2023/","timestamp":"2024-11-14T18:22:10Z","content_type":"text/html","content_length":"349102","record_id":"<urn:uuid:e9cab1b3-9740-4973-a665-70b3474e8339>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00095.warc.gz"} |
A Review of Folding Schemes
1. Introduction
A folding scheme, applicable to NP relations, involves a protocol between an untrusted prover and a verifier, each possessing two NP instances of size N. The prover also has purported witnesses for
these instances. This protocol results in a single N-sized NP instance, called the folded instance, and a corresponding witness generated privately by the prover. The essence of the scheme is that
the folded instance is satisfiable only if the original instances are. It’s considered non-trivial if it reduces the verifier’s costs and communication compared to verify witnesses for each original
instance separately. Unlike techniques in protocols like Bulletproofs, which divide and combine inner product instances, a folding scheme specifically folds NP relations.
2. Folding Schemes
Nova: Nova, a significant innovation in zero-knowledge proofs, utilizes a unique folding scheme based on relaxed R1CS statements. Specifically, a relaxed R1CS instance consists of an error vector E,
a scalar u, and public inputs and outputs x. An instance (E, u, x) is satisfied by a witness W, if
where Z = (W, x, u). Its main feature is the ability to merge two instances of the same NP problem into one while keeping the problem type intact. This process is more than just layering one problem
over another; it intricately combines them, ensuring the new, single instance upholds the complexity and attributes of both original problems. As a result, the problem becomes more efficient to solve
or verify, yet retains its original depth and integrity. This aspect is particularly vital in cryptography, where maintaining problem integrity is essential. Nova thus represents a sophisticated
method in problem consolidation, enhancing efficiency without compromising complexity, a notable advancement in zero-knowledge proofs.
SuperNova: SuperNova is an innovative recursive proof system designed for succinctly verifying the correct execution of programs on stateful machines with specific instruction sets, such as EVM or
RISC-V. A key feature of SuperNova is that the cost of proving a single step in a program is directly proportional to the size of the circuit representing the instruction executed in that step. This
approach marks a significant departure from previous methods that rely on universal circuits, where the proving cost for a program step is at least equal to the combined sizes of circuits for each
supported instruction, even if only one instruction is invoked in that step. As a result, SuperNova can accommodate a comprehensive instruction set without increasing the proving costs for individual
Sangria: In a similar vein to Nova, which employs a folding scheme based on relaxed R1CS statements, Sangria utilizes folding based on a variant of the PLONK arithmetization. More precisely, for a
standard PLONK gate
the relaxed PLONK gate equation imports a scalar u and error e:
Besides, Sangria extends the relaxed PLONK arithmetization to accept custom gates of degree 2 and circuits with higher gate arity. Let q_G be a selector vector. The gate equation can be written as
Therefore, the degree 2 custom gates can be written as
CCS: Customizable Constraint System (CCS) is a generalization of R1CS that can simultaneously capture R1CS, Plonkish, and AIR without overheads. A CCS instance consists of public input x. A CCS
witness consists of a vector w. A CCS structure-instance tuple (S, x) is satisfied by a CCS witness w if
where z = (w,1,x), M_j · z denotes matrix-vector multiplication, denotes the Hadamard product between vectors, and is a m-sized vector with entries equal to the additive identity in F.
Representing R1CS with CCS: Let t = 3; q = 2; d = 2, M_0 = A, M_1 = B, M_2 = C, S_1 = {0, 1}, S_2 = {2}, c_0 = 1, c_1 = −1, then the CCS equation becomes
Representing Plonkish with CCS: According to Hadamard production, we have the following equations:
Therefore, the standard plonk gate
can be converted to a CCS expression
Representing AIR with CCS: Let l=2 and n = m · t/2.
• Unless set to a specific value below, any entry in M_0,…, M_{t−1} equals 0.
• There is a CCS row for each of the m constraints in AIR, so, if we index the CCS rows by {0,…,m-1}, it suffices to specify how the i-th row of these matrices is set, for i = 0,…,m-1. For all j∈
{0,1,…,t — 1}, let k_j = i · t=2 + j, and we have such setups: 1) If i = 0 and j < t/2, set M_j[i][j+|w|] = 1; 2) If i = m-1 and j ≥ t/2, set M_j[i][j+|w|+t/2] = 1; 3) Otherwise, we set M_j[i]
[k_j] = 1.
HyperNova: HyperNova represents a highly efficient and recursive ZK-SNARK designed for proving incremental computations, where each computational step is represented using committed CCS (Constraint
Composition Systems). It leverages an early iteration of SuperSpartan, a framework for constructing new folding schemes adaptable to various constraints. The core concept behind HyperNova is the
creation of a polynomial that encapsulates claims about high-degree constraints along with previous claims. When subjected to a sum check, this polynomial yields claims formatted suitably for
folding, eliminating the need for cross-terms.
HyperNova can be implemented through two distinct methodologies:
• Direct Approach: This method builds directly upon the non-interactive multi-folding scheme tailored for CCS. It focuses on straightforwardly applying the principles of multi-folding to the
constraints represented within the CCS framework.
• Alternative Approach: This strategy combines the multi-folding scheme with the use of Nova as a ‘black box’. By integrating Nova’s capabilities, this approach potentially enhances the versatility
and efficiency of HyperNova, especially in contexts where Nova’s specific folding strategies are advantageous.
These methodologies underscore HyperNova’s versatility and its potential to advance the field of cryptographic proofs, particularly in scenarios requiring efficient and scalable verification of
complex computations.
ProtoStar: ProtoStar, instead of conducting a traditional sum-check reduction, simplifies an instance into a randomized sum of itself and then applies a folding process to this randomized sum. The
folding primarily involves merging random coefficients from the new sum with the existing accumulator, and validating these coefficients’ structure, specifically their formation as consecutive powers
of a challenge β. This necessitates committing to the vector of β’s powers, introducing group operations in the folding verifier, which performs scalar multiplications on these commitments.
ProtoGalaxy: ProtoGalaxy, building on ProtoStar concepts, offers a folding scheme where the recursive verifier’s additional work is minimal, involving only logarithmic field operations and a constant
number of hashes. This efficiency is further amplified when folding multiple instances in one step, reducing the verifier’s field operations per instance to a constant. Additionally, ProtoGalaxy
enhances the accumulator claim process by using a new challenge, which avoids the commitment to power vectors and ensures a concise (log N length) coefficient representation for a polynomial of
degree log(N). This approach enables recursive computation in O(n) operations, bypassing the more complex O(Nlog(N)) requirement.
CycleFold: CycleFold enhances folding-scheme-based recursive arguments by efficiently employing a cycle of elliptic curves, reducing the necessity for multiple scalar multiplications in verifiers (2
in Nova, 1 in HyperNova, and 3 in ProtoStar). It utilizes the second curve in the cycle to represent a single scalar multiplication, significantly reducing circuit sizes compared to previous methods.
This is especially advantageous in “half pairing” cycles like BN254/Grumpkin, keeping circuits minimal on the non-pairing-friendly curve. The argument includes instances on both curves, provable
using a zkSNARK over the first curve’s scalar field, optimizing efficiency and circuit complexity.
KiloNova: KiloNova is a non-uniform Proof-carrying Data (PCD) system with zero-knowledge features, inspired by HyperNova. It utilizes a variant of the CCS with linear claims on circuits and inputs to
avoid cross items. A generic folding scheme for handling multiple circuit instances was developed, ensuring zero-knowledge properties through various methods. The system incorporates optimization
techniques like circuit aggregation and decoupling, enhancing the performance of this non-uniform ZK-PCD scheme.
BaseFold: BaseFold generalizes the Fast Reed Solomon Interactive Oracle Proof of Proximity (FRI IOPP) to a wider range of linear codes, termed foldable linear codes. This involves constructing a new
family of these codes, specifically a type of randomly punctured Reed-Muller code, and establishing tight bounds on their minimum distance. Additionally, BaseFold introduces a multilinear Polynomial
Constraint System (PCS) from any foldable linear code, combining BaseFold with the sum-check protocol for multilinear polynomial evaluation. This includes a new multilinear PCS derived from FRI.
Practically, BaseFold with the new codes offers a balanced tradeoff between prover time, proof size, and verifier time, compared to previous methods.
3. Applications
Incrementally-Verifiable Computation (IVC) utilizes non-interactive folding schemes for verifying sequential computations expressed as y=F^l(x), where F is the computation, l is the number of steps,
x is the public input, and y is the output. In IVC, each step’s proof is generated by the prover, building upon the previous step’s verified proof. This involves proving an expanded circuit that
combines F’s circuit with a verifier circuit. The process is recursive, with the final proof validating the entire computation. Importantly, the verifier’s workload and proof size are independent of
the number of steps, focusing only on the final step’s proof.
4. Conclusion
The overarching aim of the zero-knowledge industry we try to achieve: significantly reducing proof generation costs, maintaining small proof sizes, and efficient verification. This has been pursued
through prover optimization and data compression techniques. In recent years, significant advancements in folding scheme techniques have been made, revolutionizing incrementally verifiable
computation (IVC) in Zero-Knowledge Proofs. This paper provides a comprehensive overview of the prevailing folding scheme in the zero-knowledge proof, outlining its principles and advantages. It also
discusses the application of folding schemes in IVC, highlighting their role in enhancing the efficiency and accuracy of computational processes within Zero-Knowledge Proofs. However, these schemes
are not the final solution, and their integration into existing ZK protocols remains an ongoing endeavor. | {"url":"https://eigenlab.medium.com/a-review-of-folding-schemes-a285a790fe2f?source=user_profile---------3----------------------------","timestamp":"2024-11-05T00:52:52Z","content_type":"text/html","content_length":"152740","record_id":"<urn:uuid:7ce47a75-2722-409a-bed1-279314fb6b48>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00146.warc.gz"} |
How to Prepare for the PSAT Math Test?
The Preliminary SAT/ National Merit Scholarship Qualifying Test (PSAT/NMSQT) is a standardized test used for college admissions in the United States.
10^th and 11^th graders take the PSAT to practice for the SAT and to secure a National Merit distinction or scholarship.
The PSAT is similar to the SAT in both format and content.
There are three sections on the PSAT:
The PSAT Math section is divided into two subsections:
A No Calculator Section contains 17 questions and students cannot use a calculator. Students have 25 minutes to complete this section.
A Calculator Section contains 31 questions. Students have 45 minutes to complete this section.
40 questions are multiple-choice questions and 8 questions are grid-ins.
PSAT Math covers the following topics:
• Coordinate Geometry
• Plane Geometry
• Data Analysis and Basic Statistics
The Absolute Best Book to Ace the PSAT Math Test
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Why should you prepare yourself for the PSAT test?
To be honest, the PSAT test is not as important as the SAT.
The PSAT test is not used for college admissions. But if so, why should you bother yourself to prepare for such an exam?
Participating in the PSAT test prepares you well for the SAT.
PSAT can be considered the gateway to the SAT. That is why it is called Preliminary SAT. These two tests are very similar to each other.
After taking the PSAT test, you can get to know your strengths and weaknesses better. As a result, you can promise yourself that you will pass a better SAT.
PSAT will still help you if you are planning to take the ACT instead of the SAT. Because ACT and SAT have a lot in common, participating in PSAT will make you more familiar with the type of questions
you need to know. Your PSAT score can somehow predict your SAT score.
The maximum score on the PSAT is different from the maximum score on the SAT.
This score on the PSAT test is 1,520 and on SAT is 1,600.
However, if someone scores 1,300 on the PSAT test, it indicates the same ability as 1,300 on the SAT.
If gaining National Merit is important to you, you should study for PSAT as you study for the SAT.
According to the National Merit Scholarship Program, all juniors who take the PSAT test will be on a waiting list for a $2,500 annual scholarship.
Of course, this scholarship is intended for those who achieve a PSAT cutoff score or higher in their state. So, if it is important for you to get this scholarship, you must take the PSAT test.
Section Overview Number of questions Testing time
Reading World Literature, Social Studies/History, Science 47 60 min
Writing Expression of Ideas, Standard English Conventions 44 35 min
Math No-Calculator Section Pre-Algebra, Algebra, Coordinate Geometry, Plane Geometry, Data analysis, and basic Statistics, Trigonometry 17 25 min
Math Calculator Section Pre-Algebra, Algebra, Coordinate Geometry, Plane Geometry, Data analysis, and basic Statistics, Trigonometry 31 45min
What are the differences between PSAT and SAT tests?
The difference between the two tests is that PSAT has fewer questions and does not have an article section. The PSAT test is a little easier.
How to study for the PSAT Math test?
Like any math test, the PSAT math test can be difficult and stressful. But with daily practice, you can overcome this anxiety.
There are ways to make even math interesting. You need to find these ways and get a better understanding of math with them. This will help you get your desired score on the test.
Here is a step-by-step guide to help you better understand the PSAT math test.
1. Choose your study programs
There are many prestigious PSAT books and study guides that can help your students prepare for the test.
Most major test preparation companies have some offerings for the PSAT, and the short-listing of the best book ends up being a puzzling phenomenon.
There are also many online PSAT courses.
If your students just started preparing for the PSAT test and you need a perfect PSAT prep book, then PSAT Math for Beginners: The Ultimate Step by Step Guide to Preparing for the PSAT Math Test is a
perfect and comprehensive prep book for students to master all PSAT topics being tested right from scratch.
It will help students brush up their math skills, boost their confidence, and do their best to succeed on the PSAT Test.
This one is an alternative book:
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You can also use this great prep book:
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If you just need a PSAT workbook to review the math topics on the test and measure your student’s exam readiness, then try: “PSAT Math Practice WorkbookThe Most Comprehensive Review for the Math
Section of the PSAT Test”
This is another great workbook for students to review all mathematics concepts being covered on the PSAT/NMSQT:
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If you think your students are good at math and just need some PSAT practice tests, then this book is the perfect PSAT test book for you:
Original price was: $20.99.Current price is: $15.99.
You can also use our FREE PSAT worksheets: PSAT Math Worksheets
Have a look at our FREE PSAT Worksheets to assess your students’ knowledge of Mathematics, find their weak areas, and learn from their mistakes.
PSAT Math FREE Resources:
2. Change your mind about math
One of the most important factors for success in any math test is having a positive outlook on math. You cannot pass a math test as long as you think negatively about it.
Try to spend some time each day learning math. You will see how effective this is in your progress and it will change the way you think about math.
Do not try to get rid of math as soon as possible. In this case, you will not learn anything. Look at it as a daily routine or a little warm-up for your mind.
3. Search and learn about the concepts of the PSAT Math test
The next step in preparing for the PSAT math test is to become more familiar with the math concepts in the test.
First, you need to do a little research on the general mathematical concepts required. Learn the basics of math required on the PSAT test.
Once you have mastered the basics, you can move on to more advanced concepts.
Do not study more advanced concepts without learning the basic concepts needed for the test. This will confuse you and make it difficult for you to understand advanced math problems.
4. Practice daily and regularly
As mentioned, daily and regular practice reduces stress, increases math skills, and increases interest in math.
Make a math schedule for each day. Do not miss a single day. Even if it is, spend a short time of the day on math and focus on it.
Do not forget that even geniuses spend the main time of the day practicing things that should not be forgotten.
5. Find your favorite way to learn math
It’s up to you which way to learn math is the best. You may prefer to stay home and prepare for the test through your PSAT prep books.
You can also take in-person or online PSAT prep classes and increase your learning speed through tutors. In such classes, you will be told tips about the PSAT test questions and their solutions.
Best PSAT Math Prep Resource
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6. Learn how to use the calculator and formula sheets correctly
There is no need to memorize formulas on the PSAT test. They usually give you the necessary formulas at the beginning of each section of the math test. It depends on you which formula is right for
which side.
But if you memorize the necessary formulas, your speed in answering the questions will increase.
Prepare yourself for the PSAT Math test so that you do not need a formula sheet to remember easy formulas.
Also, be careful about using the calculator. Although the calculator can save you time, you need to know when and how to use it.
Do not use a calculator for easy calculations. It is better to use it more to check the questions at the end of the test.
Remember from the two sections of the PSAT Math test, you are not allowed to use the calculator in one part of the test.
Check out Top Calculators for the PSAT to find the right calculator for your test.
7. Test your information
When you feel ready enough, you can use simulated PSAT Math tests for this purpose.
You can use online or written tests.
During these tests, fully simulate the conditions of the PSAT Math test.
Follow the rules of the PSAT Math test exactly.
Do not neglect time, it’s an important factor. Many PSAT and SAT test-takers fail due to a lack of time management. Do not be one of those test-takers.
Therefore, by following all the rules, especially time management, act differently than a careless test-taker.
Do not ignore your weaknesses. Correct them.
Repeat these simulated tests many times to achieve sufficient mastery.
8. How can you register?
You must register for your PSAT test through your high school.
Although the SAT is similar to the PSAT, SAT registration is done online through the SAT website.
The school usually informs you of the PSAT test time and deadline.
The high school also explains instructions on how to register and pay.
The PSAT test is offered three times each year in the autumn. This test is offered on these dates: a primary date, a Saturday date, and an alternate date.
Schools usually choose one of these three dates. Most of them choose the primary date, but if the primary date does not match the school’s schedule, they choose one of the two alternate dates.
Unfortunately, not all high schools want to administer the PSAT test, so you may have to choose another school nearby that offers the test.
You can ask your academic counselor to help you choose the right place for the test. Your school must inform you before the PSAT test date. You can talk to your counselor if you have not heard
anything by early or mid-September.
The school may ask you to sign up and pay the fee in person, or you may do so online.
The price of the test is $17 per student, depending on the school.
Some schools charge this fee and some charge extra to administer the test.
As you know, to sign up, you must provide personal and educational information such as name, address, student ID number, grade, etc.
If you are homeschooled, all you need is a school to administer the PSAT test. You can also ask an academic counselor for help.
9. Tips for taking the exam
Don’t study the day before the PSAT test.
The night before the test, make sure all the necessary equipment is ready for the test. These include a Student ID or ID card, a Registration slip or print-out, two pencils, a Sharpener, an Eraser, a
Wristwatch, a Calculator, a bottle of water, and, if necessary, a little snack to eat in the break.
The night before the exam, it is better to set your alarm clock to wake up early and go to bed early.
On test day, try to be at school about 30 minutes before the test.
Put cell phones and personal items in your home or another safe place. You are not allowed to take these items to the test room.
Keep calm during the exam. Do not be self-critical and self-judgmental. Tell yourself that you are as ready for the test as possible.
Keep in mind that the PSAT Math test consists of two parts and you have a total of 70 minutes for these two parts.
Use your time well, but do not worry about time.
In the second part, you are allowed to use the calculator, use it only when necessary to save time.
If you feel you do not know the solution to a question, do not get stuck on it, mark it with a pencil and skip it. After answering the other questions, think about them again.
Never leave any questions unanswered. In the PSAT Math test, you will not receive a negative score for incorrect answers. Therefore, if you do not know the solution to the question, you must answer
the question by guessing.
10. Check your score report
You may be worried about your PSAT score. One reason may be that the PSAT could probably indicate what your SAT score will be.
Also, if your score is good enough, your chances of receiving a National Merit Scholarship will increase.
You will receive your score in December, six to eight weeks after test day.
You can easily access your score online through your college board account.
If you have not yet created an account at College Board you can do so here.
11. How to interpret your score?
The PSAT test score is between 320 and 1,520.
In this test, mathematics, with a range of 160 to 760, holds half of the composite score section.
Test-takers also receive a more detailed score report of sections from 8 to 38. You can easily convert the score of this section to a scaled score of 760 with a very simple solution.
For the math section, simply multiply your section score by 20.
The higher your score in math, the higher your math percentile, and the better your performance than the rest of the test-takers. This percentile compares your situation with other test-takers.
A score above the 75th percentile on the PSAT can be considered a good score. This score is about 570 to 590 in each section or a total of 1,150-1,160.
A score above the 50th percentile is an average score.
A percentile above 90% can be considered an excellent score.
You may want to know who can get the National Merit Scholarship. To win this scholarship, the test-taker requires a score in the top 1% on the PSAT, which may vary from state to state.
To reach the first semi-final stage, test takers must need to get more than 1,400 points out of 1,520 points.
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PSAT FAQs:
Can you take PSAT outside of school?
No, if your school does not offer a PSAT test, you must choose another nearby school that offers the test.
Do colleges look at the PSAT score?
No. Colleges do not receive scores. The college board does not send the PSAT / NMSQT score to the colleges.
What is the difference between PSAT and SAT?
The purpose of these two tests is different. The purpose of taking the SAT is to enter college. The purpose of taking the PSAT is to prepare for the SAT and compete for a scholarship. The PSAT test
does not affect your admission to the university.
Is PSAT a good indicator of SAT?
The PSAT test is a preliminary SAT. Your score on the PSAT test can show how good your performance on the SAT could be. You should know that the maximum score is on PSAT 1,520 while this score is on
SAT 1,600.
Is 1050 PSAT score good?
In the PSAT test, a composite score higher than 1,060 is a good score, a score higher than 920 is an average score, and an excellent score is a score higher than 1,180.
Do PSAT scores get mailed?
You must have an account on the College Board website where you can go to the Student Score Report section and view your PSAT test score. Your school will also receive a paper score report, which may
be available in January.
Can I use a calculator on the PSAT Math test?
The PSAT math test consists of two sections. In the first section, you are not allowed to use the calculator. In the second section, you are allowed to use the calculator.
How many questions are in the math section of the PSAT?
There are 48 questions in the PSAT math section, which consists of two sections: 17 questions in the non-calculator section and 31 questions in the calculator section.
What kind of math is on PSAT?
This test focuses more on four areas: the heart of algebra, problem-solving and data analysis, passport to advanced math, and additional topics in math.
Is Algebra II on the PSAT?
No, unlike the SAT, which includes Algebra II, the PSAT only covers pre-algebra and basic algebra discussions.
What happens if you don’t take the PSAT junior year?
If you want to qualify for a National Merit Scholarship, you must take the PSAT. Otherwise, you will lose the chance to receive a scholarship. You will also lose the chance to take a pre-SAT practice
Is PSAT easier than SAT?
Yes, this test is easier than PSAT and it prepares you for SAT.
How can I improve my math PSAT score?
The best way to get a good PSAT score is to have enough practice. You can use the sample questions about PSAT and SAT to practice.
How many times can you take the PSAT/NMSQT?
Most students take the test once in Grade 12. They can take the test three times during high school, but only once a year. The scholarship program only looks at the junior year PSAT/NMSQT score.
College Entrance Tests
The Best Books to Ace the PSAT Test
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How to use \dagger ?
General questions (367)
We are in the process of making this properly supported, but in the meantime, give this function a shot (you need to build Cadabra from source for this, the functionality is not yet in any of the
binary packages):
def expand_conjugate(ex):
tst:= (A??)^{\dagger};
for node in ex:
if tst.matches( node ):
for prod in node["\\prod"]:
for factor in prod.factors():
lst.append($ @(factor) $)
for factor in list(reversed(lst)):
rep.top().append_child($ @(factor)^{\dagger} $)
return ex
If you stick the above in a cell and evaluate it, you can then do
ex:= (A B C)^{\dagger} + Q + (D E)^{\dagger};
to produce
$$C^{\dagger} B^{\dagger} A^{\dagger} + Q + E^{\dagger} D^{\dagger}$$
There will be better support for generic conjugation operations in Cadabra soon.
For the other two lines, try simple substitution with a rule of the type
rl:= { (A?^{\dagger})^{\dagger} = A?,
( (A??)^{\dagger} )^{\dagger} = A??,
A?^{\dagger} A? = 1,
A? A^{\dagger} = 1,
(A??)^{\dagger} A?? = 1,
A?? (A??)^{\dagger} = 1 };
You can then do e.g.
ex:= ( U^{\dagger} )^{\dagger} + U^{\dagger} U
+ (A B)^{\dagger} + ( (A B)^{\dagger} )^{\dagger};
substitute(ex, rl);
Hope this helps.
Thanks, Kasper. There seem to be some bugs in your code , Cadabra reports "kernel crash". PS1: I also wish that Cadabra could know following rule in next version:
where H is Hermitian operator( including real number).
PS2: By the way, do you konw how to achieve following operation?
a b c... d e...->d e... a b c...
Following is my attempt:
ex:=a b c... d e...;
substitute(_,$A?? d B??-> d B?? A??$);
it does't work. I want to implement cyclicity of trace
tr(a b c)=tr(c a b)=tr(b c a)
PS3: I don't find any easy ways to compute the traces of gamma matrices in cadabra, are there any plans to do this work? | {"url":"https://cadabra.science/qa/1071/how-to-use-dagger?show=1073","timestamp":"2024-11-13T19:43:38Z","content_type":"text/html","content_length":"27698","record_id":"<urn:uuid:f6f29c9c-c0c5-46fe-bd97-23e13c050799>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00575.warc.gz"} |
Econometric Sense
Anyone that has had a basic statistics course is familiar with the idea that the mean can be biased, or influenced by outliers. Basic mathematical statistics shows that the estimator for the mean can
be derived by the method of least squares i.e. min ∑ (x-c)^2 where c* = mean(x), or if we specify a likelihood function using the Gaussian distribution, the estimator for the mean can be derived
using maximum likelihood. M-estimators generalize least squares and maximum likelihood, having certain properties that minimize sensitivity to outliers. M-estimators address sensitivity to outliers
by replacing (x-c)^2 with another function that gives less weight to extreme values, which otherwise can exert leverage or 'pull' mean estimates in the direction of the tail of a distribution. Wilcox
provides a basic expression for deriving an M-estimator: Choose μ [m] to satisfy ∑ ψ (x[i] -μ [m] )/ τ =0 τ = a measurement of scale In Hoaglin, Mosteller, & Tukey a more detailed expression is
given: M-estimators are a family of estimators, all involving the same scale measure or auxiliary scale parameter, the value of the tuning constant determining the individual member of the family.
Choose T[n] to satisfy ∑ ψ (x[i] -T[n] )/ cS[n] = 0 S[n] = auxillary estimator of scale c = tuning constant Properties of M-Estimators: In Hoaglin, Mosteller, and Tukey, Goodall describes two
important properties of robust M-estimators: Resistance: an estimator is resistant if it is altered to only a limited extent by a small proportion of outliers. The estimator breaks down if the
proportion becomes too large. Robustness of Efficiency: over a range of distributions, the variance (or mean squared error) is close to minimum for each distribution. This guarantees an estimator is
good when repeated samples are drawn from a distribution that is not known precisely. Goodall give examples of M-estimators including Tukey's biweight, Huber's, or Andrew's Wave. References:
Introduction to Robust Estimation and Hypothesis Testing. 2nd Edition. Rand R. Wilcox. Elsevier. 2005. Understanding Robust and Explorotory Data Analysis. David C. Hoaglin. Frederick Mosteller. John
W. Tukey. Wiley. 1983
'Statistical Modeling: The Two Cultures' by L. Breiman (Statistical Science
2001, Vol. 16, No. 3, 199–231) is an interesting paper that is a must read for anyone traditionally trained in statistics, but new to the concept of machine learning. It gives perspective and context
to anyone that may attempt to learn to use data mining software such as SAS Enterprise Miner or who may take a course in machine learning (like Dr. Ng's (Stanford) youtube lectures in machine
learning .) The algorithmic machine learning paradigm is in great contrast to the traditional probabilistic approaches of 'data modeling' in which I had been groomed both as an undergraduate and in
graduate school.
From the article, two cultures are defined:
"There are two cultures in the use of statistical modeling to reach conclusions from data.
Classical Statistics/Stochastc Data Modeling Paradigm:
" assumes that the data are generated by a given stochastic data model. "
Algorithmic or Machine Learning Paradigm:
"uses algorithmic models and treats the data mechanism as unknown."
In a lecture for Eco 5385
Data Mining Techniques for Economists
, Professor Tom Fomby
of Southern Methodist University distinguishes machine learning from classical statistical techniques:
Classical Statistics: Focus is on hypothesis testing of causes and effects and interpretability of models. Model Choice is based on parameter significance and In-sample Goodness-of-fit.
Machine Learning: Focus is on Predictive Accuracy even in the face of lack of interpretability of models. Model Choice is based on Cross Validation of Predictive Accuracy using Partitioned Data Sets.
For some, this distinction may be made more transparent by comparing the methods used under each approach. Professor Fomby makes these distinctions:
Methods Classical Statistics: Regression, Logit/Probit, Duration Models, Principle Components, Discriminant Analysis, Bayes Rules
Artificial Intelligence/Machine Learning/Data Mining: Classification and Regression Trees, Neural Nets, K-Nearest Neighbors, Association Rules, Cluster Analysis
From the standpoint of econometrics, the data modeling culture is described very well in this post by Tim Harford:
"academic econometrics is rarely used for forecasting. Instead, econometricians set themselves the task of figuring out past relationships. Have charter schools improved educational standards? Did
abortion liberalisation reduce crime? What has been the impact of immigration on wages?"
The methodologies referenced in the article (like logistic regression) that are utilized under the data modeling or classical statistics paradigm are a means to fill what Brieman refers to as a black
box. Under this paradigm analysts are attempting to characterize an outcome by estimating parameters and making inferences about them based on some assumed data generating process. It is not to say
that these methods are never used under the machine learning paradigm, but how they are used. The article provides a very balanced 'ping-pong' discussion citing various experts from both cultures,
including some who seem to promote both including the authors of The Elements of Statistical Learning: Data Mining, Inference, and Prediction.
In my first econometrics course, the textbook cautioned against 'data mining,' described as using techniques such as stepwise regression. It insisted on letting theory drive model development, rating
the model on total variance explained, and the significance of individual coefficients. This advice was certainly influenced by the 'data modeling' culture. The text was published in the same year as
the Breiman article.
Of course, as the article mentions, if what you are interested in is theory and the role of particular variables in underlying processes, then traditional inference seems to be appropriate.
When algorithmic models are more appropriate (especially when the goal is prediction) a stoachastic model designed to make inferences about specific model co-efficients may provide "the right answer
to the wrong question" as Emanuel Parzen puts it in his comments on Breiman.
Keeping an Open Mind: 'Multiculturalism' in Data Science
As Breiman states:
"Approaching problems by looking for a data model imposes an apriori straight jacket that restricts the ability of statisticians to deal with a wide range of statistical problems."
As Parzen states "I believe statistics has many cultures." He points out that many practitioners are well aware of the divides that exist between Bayesians and frequentists, algorithmic approaches
aside. Even if we restrict our tool box to stochastic methods, we can often find our hands tied if we are not open minded or understand the social norms that distinguish theory from practice. And
there are plenty of divisive debates, like the use of linear probability models for one.
This article was updated and abridged on December 2, 2018.
Often, after conducting an analysis, I've been asked which variables are the most important? While a simple and practical question, the ways of answering it are not always simple or practical.
Professor David Firth of Oxford assembled a very useful literature review addressing this topic which can be found here.
In a critique of The Bell Curve Goldberger and Manski offer the following:
Artheur S. Goldberger and Charles F. Manski in Journal of Economic Literature Vol. XXXIII (June 1995), pp. 762-776
"We must still confront HM's interpretation of the relative slopes of the two curves in Figure 1 as measuring the "relative importance" of cognitive ability and SES in determining poverty status.
Standardization in the manner of HM-essentially using "beta weights"-has been a common practice in sociology, psychology, and education. Yet it is very rarely encountered in economics. Most
econometrics text- books do not even mention the practice. The reason is that standardization accomplishes nothing except to give quantities in noncom- parable units the superficial appearance of
being in comparable units. This accomplishment is worse than useless-it yields misleading inferences."
"We find no substantively meaningful way to interpret the empirical analysis in Part II of The Bell Curve as showing that IQ is "more important" than SES as a determinant of social behaviors....The
Coleman Report sought to measure the "strength" of the relationship between various school factors and pupil achievement through the percent of variance explained by each factor, an approach similar
to that of HM. Cain and Watts write (p. 231): "this measure of strength is totally inappropriate for the purpose of informing policy choice, and cannot provide relevant information for the policy
Again, this goes back to Leamer's discussion of the current state of econometrics on EconTalk, as well as Brieman's comments of variable importance in Statistical Modeling: The Two Cultures
"My definition of importance is based on prediction. A variable might be considered important if deleting it seriously affects prediction accuracy...Importance does not yet have a satisfactory
theoretical definition...It depends on the dependencies between the output variable and the input variables, and on the dependencies between the input variables. The problem begs for research."
In A Guide to Econometrics Kennedy presents multiple regression in the context of the Ballentine Venn diagram and explains: (click to enlarge)
“.. the blue-plus-red-plus-green area represents total variation in Y explained by X and Z together (X1 & X2 in my adapted diagram above)….the red area is discarded only for the purpose of estimating
the coefficients, not for predicting Y; once the coefficients are estimated, all variation in X and Z is used to predict Y…Thus, the R^2 resulting from multiple regression is given by the ratio of
the blue-plus-red-plus-green area to the entire Y circle. Notice there is no way of allocating portions of total R^2 to X and Z because the red area variation is explained by both, in a way that
cannot be disentangled. Only if X and Z are orthogonal, and the red area disappears, can total R^2 be allocated unequivocally to X and Z separately.”
So on a theoretical basis, total variance explained is out. Probably better answers revolve around the notion that all significant variables are important. However, Ziliak and McCloskey argue against
this approach:
"Significant does not mean important and insignificant does not mean unimportant"
The thing to bear in mind, in absence of theoretical justification, there are questions that need answering. Perhaps a number of practical approaches should be embraced, including some of those
criticized above.
(for an interesting discussion of this topic on google groups see here)
This is a question that puzzled me. After going through several undergraduate courses in statistical inference, and a rigorous theoretical treatment in graduate school, I felt at a loss. Suddenly,
(so I thought) I was dealing with population data, not sample data. Should I eschew the use of confidence intervals and levels of significance? Do calculated differences between groups using
population data represent 'the' difference between groups?
From a post entitled “How does statistical analysis differ when analyzing the entire population rather than a sample? professor Andrew Gelman states:
“So, one way of framing the problem is to think of your "entire population" as a sample from a larger population, potentially including future cases. Another frame is to think of there being an
underlying probability model. If you're trying to understand the factors that predict case outcomes, then the implicit full model includes unobserved factors (related to the notorious "error term")
that contribute to the outcome. If you set up a model including a probability distribution for these unobserved outcomes, standard errors will emerge.”
'Statistical Modeling: The Two Cultures' by L. Breiman (Statistical Science
2001, Vol. 16, No. 3, 199–231) is an interesting paper that is a must read for anyone traditionally trained in statistics, but new to the concept of machine learning. It gives perspective and context
to anyone that may attempt to learn to use data mining software such as SAS Enterprise Miner or who may take a course in machine learning (like Dr. Ng's (Stanford) youtube lectures in machine
learning .) The algorithmic machine learning paradigm is in great contrast to the traditional probabilistic approaches of 'data modeling' in which I had been groomed both as an undergraduate and in
graduate school.
From the article, two cultures are defined:
"There are two cultures in the use of statistical modeling to reach conclusions from data.
Classical Statistics/Stochastc Data Modeling Paradigm:
" assumes that the data are generated by a given stochastic data model. "
Algorithmic or Machine Learning Paradigm:
"uses algorithmic models and treats the data mechanism as unknown."
In a lecture for Eco 5385
Data Mining Techniques for Economists
, Professor Tom Fomby
of Southern Methodist University distinguishes machine learning from classical statistical techniques:
Classical Statistics: Focus is on hypothesis testing of causes and effects and interpretability of models. Model Choice is based on parameter significance and In-sample Goodness-of-fit.
Machine Learning: Focus is on Predictive Accuracy even in the face of lack of interpretability of models. Model Choice is based on Cross Validation of Predictive Accuracy using Partitioned Data Sets.
For some, this distinction may be made more transparent by comparing the methods used under each approach. Professor Fomby does a great job making these distinctions:
Methods Classical Statistics: Regression, Logit/Probit, Duration Models, Principle Components, Discriminant Analysis, Bayes Rules
Artificial Intelligence/Machine Learning/Data Mining: Classification and Regression Trees, Neural Nets, K-Nearest Neighbors, Association Rules, Cluster Analysis
From the standpoint of econometrics, the data modeling culture is described very well in this post by Tim Harford:
"academic econometrics is rarely used for forecasting. Instead, econometricians set themselves the task of figuring out past relationships. Have charter schools improved educational standards? Did
abortion liberalisation reduce crime? What has been the impact of immigration on wages?"
This is certainly consistent with the comparisons presented in the Statistical Science article. Note however, that the methodologies referenced in the article (like logistic regression) that are
utilized under the data modeling or classical statistics paradigm are a means to fill what Brieman refers to as a black box. Under this paradigm analysts are attempting to characterize an outcome by
estimating parameters and making inferences about them based on some assumed data generating process. It is not to say that these methods are never used under the machine learning paradigm, but how
they are used. The article provides a very balanced 'ping-pong' discussion citing various experts from both cultures, including some who seem to promote both including the authors of The Elements of
Statistical Learning: Data Mining, Inference, and Prediction.
In my first econometrics course, the textbook cautioned against 'data mining,' described as using techniques such as stepwise regression. It insisted on letting theory drive model development, rating
the model on total variance explained, and the significance of individual coefficients. This advice was certainly influenced by the 'data modeling' culture. The text was published in the same year as
the Breiman article. ( I understand this caution has been moderated in contemporary editions).
Of course, as the article mentions, if what you are interested in is theory and the role of particular variables in underlying processes, then traditional inference seems to be the appropriate
direction to take. (Breiman of course still takes issue, arguing that we can't trust the significance of an estimated co-efficient if the model overall is a poor predictor).
"Higher predictive accuracy is associated with more reliable information about the underlying data mechanism. Weak predictive accuracy can lead to questionable conclusions."
"Algorithmic models can give better predictive accuracy than data models,and provide better information about the underlying mechanism."
"The goal is not interpretability, but accurate information."
When algorithmic models are more appropriate (especially when the goal is prediction) a stoachastic model designed to make inferences about specific model co-efficients may provide "the right answer
to the wrong question" as Emanuel Parzen puts it in his comments on Breiman.
I even find a hint of this in Greene, a well known econometrics textbook author:
"It remains an interesting question for research whether fitting y well or obtaining good parameter estimates is a preferable estimation criterion. Evidently, they need not be the same thing."
p. 686 Greene, Econometric Analysis 5^th ed
Keeping an Open Mind: Multiculturalism in Data Science
As Breiman states:
"Approaching problems by looking for a data model imposes an apriori straight jacket that restricts the ability of statisticians to deal with a wide range of statistical problems."
A multicultural approach to analysis (stochastic or algorithmic) seems to be the take away message of the Breiman article and discussions that follow. The field data science, and as clearly depicted
in Drew Conway's data science Venn diagram is multi-pillared:
Parzen points out that many practitioners are well aware of the divides that exist between Bayesians and frequentists, algorithmic approaches aside. Even if we restrict our tool box to stochastic
methods, we can often find our hands tied if we are not open minded or understand the social norms that distinguish theory from practice. And there are plenty of divisive debates, like the use of
linear probability models for one.
As Parzen states "I believe statistics has many cultures." We will be a much more effective in our work and learning if we have this understanding, and embrace rather than fight the diversity of
thought across various fields of science and each discipline's differing social norms and practices. Data science at best, is an interdisciplinary field of study and work.
This article was updated and abridged on December 2, 2014.
Earlier I had a problem that required merging 3 years of trade data, with about 12 csv files per year. Merging all of these data sets with pairwise left joins using the R merge statement worked
(especially after correcting some errors pointed out by Hadley Wickham However, in both my hobby hacking and on the job, I was curious if there might be a better way to do this than countless sets of
merge statements (not to mention the multiple lines of code required for reading in the csv files)
So, I sent a tweet to the #rstats followers with a link to where I posted the problem on this blog to see if I could get a hint. (twitter has been a very valuable networking tool for me, I've learned
a lot about data mining, machine learning, and R from the tweet-stream. Tweets and blog posts from people like Hadley Wickham, Drew Conway, and J.D. Long have been tremendously helpful to me as I've
taken up R.
Back to the topic at hand, below is my new code, based on suggestions from Hadley Wickham and a comment (from Harlan) that lead me to some answers to a similar question on stack overflow. The code
below requires he reshape library as well as plyr, which I should mention appears to have been created by Hadley Wickham himself.
# read all Y2000 csv files
filenames <- list.files(path="/Users/wkuuser/Desktop/R Data Sets/TRADE_DATA/TempData00", full.names=TRUE)
import.list <- llply(filenames, read.csv)
# left join all Y2000 csv files
AllData00 <- Reduce(function(x, y) merge(x, y, all=FALSE,by.x="Reporting.Countries.Partner.Countries",by.y="Reporting.Countries.Partner.Countries",all.x =TRUE, all.y =FALSE),import.list,accumulate=F)
dim(AllData00) # n = 211 211
# rename common key variable to something less awkward and change World to World00
AllData00 <- rename(AllData00, c(Reporting.Countries.Partner.Countries="Partner", World = "World00"))
names(AllData00) # list all variable names
# keep only the partner name variable and total world trade
AllData00 <-AllData00[c("Partner","World00")]
dim(AllData00) # data dimensions
names(AllData00) # variable names
fix(AllData00) # view in data editor
Created by Pretty R at inside-R.org
That pretty well gives me the data set I need, for year 2000 data. I repeated the process for 2004 and 2008 data sets I had and then merged them with left joins to get the final data set. All I am
after at this point is the total world trade for each of the countries/groups listed. This could probably be made even more efficient, but is is a lot less coding than what I initially used for the
project. (see below- and this doesn't even include some of the renaming and sub-setting functions I performed above) And, this process would have to be repeated 2 more times for 2004 and 2008 data.
To say that the above code is much more efficient is an understatement. (note the code below actually contains some mistakes as Hadley Wickham pointed out. For instance, in the merge statement, I
have by.a and by.b, or by.'dataset', while in every case it should be by.x and by.y. I guess x and y are alias's for the data sets being merged, sort of like a and b would be in SQL, if you were to
create table newdataset as
select *
from dat1 a left join dat2 b
on a.partner=b.partner
So, I'm not sure why my code even worked to begin with. I do realize that instead of the merge statement in R I could have used the sqldf package in R, but I have had issues with my mac crashing when
I try to load the library. Still, I don't think SQL would have made things any better, as I would still be doing a series of left joins vs. the more compact code using the reduce function in R. I've
used sqldf in a windows environment before and it worked great by the way.
The code below first reads in each data file individually, and then executes the endless number of left joins to give me the same data set I got above with a fraction of the amount of required code.
# a
a <- read.csv("X_A.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
a <- rename(a, c(Reporting.Countries.Partner.Countries="Partner"))
# b
b <- read.csv("X_B.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
b <- rename(b, c(Reporting.Countries.Partner.Countries="Partner"))
# c
c <- read.csv("X_C.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
c <- rename(c, c(Reporting.Countries.Partner.Countries="Partner"))
# de
de <- read.csv("X_DE.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
de <- rename(de, c(Reporting.Countries.Partner.Countries="Partner"))
# fgh
fgh <- read.csv("X_FGH.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
fgh <- rename(fgh, c(Reporting.Countries.Partner.Countries="Partner"))
# ijk
ijk <- read.csv("X_IJK.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
ijk <- rename(ijk, c(Reporting.Countries.Partner.Countries="Partner"))
# lm
lm <- read.csv("X_LM.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
lm <- rename(lm, c(Reporting.Countries.Partner.Countries="Partner"))
# nop
nop <- read.csv("X_NOP.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
nop <- rename(nop, c(Reporting.Countries.Partner.Countries="Partner"))
# qr
qr <- read.csv("X_QR.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
qr <- rename(qr, c(Reporting.Countries.Partner.Countries="Partner"))
# s odd name changed to 'SaloTomaPrincip' manaully in excel
s <- read.csv("X_S.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
s <- rename(s, c(Reporting.Countries.Partner.Countries="Partner"))
# tuv
tuv <- read.csv("EX_TUV.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
tuv <- rename(tuv, c(Reporting.Countries.Partner.Countries="Partner"))
# wxyz
wxyz <- read.csv("X_WXYZ.csv", na.strings=c(".", "NA", "", "?"), encoding="UTF-8")
wxyz <- rename(wxyz, c(Reporting.Countries.Partner.Countries="Partner"))
# ------------------------------------------------------------------
# sequentially left join data sets
# ------------------------------------------------------------------
# a & b
ab <- merge(a,b, by.a = Partner, by.b =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
14 + 18 - 1 # r = 211 c = 31
# abc
abc <- merge(ab,c, by.ab = Partner, by.c =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
31 + 24 -1 # n = 54
# a_e
a_e <- merge(abc,de, by.abc = Partner, by.de =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
54 + 18 -1 # n = 71
# a_h
a_h <- merge(a_e,fgh, by.a_e = Partner, by.fgh =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
71 + 23 -1 # n = 93
# a_k
a_k <- merge(a_h,ijk, by.a_h = Partner, by.ijk =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
93 + 16 -1 # n = 108
# a_m
a_m <- merge(a_k,lm, by.a_k = Partner, by.lm =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
108 + 26 - 1 # n = 133
# a_p
a_p <- merge(a_m,nop, by.a_m = Partner, by.nop =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
133 + 20 - 1 # n = 152
# a_r
a_r <- merge(a_p,qr, by.a_p = Partner, by.qr =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
152 + 6 -1 # n = 157
# a_s
a_s <- merge(a_r,s, by.a_r = Partner, by.s =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
157 + 27 - 1 # n = 183
# a_v
a_v <- merge(a_s,tuv, by.a_s = Partner, by.tuv =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
183 + 21 - 1 # n = 203
# a_z (complete data set after this merge)
a_z <- merge(a_v,wxyz, by.a_v = Partner, by.wxyz =Partner, all = FALSE, all.x = TRUE, all.y = FALSE)
dim(a_z) # n = 211
Created by Pretty R at inside-R.org
(note scroll all the way down to see 'old code' and 'new more flexible code'
Recall and older post that presented overlapping density plots using R (Visualizing Agricultural Subsidies by KY County) see image below.
The code I used to produce this plot makes use of the rbind and data.frame functions (see below)
library(colorspace) # package for rainbow_hcl function
ds <- rbind(data.frame(dat=KyCropsAndSubsidies[,][,"LogAcres"], grp="All"),
data.frame(dat=KyCropsAndSubsidies[,][KyCropsAndSubsidies$subsidy_in_millions > 2.76,"LogAcres"], grp=">median"),
data.frame(dat=KyCropsAndSubsidies[,][KyCropsAndSubsidies$subsidy_in_millions <= 2.76,"LogAcres"], grp="<=median"))
# histogram and density for all ears
hs <- hist(ds[ds$grp=="All",1], main="", xlab="LogAcres", col="grey90", ylim=c(0, 25), breaks="fd", border=TRUE)
dens <- density(ds[ds$grp=="All",1], na.rm=TRUE)
rs <- max(hs$counts)/max(dens$y)
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(3)[1])
# density for above median subsidies
dens <- density(ds[ds$grp==">median",1], na.rm=TRUE)
rs <- max(hs$counts)/max(dens$y)
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(3)[2])
# density for below median subsidies
dens <- density(ds[ds$grp=="<=median",1], na.rm=TRUE)
rs <- max(hs$counts)/max(dens$y)
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(3)[3])
# Add a rug to illustrate density.
rug(ds[ds$grp==">median", 1], col=rainbow_hcl(3)[2])
rug(ds[ds$grp=="<=median", 1], col=rainbow_hcl(3)[3])
# Add a legend to the plot.
legend("topright", c("All", ">median", "<=media"), bty="n", fill=rainbow_hcl(3))
# Add a title to the plot.
title(main="Distribution of Acres Planted by Subsidies Recieved Above or Below Median", sub=paste("Created Using R Statistical Package"))
Created by Pretty R at inside-R.org
I really don't understand the ins and outs of the rbind or data.frame functions, and in another project, when I tried to repeat a similar analysis, it wouldn't work. I could not figure out what my
error was, but I new enough about R to create the plots with an alternative implementation. It is not as compact, but more general, and it worked. (see code below, although it references a new data
set with new vars and produces 4 density curves vs. 3)
# histogram and density estimates for all data
hs <- hist(trade_by_yr$logTrade,main="", xlab="trade", col="grey90", ylim=c(0, 95), breaks="fd", border=TRUE) # histogram
dens <- density(trade_by_yr$logTrade) # density
rs <- max(hs$counts)/max(dens$y) # rescale/mormalize density
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(4)[1]) # plot densiy
# density estimates for year 2000 trade data
y2000 <- trade_by_yr[trade_by_yr$year==2000,] # subset data for year
dens <- density(y2000$logTrade) # density
rs <- max(hs$counts)/max(dens$y) # rescale/mormalize density
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(4)[2]) # plot densiy
# density estimates for year 2004 trade data
y2004 <- trade_by_yr[trade_by_yr$year==2004,] # subset data for year
dens <- density(y2004$logTrade) # density
rs <- max(hs$counts)/max(dens$y) # rescale/mormalize density
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(4)[3]) # plot densiy
# densty estimates for year 2008 trade data
y2008 <- trade_by_yr[trade_by_yr$year==2008,] # subset data for year
dens <- density(y2008$logTrade) # density
rs <- max(hs$counts)/max(dens$y) # rescale/mormalize density
lines(dens$x, dens$y*rs, type="l", col=rainbow_hcl(4)[4]) # plot densiy
# Add a legend to the plot.
legend("topright", c("All", "2000", "2004", "2008"), bty="n", fill=rainbow_hcl(4))
# Add a title to the plot.
title(main="Distribution of Total World Trade Volume by Country by Year", sub=paste("Created Using R Statistical Package"))
Created by Pretty R at inside-R.org
See graph below: | {"url":"https://econometricsense.blogspot.com/2011/01/","timestamp":"2024-11-11T11:23:02Z","content_type":"text/html","content_length":"206690","record_id":"<urn:uuid:22872a39-1389-4cac-a57b-de9007da037d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00067.warc.gz"} |
Browsing Codes
The ZENO software package integrates N-body and SPH simulation codes with a large array of programs to generate initial conditions and analyze numerical simulations. Written in C, the ZENO system is
portable between Mac, Linux, and Unix platforms. It is in active use at the Institute for Astronomy (IfA), at NRAO, and possibly elsewhere.
Zeno programs can perform a wide range of simulation and analysis tasks. While many of these programs were first created for specific projects, they embody algorithms of general applicability and
embrace a modular design strategy, so existing code is easily applied to new tasks. Major elements of the system include structured data file utilities facilitate basic operations on binary data,
including import/export of ZENO data to other systems; snapshot generation routines to create particle distributions with various properties; systems with user-specified density profiles can be
realized in collisionless or gaseous form; multiple spherical and disk components may be set up in mutual equilibrium; and snapshot manipulation routines permit the user to sift, sort, and combine
particle arrays, translate and rotate particle configurations, and assign new values to data fields associated with each particle.
Simulation codes include both pure N-body and combined N-body/SPH programs. Pure N-body codes are available in both uniprocessor and parallel versions. SPH codes offer a wide range of options for gas
physics, including isothermal, adiabatic, and radiating models. Snapshot analysis programs calculate temporal averages, evaluate particle statistics, measure shapes and density profiles, compute
kinematic properties, and identify and track objects in particle distributions. Visualization programs generate interactive displays and produce still images and videos of particle distributions; the
user may specify arbitrary color schemes and viewing transformations. | {"url":"http://www.ascl.net/code/all/page/24/limit/3572/order/title/listmode/full/dir/asc","timestamp":"2024-11-13T00:08:17Z","content_type":"text/html","content_length":"58803","record_id":"<urn:uuid:59487e18-aff2-45f6-b762-164d9b7faca1>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00875.warc.gz"} |
Heat Online Free Calculator | ORCHIDS
Welcome to the Heat Calculator , your go-to tool for simplifying heat-related calculations. Whether you're a student, scientist, or simply interested in understanding heat transfer, our calculator
provides a user-friendly platform for accurate and efficient computations. Heat plays a crucial role in various fields such as physics, engineering, and everyday activities, and our calculator is
designed to make heat calculations accessible to all.
What is a Heat Calculator?
The Heat Calculator is an online tool designed to help users calculate heat transfer in different scenarios. It considers factors such as mass, specific heat, and temperature changes to provide
accurate results. Whether you're dealing with thermal physics problems, engineering applications, or cooking-related calculations, our calculator is here to simplify the process.
Key Features of Heat Calculator:
• User-friendly Interface: Our calculator offers a simple and intuitive design, making it easy for users to input values and obtain accurate heat transfer calculations
• Comprehensive Formula: The calculator uses a comprehensive formula that incorporates mass, specific heat, and temperature changes, ensuring accurate results for a wide range of scenarios.
• Real-time Results: Enjoy instant results as you input values, allowing you to quickly analyze and understand heat transfer in different situations.
What does the Heat Calculator calculate?
The calculator computes heat transfer based on the mass of an object, its specific heat, and the temperature change.
What scenarios can the Heat Calculator be used for?
The calculator is versatile and can be used for thermal physics problems, engineering applications, cooking-related heat calculations, and more.
Why use the Heat Calculator?
Our calculator simplifies heat transfer calculations, providing users with quick and accurate results without the need for complex manual computations.
How does the Heat Calculator work?
The calculator utilizes a formula that considers mass, specific heat, and temperature changes to calculate heat transfer in real-time.
Solved Illustrated Examples:
Mass (m): 2 kg
Specific Heat (c): 500 J/kg°C
Temperature Change (ΔT): 30°C
Δ =2 × 500 × 30=30,000
Q=2 × 500 × 30=30,000J
Mass (m): 5 kg
Specific Heat (c): 800 J/kg°C
Temperature Change (ΔT): 15°C
Q=5 × 800 × 15=60,000 j
Mass (m): 3 kg
Specific Heat (c): 1,200 J/kg°C
Temperature Change (ΔT): 20°C
Q=3 × 1,200 × 20=72,000J | {"url":"https://www.orchidsinternationalschool.com/calculators/physics-calculators/heat-calculator","timestamp":"2024-11-06T03:07:22Z","content_type":"text/html","content_length":"27526","record_id":"<urn:uuid:13e386b7-c8fe-4d76-a511-298bfb2fd99d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00206.warc.gz"} |
Fractional Greedy Algorithms | CodingDrills
Introduction to Greedy Algorithms
Greedy Algorithms for Optimization Problems
Greedy Algorithms for Shortest Path
Greedy Algorithms for Graphs
Applications of Greedy Algorithms
Greedy Algorithms in Real Life
Greedy Algorithms for Combinatorial Problems
Greedy Algorithms in Artificial Intelligence
Case Studies and Examples
Fractional Greedy Algorithms
Applications of Greedy Algorithms: Fractional Greedy Algorithms
In the realm of computer science and algorithm design, the concept of Greedy Algorithms plays a prominent role. These algorithms aim to make locally optimal choices in the hope of achieving a
globally optimal solution. One particular class of Greedy Algorithms that often finds applicability in various scenarios is the Fractional Greedy Algorithms. In this tutorial, we will delve deeper
into Fractional Greedy Algorithms, explore their applications, and provide code snippets to illustrate their usage.
Understanding Greedy Algorithms
Before we dive into Fractional Greedy Algorithms, let's briefly recap the fundamentals of Greedy Algorithms. A Greedy Algorithm is an algorithmic paradigm that follows the "greedy" strategy of making
the locally best choice at each stage in the hope of achieving the globally optimal solution. In other words, at each step, the algorithm chooses the option that appears to be the best at that
moment, without considering the larger picture.
Applications of Greedy Algorithms
Greedy Algorithms find extensive application in numerous real-world problems across various domains. Some of the common applications include:
1. Activity Selection: Given a set of activities, each with a start and finish time, the task is to select the maximum number of activities that can be performed without overlapping.
activities = [(1, 2), (3, 4), (0, 6), (5, 7), (8, 9), (5, 9)]
def select_activities(activities):
activities.sort(key=lambda x: x[1])
selected_activities = [activities[0]]
previous_finish_time = activities[0][1]
for activity in activities[1:]:
start_time, finish_time = activity
if start_time >= previous_finish_time:
previous_finish_time = finish_time
return selected_activities
selected = select_activities(activities)
print(selected) # Output: [(1, 2), (3, 4), (5, 7), (8, 9)]
2. Huffman Coding: Huffman coding is a lossless data compression algorithm that compresses data by assigning variable-length codes to different characters. The most frequent characters are assigned
shorter codes, resulting in efficient encoding and decoding.
3. Interval Scheduling: Given a set of intervals with start and end times, the goal is to find the maximum number of non-overlapping intervals.
intervals = [(3, 9), (1, 4), (7, 10), (2, 5), (8, 11), (12, 14)]
def select_intervals(intervals):
intervals.sort(key=lambda x: x[1])
selected_intervals = [intervals[0]]
previous_end_time = intervals[0][1]
for interval in intervals[1:]:
start_time, end_time = interval
if start_time >= previous_end_time:
previous_end_time = end_time
return selected_intervals
selected = select_intervals(intervals)
print(selected) # Output: [(1, 4), (7, 10), (12, 14)]
Fractional Greedy Algorithms
Fractional Greedy Algorithms, also known as Greedy Algorithms with Fractional Solutions, are a variant of Greedy Algorithms that allow the selection of fractions of available items to form the
optimal solution. This type of algorithm comes into play when the items to be selected cannot be chosen as a whole but can be divided or broken down into fractions to form an optimal solution.
Let's understand this concept with an example:
Consider a scenario where we have a knapsack with a limited capacity and a set of items, each having a weight and a value. The goal is to maximize the total value of the items that can fit into the
knapsack without exceeding its capacity. In this case, Fractional Greedy Algorithms can be used to determine the best fraction of each item to include.
Here's a code snippet showcasing the Fractional Knapsack problem:
def fractional_knapsack(items, capacity):
items.sort(key=lambda x: x[1] / x[0], reverse=True)
total_value = 0
knapsack = []
for item in items:
weight, value = item
if capacity >= weight:
capacity -= weight
total_value += value
fraction = capacity / weight
knapsack.append((weight * fraction, value * fraction))
total_value += value * fraction
return knapsack, total_value
items = [(10, 60), (20, 100), (30, 120)]
capacity = 50
selected_items, total_value = fractional_knapsack(items, capacity)
print("Selected Items:", selected_items)
print("Total Value:", total_value)
The code snippet above demonstrates the Fractional Knapsack problem, where we select items based on their weight-to-value ratio, prioritizing the most valuable items within the knapsack's capacity.
In conclusion, Greedy Algorithms, and more specifically Fractional Greedy Algorithms, provide powerful techniques for solving optimization problems. By making locally optimal choices, these
algorithms offer efficient solutions to a wide range of real-world problems. We explored applications such as activity selection and interval scheduling, along with a code example for the Fractional
Knapsack problem. By implementing and understanding these algorithms, programmers can enhance their problem-solving abilities and devise efficient solutions for various scenarios. | {"url":"https://www.codingdrills.com/tutorial/introduction-to-greedy-algorithms/fractional-greedy","timestamp":"2024-11-08T04:43:15Z","content_type":"text/html","content_length":"315037","record_id":"<urn:uuid:f2a01da8-9287-437a-a616-fd9fe3fb9435>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00477.warc.gz"} |
Note: The SubcurveTraits_2 can comprise of Line_segments, Conic_arcs, Circular_arc, Bezier_curves, or Linear_curves.
A portion or a part of any of the above mentioned geometric traits is called a subcurve.
The traits class Arr_polycurve_traits_2 handles piecewise curves that are not necessarily linear, such as conic arcs, circular arcs, Bezier curves, or line segments. We call such a compound curve a
polycurve. A polycurve is a chain of subcurves, where each two neighboring subcurves in the chain share a common endpoint; that is, the polycurve is continuous. Furthermore, the target of the \(i\)th
segement of a polycurve has to coincide with the source of the \(i+1\)st segment; that is, the polycurve has to be well-oriented. Note that it is possible to construct general polycurves that are
neither continuous nor well-oriented, as it is impossible to enforce this precondition (using the set of predicates required by the relevant concepts, see below). However, such polycurves cannot be
used for the actual computation of arrangements. The traits class template exploits the functionality of the SubcurveTraits_2 template-parameter to handle the subcurves that compose the polycurve.
The type substituting the template parameter SubcurveTraits_2 when the template Arr_polycurve_traits_2 is instantiated must be a model of the concepts
If, in addition, the SubcurveTraits_2 models the concept ArrangementApproximateTraits_2 then Arr_polycurve_traits_2 models this concept as well. The same holds for the concept
ArrangementOpenBoundaryTraits_2. If no type is provided, then Arr_segment_traits_2 (instantiated with Exact_predicates_exact_constructions_kernel as the kernel) is used. Otherwise,
Arr_algebraic_segment_traits_2<Coefficient>, Arr_Bezier_curve_traits_2<RatKernel, AlgKernel, NtTraits>, Arr_circle_segment_traits_2<Kernel>, Arr_conic_traits_2<RatKernel, AlgKernel, NtTraits>,
Arr_linear_traits_2<Kernel>, Arr_non_caching_segment_traits_2<Kernel>, Arr_segment_traits_2<Kernel>, Arr_rational_function_traits_2<AlgebraicKernel_d_1>, or any other model of the concepts above can
be used.
The number type used by the injected subcurve traits should support exact rational arithmetic (that is, the number type should support the arithmetic operations \( +\), \( -\), \( \times\) and \( \
div\) carried out without loss of precision), in order to avoid robustness problems, although other inexact number types could be used at the user's own risk.
A polycurve that comprises \(n > 0\) subcurves has \( n+1 \) subcurve end-points, and they are represented as objects of type SubcurveTraits_2::Point_2. Since the notion of a vertex is reserved to
0-dimensional elements of an arrangement, we use, in this context, the notion of points in order to refer to the vertices of a polycurve. For example, an arrangement induced by a single non-self
intersecting polycurve has exactly two vertices regardless of the number of subcurve end-points. Finally, the types Subcurve_2 and X_monotone_subcurve_2 nested in Arr_polycurve_traits_2 are nothing
but SubcurveTraits_2::Curve_2 and SubcurveTraits_2::X_monotone_curve_2, respectively.
A note on Backwards compatibility
In CGAL version 4.2 (and earlier) any object of the X_monotone_curve_2 type nested in Arr_polycurve_traits_2 which in that version was called Arr_polyline_tratis_2 maintained a direction invariant;
namely, its vertices were ordered in an ascending lexicographical \((xy)\)-order. This restriction is no longer imposed and X_monotone_curve_2 can be now directed either from right-to-left or
left-to-right. If you wish to maintain a left-to-right orientations of the \(x\)-monotone polycurve, set the macro CGAL_ALWAYS_LEFT_TO_RIGHT to 1 before any CGAL header is included.
ArrangementApproximateTraits_2 (if the type that substitutes the template parameter SubcurveTraits_2 models the concept as well)
See also | {"url":"https://doc.cgal.org/5.0.2/Arrangement_on_surface_2/classCGAL_1_1Arr__polycurve__traits__2.html","timestamp":"2024-11-01T18:59:16Z","content_type":"application/xhtml+xml","content_length":"33066","record_id":"<urn:uuid:4c5940d0-da9c-40ab-a7e1-4aca97d82025>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00351.warc.gz"} |
How to Create a Configurable Filter Using a Kaiser Window
This article explains how to create a windowed-sinc filter with a Kaiser (or Kaiser-Bessel) window. The windowed-sinc filters in previous articles such as How to Create a Simple Low-Pass Filter
typically had two parameters, the cutoff frequency \(f_c\) and the transition bandwidth (or rolloff) \(b\). With a Kaiser window, there is a third input parameter \(\delta\), the ripple. All three
parameters are illustrated in Figure 1.
For the specific case of the Kaiser window, the same \(\delta\) is used in the passband and in the stopband. The impulse response of the underlying sinc filter is, as in How to Create a Simple
Low-Pass Filter, still defined as
where \(f_c\) is the cutoff frequency, and where \(n\) runs from \(-\infty\) to \(+\infty\). Remember that this range of \(n\) is actually the reason that a window is necessary. The impulse response
of the ideal filter is infinite, and needs to be “shortened” in a way that is better than simple truncation.
The main advantage of the Kaiser window is that it is more configurable than, e.g., the Blackman window. The acceptable ripple of the filter can be specified, while it is fixed (but known) for the
basic window types.
Kaiser Window
The definition of the Kaiser window is given by
with \(N\) the length of the filter, \(n\) running from zero to \(N-1\), and \(\beta\) a parameter that can be chosen freely (see below). \(I_0(\cdot)\) is the zeroth-order modified Bessel function
of the first kind. Especially with the Bessel function, this expression looks more complicated than the expressions for windows such as Blackman. However, Bessel functions have many applications and
are directly available in Python and MATLAB.
By varying the length \(N\) and the parameter \(\beta\), the properties of the final filter, which is the product of \(h[n]\) (shifted to the range \([0,N-1]\)) and \(w[n]\), can be determined. This
results in the three mentioned tunable parameters, the cutoff frequency \(f_c\), the transition bandwidth \(b\), and the ripple \(\delta\). Traditionally, \(\delta\) is seen as the tunable parameter,
and a new parameter \(A\) is then computed as
Personally, I would suggest that you take this \(A\) to be the tunable parameter, since it is simply the attenuation of the filter in the stopband in dB. If the ripple in the passband is very
important, you might not be able to do that. However, for a typical filter that is used to remove a range of frequencies more or less completely, you’ll need to make \(A\) relatively large, resulting
in a small ripple in the passband anyway.
Kaiser then empirically, through numerical experimentation, determined which value of \(\beta\) to use to end up with a given value for \(A\), as
0.1102(A-8.7),&A\gt 50\\[0.3em]
0.5842(A-21)^{0.4}+0.07886(A-21),&21\leq A\leq 50\\[0.3em]
0,&A\lt 21
The final parameter that you need is then the filter length \(N\), also determined empirically by Kaiser as
\[N=\dfrac{A-8}{2.285\cdot2\pi b}+1.\]
I’ve added the term \(+1\) because the formula of Kaiser estimates the filter order, which is one less than the filter length. Additionally, you often still want the filter to have an odd length, so
that its delay is an integer number of samples, so you might have to do another \(+1\) if the original estimate is even…
And that’s it. You can then use these parameters \(\beta\) and \(N\) in the expression for \(w[n]\) above to create a Kaiser window with the given transition bandwidth \(b\) and attenuation in the
stopband \(A\).
For the parameters \(f_c=0.25\), \(b=0.1\), and \(A=40\), the result is given in Figure 2. In this case, \(N=25\) and \(\beta=3.395\).
For the parameters \(f_c=0.125\), \(b=0.05\), and \(A=60\), the result is given in Figure 3. In this case, \(N=75\) and \(\beta=5.653\).
To create other types of filters such as high-pass or band-pass, the standard techniques can be used, for example as described in How to Create a Simple High-Pass Filter and How to Create Simple
Band-Pass and Band-Reject Filters. Figure 4 shows an example of a high-pass filter that was created through the technique described in Spectral Reversal to Create a High-Pass Filter, from the same
parameters that were used for Figure 3.
Python Code
In Python, all these formulas can be implemented concisely.
from __future__ import division
import numpy as np
fc = 0.25 # Cutoff frequency as a fraction of the sampling rate (in (0, 0.5)).
b = 0.1 # Transition band, as a fraction of the sampling rate (in (0, 0.5)).
A = 40 # Attenuation in the stopband [dB].
N = int(np.ceil((A - 8) / (2.285 * 2 * np.pi * b))) + 1
if not N % 2: N += 1 # Make sure that N is odd.
if A > 50:
beta = 0.1102 * (A - 8.7)
elif A <= 50 and A >= 21:
beta = 0.5842 * (A - 21) ** 0.4 + 0.07886 * (A - 21)
beta = 0
n = np.arange(N)
# Compute sinc filter.
h = np.sinc(2 * fc * (n - (N - 1) / 2))
# Compute Kaiser window.
w = np.i0(beta * np.sqrt(1 - (2 * n / (N - 1) - 1) ** 2)) / np.i0(beta)
# Multiply sinc filter by window.
h = h * w
# Normalize to get unity gain.
h = h / np.sum(h)
Applying the filter \(h\) to a signal \(s\) by convolving both sequences can then be as simple as writing the single line:
In the Python script above, I compute everything in full to show you exactly what happens, but, in practice, shortcuts are available. For example, the Kaiser window can be computed with w = np.kaiser
(N, beta).
Filter Design Tool
This article is complemented with a Filter Design tool. With respect to Kaiser windows, it allows building low-pass, high-pass, band-pass, and band-reject filters. Try it now!
Hello Tom, if you're still there! I hit this page on a search for sinc filter coefficients sox since the sox manual page doesn't give a reference for the use of the sinc function. Thanks for
providing this information, I think it's probably just what I need. Best regards, Tim S. (My web site is always years out of date and barely functional.)
Yeah, I'm still here! And also: uh oh, people are starting to notice that I haven't posted anything in over a year... :-) However, it's a pause, and not a final stop!
The content of this field is kept private and will not be shown publicly.
Spam avoidance measure, sorry for this. | {"url":"https://tomroelandts.com/articles/how-to-create-a-configurable-filter-using-a-kaiser-window","timestamp":"2024-11-01T20:31:03Z","content_type":"text/html","content_length":"35527","record_id":"<urn:uuid:63b8a030-d7a8-4afa-8010-d95ec56d3a6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00864.warc.gz"} |
3i-Infotech Numerical Ability Interview Questions with Answers
Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a
Matematics is like Love,a simple idea but can get complicated.
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations. | {"url":"https://m4maths.com/placement-puzzles.php?SOURCE=3i-Infotech&TOPIC=Numerical%20Ability","timestamp":"2024-11-06T15:46:41Z","content_type":"text/html","content_length":"95619","record_id":"<urn:uuid:74305642-7275-4caf-bf30-0a3facacb91e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00340.warc.gz"} |
Can anyone teach me what is a graph and how it works?thank you.. | Socratic
Can anyone teach me what is a graph and how it works?thank you..
1 Answer
Sample plot of line $y = x - 4$
Notice Slope $m = 1 ,$ For $\theta = {45}^{\circ}$, $c = - 4$ is the $y$- intercept.
First step: Express the given equations in the slope-intercept form.
$y = m x + c$
Here $m$ is the slope and $c$ is the $y$- intercept.
Slope of a line is $\tan \theta$, where $\theta$ is the angle the line makes with the $x$-axis.
The intercept is the point where the line intersects the $y$-axis
Second step: Plot both the lines on a graph paper.
Third step: Locate the point of intersection of both lines. Note down point $\left(x , y\right)$, the point of intersection. This is the solution for set of linear equations.
To learn how to plot straight line on a graph, one can make use of Graph tool given at the top of Answer box.
graph{y=x-4 [-10.14, 10.13, -5.07, 5.07]}
Impact of this question
1244 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/can-anyone-teach-me-what-is-a-graph-and-how-it-works-thank-you","timestamp":"2024-11-07T01:15:21Z","content_type":"text/html","content_length":"33503","record_id":"<urn:uuid:c35e91c4-f03d-443b-addc-c2402788ec00>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00280.warc.gz"} |
High Low Breaker Backtest Strategy
1. High Low Breaker Backtest Strategy
High Low Breaker Backtest Strategy
, Date: 2023-11-27 15:37:13
The High Low Breaker Backtest strategy is a trend-following strategy that uses the historical highs and lows of a stock to determine if the price breaks out of these high-low ranges. It calculates
the highest price and lowest price over a certain period, and generates buy signals when the current period’s price exceeds the highest price over a recent period, and sell signals when the price
breaks below the lowest price over a recent period. As a type of trend-following strategy, it can capture some trending characteristics of stock prices and has practical value for live trading.
Strategy Logic
The core logic of this strategy is to calculate the highest price and lowest price over a certain number of bars (default 50 bars). When calculating highest/lowest prices, it allows using close
prices or actual high/low prices (default to use high/low prices). Then it checks if the current bar’s closing price or high price exceeds the highest price over the recent period. If yes and it’s
been more than a minimum number of bars (default 30 bars) since the last highest price bar, it generates a buy signal. Likewise, if the current bar’s closing price or low price breaks the lowest
price over the recent period and a minimum number of bars passed since last lowest price bar, it generates a sell signal.
Upon generating buy signals, the strategy enters long positions at that price, with a stop loss price and take profit price set. It exits the position with a stop loss when stop loss price is
touched, and exits with a take profit when take profit price is touched. The logic for sell signals is similar.
Advantage Analysis
This high low breaker backtest strategy has the following advantages:
1. The logic is simple and easy to understand/implement.
2. It can capture some trending characteristics of stock prices.
3. Parameters can be optimized to find best parameter combinations.
4. Built-in stop loss and take profit controls risk.
5. Visualizations greatly facilitate parameter tuning and results analysis.
Risk Analysis
This strategy also has some risks:
1. Prone to multiple flip-flop trades and over-trading.
2. Frequent position opening when price oscillates.
3. Missing major trend opportunities if parameters not properly set.
4. Not considering price fluctuations frequency and magnitude.
5. No signal validation with other indicators.
The following aspects can help mitigate these risks:
1. Reduce stop loss distance to increase holding time.
2. Add more entry criteria to avoid frequent entries.
3. Optimize parameters to find optimum combinations.
4. Add filter conditions with other indicators.
Optimization Directions
This strategy can be improved in the following ways:
1. Parameter optimization using more systematic testing.
2. Add signal filters with other indicators e.g. moving averages.
3. Consider price volatility using ATR to adapt breakout thresholds.
4. Differentiate trending vs oscillating markets to adapt parameters.
5. Enhance position sizing rules e.g. stop opening new positions after significant loss.
In summary, the High Low Breaker Backtest Strategy is a simple and practical trend-following strategy. It generates trading signals based on price breaking periodic highest/lowest prices. The
strategy has advantages like simplicity, trend-following, and parameter optimizability, but also risks like over-trading and inability to handle oscillating markets. Further optimizations can be done
around parameters, signal filters, position sizing etc. to improve its performance.
start: 2023-11-25 00:00:00
end: 2023-11-26 00:00:00
period: 1m
basePeriod: 1m
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
strategy("High/Low Breaker Backtest 1.0", overlay=true, initial_capital=1000, default_qty_type=strategy.percent_of_equity, default_qty_value=100, max_bars_back=700)
// Strategy Settings
takeProfitPercentageLong = input(.1, title='Take Profit Percentage Long', type=float)/100
stopLossPercentageLong = input(0.15, title='Stop Loss Percentage Long', type=float)/100
takeProfitPercentageShort = input(.1, title='Take Profit Percentage Short', type=float)/100
stopLossPercentageShort = input(0.15, title='Stop Loss Percentage Short', type=float)/100
candlesBack = input(title="Number of candles back", defval=50)
useHighAndLows = input(true, title="Use high and lows (uncheck to use close)", defval=true)
lastBarsBackMinimum = input(title="Number of candles back to ignore for last high/low", defval=30)
showHighsAndLows = input(true, title="Show high/low lines", defval=true)
getIndexOfLowestInSeries(series, period) =>
index = 0
current = series
for i = 1 to period
if series[i] <= current
index := i
current := series[i]
getIndexOfHighestInSeries(series, period) =>
index = 0
current = series
for i = 1 to period
if series[i] >= current
index := i
current := series[i]
indexOfHighestInRange = getIndexOfHighestInSeries(useHighAndLows ? high : close, candlesBack)
indexOfLowestInRange = getIndexOfLowestInSeries(useHighAndLows ? low : close, candlesBack)
max = useHighAndLows ? high[indexOfHighestInRange] : close[indexOfHighestInRange]
min = useHighAndLows ? low[indexOfLowestInRange] : close[indexOfLowestInRange]
barsSinceLastHigh = indexOfHighestInRange
barsSinceLastLow = indexOfLowestInRange
isNewHigh = (useHighAndLows ? high > max[1] : close > max[1]) and (barsSinceLastHigh[1] + 1 > lastBarsBackMinimum)
isNewLow = (useHighAndLows ? low < min[1] : close < min[1]) and (barsSinceLastLow[1] + 1 > lastBarsBackMinimum)
alertcondition(condition=isNewHigh, title="New High", message="Last High Broken")
alertcondition(condition=isNewLow, title="New Low", message="Last Low Broken")
if high > max
max := high
barsSinceLastHigh := 0
if low < min
min := low
barsSinceLastLow := 0
plot( showHighsAndLows ? max : na, color=red, style=line, title="High", linewidth=3)
plot( showHighsAndLows ? min : na, color=green, style=line, title="Low", linewidth=3)
// Strategy Entry/Exit Logic
goLong =isNewHigh
longStopLevel = strategy.position_avg_price * (1 - stopLossPercentageLong)
longTakeProfitLevel = strategy.position_avg_price * (1 + takeProfitPercentageLong)
goShort = isNewLow
shortStopLevel = strategy.position_avg_price * (1 + stopLossPercentageShort)
shortTakeProfitLevel = strategy.position_avg_price * (1 - takeProfitPercentageShort)
strategy.entry("Long", strategy.long, when=goLong)
strategy.exit("Long Exit", "Long", stop=longStopLevel, limit=longTakeProfitLevel)
strategy.entry("Short", strategy.short, when=goShort)
strategy.exit("Short Exit", "Short", stop=shortStopLevel, limit=shortTakeProfitLevel)
plot(goShort ? shortStopLevel : na, color=yellow, style=linebr, linewidth=2)
plot(goShort ? shortTakeProfitLevel : na, color=blue, style=linebr, linewidth=2)
plot(goLong ? longStopLevel : na, color=yellow, style=linebr, linewidth=2)
plot(goLong ? longTakeProfitLevel : na, color=blue, style=linebr, linewidth=2) | {"url":"https://www.fmz.com/strategy/433426","timestamp":"2024-11-06T20:18:03Z","content_type":"text/html","content_length":"16103","record_id":"<urn:uuid:a85e9533-44a9-410d-8165-234bda890e2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00540.warc.gz"} |
Proof of derivative rules
Registered: 2022-05-10
Posts: 52
Proof of derivative rules
Sum rule:
Product rule:
Reciprocal rule:
Quotient rule:
Power rule:
Chain rule:
Inverse function rule:
Natural exponential rule:
Natural logarithm rule:
Exponential rule:
Logarithm rule:
Derivatives of the Trigonometric Functions
Inverse sine rule:
Inverse cosine rule:
Inverse tangent rule:
Derivative of advanced operators:
Derivative of function equations:
Derivative of statistical function:
n-dimensional hyper-surface area of (n+1)-dimensional frustum of hyper-cone:
When h tends to zero:
Last edited by lanxiyu (2024-09-17 02:01:36) | {"url":"https://mathisfunforum.com/viewtopic.php?pid=435392","timestamp":"2024-11-11T18:29:47Z","content_type":"application/xhtml+xml","content_length":"52641","record_id":"<urn:uuid:25395eb4-83fb-4aa3-93c9-fe507b67b24f>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00103.warc.gz"} |
19.1: Enough with the Zooming Already (10 minutes)
Provide access to graphing technology. It may be desirable for students to work in groups of four, so that they can view all four graphing windows on four different devices at the same time.
If needed, demonstrate using technology to graph the function and set the graphing window for the first given window. Then, allow students to set the remaining graphing windows and answer the
Student Facing
Use graphing technology to create a graph of \(y=10^x\). Here are some different graphing windows to try:
• A: \(\text-10 \le x \le 10\) and \(\text-10 \le y \le 10\)
• B: \(\text-1 \le x \le 1\) and \(\text-10 \le y \le 10\)
• C: \(\text-50 \le x \le 50\) and \(\text-10 \le y \le 10\)
• D: \(\text-1 \le x \le 1\) and \(\text-50 \le y \le 50\)
1. Which graphing window makes the graph look . . .
1. the steepest?
2. the flattest?
2. Come up with a new graphing window that makes the graph look even steeper than the steepest one you identified.
3. Come up with a new graphing window that makes the graph look even flatter than the flattest one you identified.
Activity Synthesis
The main purpose of this activity is to learn or revisit how to specify a graphing window using the available technology. Once students show proficiency with that, quickly move on to the next
19.2: How to Lie with Graphing Windows (20 minutes)
In this activity, students get more practice adjusting their graphing windows. Then, they explore why someone might want to choose a different graphing window to tell a different story.
Display the 4 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to
discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
These graphs represent a function modeling the amount of a harmful chemical in drinking water in parts per billion over time in years.
1. Here is Graph A, with rectangles approximating the graphing windows of B, C, and D superimposed. Which rectangle matches which graph?
2. Use graphing technology to create a graph of \(f(x)=0.2 \boldcdot (1.045)^x\). Practice adjusting the graphing window so your graph looks like A, B, C, and D.
3. Imagine you are a public health official worried about this model, and you want to convince others that they should be worried, too. Which graphing window would you use, and why?
4. Imagine you are a public relations official for the company responsible for the chemical in the drinking water, and you want to convince others not to worry. Which graphing window would you use,
and why?
5. What questions should a journalist writing about the issue ask each person about their graphs, and which graph should she publish with the article?
Activity Synthesis
After demonstrating or asking a student to share how to adjust the window to display each version of the graph, focus discussion on the last three questions. Emphasize that all of the graphs are
showing the same information, but the choice of graphing window makes the information look either harmless or alarming. Questions for discussion:
• “What are some reasons graphing window C might make people worry?” (It appears very steep, like the concentration of the harmful chemical is shooting up.)
• “What are some reasons someone might decide not to worry about graphing window C?” (The time span is 80 years, so perhaps the trend can be interrupted before then. Also, we would need more
information about how concentrated the chemical needs to be before it is harmful. Maybe 8 parts per billion isn’t actually anything to worry about.)
• “What are some reasons graphing window B might look like nothing to worry about?” (It appears very flat, like the concentration of the chemical is hardly increasing at all.)
• “What are some reasons someone might decide to dig deeper, if they were only presented with window B?” (It only shows 8 years, so someone might wonder what happens after that. Also, if you knew
the model was exponential, you would suspect it might shoot up eventually.)
19.3: Renting a Car (10 minutes)
In this activity, students choose a graphing window to tell a particular story.
Continue access to graphing technology. Students may be ready to dive into the activity without much assistance, or may need to engage in a notice and wonder to familiarize themselves with the
context and the information encoded in the equation and the graph.
Student Facing
Suppose \(y=0.50d + 4\) represents the cost of renting a car, \(y\), as a function of miles driven, \(d\). Here’s a graph representing the function.
1. What is the graphing window used for the given graph?
2. Find a graphing window so that:
1. It gives the impression that the charge per mile driven is very high and the total rental cost gets really expensive really fast.
2. It gives the impression that the charge for every mile driven is close to nothing and the total rental cost will be pretty low even if the car is driven many miles.
Activity Synthesis
Invite selected students to share their responses. Questions for discussion:
• “What adjustments to the graphing window make the cost seem lower? What adjustments make the cost seem higher?” (The graph in the second question make the cost seem lower, because the graph looks
quite flat and close to the horizontal axis. The graph is the first question make the cost seem higher, because the graph looks quite steep.)
• “For each situation, how did you decide whether to make the graph look flatter or steeper?” (I thought about whether the situation suggested the function should be increasing slowly or quickly.)
• “What were some strategies to make the graph look flatter?” (Keep the domain the same or smaller, and make the range larger.)
• “What were some strategies to make the graph look steeper?” (Keep the range the same, and make the domain larger.)
• “In most graphing technology, you can zoom in or zoom out. Why might you want to set a specific graphing window, instead of zooming in or out?” (Zooming in or out often doesn’t change the overall
shape of the graph, because the domain and range are being scaled by the same amount. Choosing a specific graphing window allows you more control over what is shown.) | {"url":"https://im-beta.kendallhunt.com/HS/teachers/4/5/19/index.html","timestamp":"2024-11-03T05:53:02Z","content_type":"text/html","content_length":"100833","record_id":"<urn:uuid:fdb95652-8435-4163-bc3a-c3631e19ca36>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00114.warc.gz"} |
How do you find the center and radius of a circle?
In order to find the center and radius, we need to change the equation of the circle into standard form, ( x − h ) 2 + ( y − k ) 2 = r 2 (x-h)^2+(y-k)^2=r^2 (x−h)2+(y−k)2=r2, where h and k are the
coordinates of the center and r is the radius.
What is a center radius?
The center is a fixed point in the middle of the circle; usually given the general coordinates (h, k). The fixed distance from the center to any point on the circle is called the radius.
What is the formula for equation of a circle?
We know that the general equation for a circle is ( x – h )^2 + ( y – k )^2 = r^2, where ( h, k ) is the center and r is the radius.
How do you find the center of a circle from an equation?
From the standard equation of a circle (x−a)2+(y−b)2=r2, the center is the point (a,b) and the radius is r.
How do you find the coordinates of the center of a circle?
Explanation: The formula for the equation of a circle is (x – h)2+ (y – k)2 = r2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle. If a
circle is tangent to the x-axis at (3,0), this means it touches the x-axis at that point.
How do you find the radius of a circle using coordinates?
The formula for the equation of a circle is (x – h)2+ (y – k)2 = r2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle.
What is the equation for finding diameter?
Formula. The formula to find the diameter states the relationship between the diameter and the radius. The diameter is made up of two segments that are each a radius. Therefore, the formula is:
Diameter = 2 * the measurement of the radius. You can abbreviate this formula as d=2r.
How do you find the standard equation of a circle?
The standard equation of a circle is a way to describe all points lying on a circle with just one formula: (x – A)² + (y – B)² = r². (x, y) are the coordinates of any point lying on the circumference
of the circle.
What is the formula for finding the diameter of a circle?
If you know the area of the circle, divide the result by π and find its square root to get the radius; then multiply by 2 to get the diameter. This goes back to manipulating the formula for finding
the area of a circle, A = πr 2, to get the diameter. You can transform this into r = √(A/π) cm.
What is the equation for the radius of a circle?
This gives us the radius of the circle. Using the center point and the radius, you can find the equation of the circle using the general circle formula (x-h)* (x-h) + (y-k)* (y-k) = r*r, where (h,k)
is the center of your circle and r is the radius. | {"url":"https://www.heyiamindians.com/how-do-you-find-the-center-and-radius-of-a-circle/","timestamp":"2024-11-08T08:28:47Z","content_type":"text/html","content_length":"43619","record_id":"<urn:uuid:2feae2d7-9b4c-4508-852a-41156f64d2f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00359.warc.gz"} |
Statistical Physics and Modeling of Human Mobility
Gallotti, Riccardo
Statistical Physics and Modeling of Human Mobility
, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in
, 25 Ciclo. DOI 10.6092/unibo/amsdottorato/5198.
Documenti full-text disponibili:
Documento PDF (English) - Richiede un lettore di PDF come Xpdf o Adobe Acrobat Reader
Download (4MB) | Anteprima
In this thesis, we extend some ideas of statistical physics to describe the properties of human mobility. By using a database containing GPS measures of individual paths (position, velocity and
covered space at a spatial scale of 2 Km or a time scale of 30 sec), which includes the 2% of the private vehicles in Italy, we succeed in determining some statistical empirical laws pointing out
"universal" characteristics of human mobility. Developing simple stochastic models suggesting possible explanations of the empirical observations, we are able to indicate what are the key quantities
and cognitive features that are ruling individuals' mobility. To understand the features of individual dynamics, we have studied different aspects of urban mobility from a physical point of view. We
discuss the implications of the Benford's law emerging from the distribution of times elapsed between successive trips. We observe how the daily travel-time budget is related with many aspects of the
urban environment, and describe how the daily mobility budget is then spent. We link the scaling properties of individual mobility networks to the inhomogeneous average durations of the activities
that are performed, and those of the networks describing people's common use of space with the fractional dimension of the urban territory. We study entropy measures of individual mobility patterns,
showing that they carry almost the same information of the related mobility networks, but are also influenced by a hierarchy among the activities performed. We discover that Wardrop's principles are
violated as drivers have only incomplete information on traffic state and therefore rely on knowledge on the average travel-times. We propose an assimilation model to solve the intrinsic scattering
of GPS data on the street network, permitting the real-time reconstruction of traffic state at a urban scale.
In this thesis, we extend some ideas of statistical physics to describe the properties of human mobility. By using a database containing GPS measures of individual paths (position, velocity and
covered space at a spatial scale of 2 Km or a time scale of 30 sec), which includes the 2% of the private vehicles in Italy, we succeed in determining some statistical empirical laws pointing out
"universal" characteristics of human mobility. Developing simple stochastic models suggesting possible explanations of the empirical observations, we are able to indicate what are the key quantities
and cognitive features that are ruling individuals' mobility. To understand the features of individual dynamics, we have studied different aspects of urban mobility from a physical point of view. We
discuss the implications of the Benford's law emerging from the distribution of times elapsed between successive trips. We observe how the daily travel-time budget is related with many aspects of the
urban environment, and describe how the daily mobility budget is then spent. We link the scaling properties of individual mobility networks to the inhomogeneous average durations of the activities
that are performed, and those of the networks describing people's common use of space with the fractional dimension of the urban territory. We study entropy measures of individual mobility patterns,
showing that they carry almost the same information of the related mobility networks, but are also influenced by a hierarchy among the activities performed. We discover that Wardrop's principles are
violated as drivers have only incomplete information on traffic state and therefore rely on knowledge on the average travel-times. We propose an assimilation model to solve the intrinsic scattering
of GPS data on the street network, permitting the real-time reconstruction of traffic state at a urban scale.
Altri metadati
Statistica sui download | {"url":"http://amsdottorato.unibo.it/5198/","timestamp":"2024-11-02T11:29:09Z","content_type":"application/xhtml+xml","content_length":"41433","record_id":"<urn:uuid:94934286-7304-478e-a945-64fd7a2e2da6>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00391.warc.gz"} |
Taylor Rule - Breaking Down Finance
Taylor Rule
The Taylor Rule is a monetary policy rule in economics. The rule is called the Taylor Rule because it was proposed by John B. Taylor in 1993. It describes a central bank’s monetary policy when the
bank determines its monetary policy based on price stability and economic output.
Taylor Rule definition
The Taylor rule is based on the observation that, in the United States at least, the central bank has a “dual mandate”. In particular, the Federal Reserve (FED) tries to maintain price stability and
maximum employment. Such a mutual mandate can be summarized using the Taylor Rule for monetary policy. The rule states that the nominal interest rate (R[f]) can be approximated as follows:
where the output gap is typically defined as the “percentage deviation of real GDP from its target”. The output gap measures whether output is above or below its ‘potential’.
Taylor rule formula interpretation
The Taylor rule formula above clearly shows that nominal interest rate is determined both by inflation (price stability) and output gat (employment and growth). The Taylor rule formula therfore
clearly reflects the dual mandate of the Fed.
To see this, let’s first assume that inflation is at 2% and that the output gap is at zero. In that case, the Fed is expected to set a nominal interest rate of 4%. Given an inflation rate of 2%,
this corresponds to a a real interest rate (R[f]—inflation) of 2%.
Next, imagine the inflation rate rises above 2%. In that case, the Fed is expected to raise nominal interest rates more than one-for-one. This is called the the “Taylor principle”. In particular, if
inflation rate goes up to 3%, then the Fed is expected to increase the nominal rate to 5.5%. The real interest rate goes up to 2.5%. This exceeds the inflation rate and is meant to cool the economy.
This should bring the inflation back down toward its target. Similarly, a negative output gap, i.e., high unemployment, results in lower interest rates that stimulate the economy.
Below, we plot a Taylor rule graph for the the Fed. The graph shows the expected interest rate based on the inflation rate and the output gap for the US.
The data used to construct this figure is provided by the FRED and can be downloaded here.
The Taylor rule is, of course, only an approximation. It also only describes the possible behavior of central banks that have a dual mandate. The European Central Bank (ECB) for example only targets
price stability and does not consider the output gap when setting the short term interest rate.
We discussed a very simple formula that can be used to model a central bank’s monetary policy decisions. This rule, called the Taylor rule, worked well historically. Since the financial crisis,
however, the rule no longer fits monetary policy that well.
The following Excel file implements a Taylor Rule calculator: | {"url":"https://breakingdownfinance.com/finance-topics/finance-basics/taylor-rule/","timestamp":"2024-11-09T14:18:39Z","content_type":"text/html","content_length":"236490","record_id":"<urn:uuid:3684a340-988c-4326-900f-ca421976c67e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00095.warc.gz"} |
Advice for students
I am happy to receive applications from students asking for internships, for thesis guidance and project assistant positions in scientific computing. All of my work involves computations and requires
proficiency in working with computers and coding. Here is some advice on what you can do to improve your chances of success in scientific computing.
• You need to know some theory. Here is a list of books I recommend.
• Learn to work on Unix/Linux and the command line. Use a proper text editor like Vi/Vim or Emacs.
• Learn a programming language really well, e.g., Fortran and C/C++. I would recommend learning all three of these. (Matlab is not a programming language. I dont have any projects that can be done
in Matlab.)
• Learn Python, which is great for scripting, data analysis and visualization.
• Learn to use some visualization tools like gnuplot, Paraview and VisIt. Python also has good support for visualization, see also PyVista.
• Learn to use a version control system like git.
• Learn parallel programming concepts and MPI.
• Learn linear algebra libraries like Petsc and/or Trilinos.
• Learn Latex. Write your reports/cv/application using Latex. (Video)
• Put up your scientific computing project work on github, gitlab or Bitbucket.
• Take the many online courses being offered these days on scientific computing topics. Do the assignments and exams.
• Finally, send your cv/application in PDF format ONLY.
When you write to me, it is highly desirable if you can show me some proof for all/some of the above in terms of actual code, results and project reports. In your CV, mention all the courses, in
person or online, that you have done in Physics, Applied Math, Numerical Methods, Scientific Computing, etc.
To work with me
The above advice is fairly general for anybody wanting to work in scientific computing, numerical solution of PDE, finite element methods, computational fluid dynamics, etc. If you want to work with
me, it will help if you are able to satisfy some of the above requirements. In addition, you will have a better chance if you have worked with any of the following. | {"url":"https://cpraveen.github.io/forstudents.html","timestamp":"2024-11-15T04:47:34Z","content_type":"text/html","content_length":"8663","record_id":"<urn:uuid:0a41ecc4-dcfa-48e5-8404-94d56dd91cb4>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00744.warc.gz"} |
American Mathematical Society
The character of $\omega _{1}$ in first countable spaces
HTML articles powered by AMS MathViewer
Proc. Amer. Math. Soc. 62 (1977), 149-155
DOI: https://doi.org/10.1090/S0002-9939-1977-0438272-7
PDF | Request permission
We define a cardinal function $\chi (P,Q)$, where P and Q are properties of topological spaces. We show that it is consistent and independent that $\chi ({\omega _1},\;{\text {first}}\;{\text
{countable}}) = {\omega _1}$. References
• Keith J. Devlin, Aspects of constructibility, Lecture Notes in Mathematics, Vol. 354, Springer-Verlag, Berlin-New York, 1973. MR 0376351, DOI 10.1007/BFb0059290
• Keith J. Devlin, Kurepa’s hypothesis and the continuum, Fund. Math. 89 (1975), no. 1, 23–31. MR 398826, DOI 10.4064/fm-89-1-23-31
• William Fleissner, Normal Moore spaces in the constructible universe, Proc. Amer. Math. Soc. 46 (1974), 294–298. MR 362240, DOI 10.1090/S0002-9939-1974-0362240-4
• William G. Fleissner, A normal collectionwise Hausdorff, not collectionwise normal space, General Topology and Appl. 6 (1976), no. 1, 57–64. MR 391032, DOI 10.1016/0016-660X(76)90008-8
• Stephen H. Hechler, On the existence of certain cofinal subsets of $^{\omega }\omega$, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif.,
1967) Amer. Math. Soc., Providence, R.I., 1974, pp. 155–173. MR 0360266
• Thomas J. Jech, Trees, J. Symbolic Logic 36 (1971), 1–14. MR 284331, DOI 10.2307/2271510 I. Juhász, Consistency results in topology, Logic Handbook (to appear).
• I. Juhász and William Weiss, On a problem of Sikorski, Fund. Math. 100 (1978), no. 3, 223–227. MR 509548, DOI 10.4064/fm-100-3-223-227
• William Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21–46. MR 313057, DOI 10.1016/0003-4843(72)90017-4
• Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23, Published for the Conference Board of the
Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August
12–16, 1974. MR 0367886, DOI 10.1090/cbms/023 S. Shelah, Decomposing uncountable squares into countably many chains (to appear). J. H. Silver, The independence of Kurepa’s conjecture and
two-cardinal conjectures in model theory, Proc. Sympos. Pure Math., vol. 13, part 1, Amer. Math. Soc., Providence, R.I., 1971, pp. 383-390. MR 43 #3112.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54A25, 04A20
• Retrieve articles in all journals with MSC: 54A25, 04A20
Bibliographic Information
• © Copyright 1977 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 62 (1977), 149-155
• MSC: Primary 54A25; Secondary 04A20
• DOI: https://doi.org/10.1090/S0002-9939-1977-0438272-7
• MathSciNet review: 0438272 | {"url":"https://www.ams.org/journals/proc/1977-062-01/S0002-9939-1977-0438272-7/home.html","timestamp":"2024-11-14T04:39:14Z","content_type":"text/html","content_length":"61516","record_id":"<urn:uuid:1436795e-0817-4779-aa42-6af386878986>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00880.warc.gz"} |
Printable 50×50 Multiplication Chart | Multiplication Chart Printable
Printable 50×50 Multiplication Chart
Multiplication Chart 50×50 Leonard Burton s Multiplication Worksheets
Printable 50×50 Multiplication Chart
Printable 50×50 Multiplication Chart – A Multiplication Chart is a valuable tool for children to discover just how to increase, separate, as well as locate the tiniest number. There are lots of uses
for a Multiplication Chart.
What is Multiplication Chart Printable?
A multiplication chart can be made use of to help youngsters discover their multiplication truths. Multiplication charts can be found in several kinds, from complete web page times tables to solitary
page ones. While private tables serve for presenting portions of details, a complete page chart makes it less complicated to examine realities that have currently been understood.
The multiplication chart will generally feature a left column as well as a top row. When you desire to locate the item of two numbers, select the first number from the left column and also the second
number from the top row.
Multiplication charts are useful discovering tools for both kids and also grownups. Children can use them in the house or in school. Printable 50×50 Multiplication Chart are readily available online
and can be printed out and also laminated for longevity. They are a terrific tool to make use of in math or homeschooling, as well as will certainly provide a visual pointer for kids as they learn
their multiplication truths.
Why Do We Use a Multiplication Chart?
A multiplication chart is a layout that shows how to multiply two numbers. It generally contains a top row and a left column. Each row has a number representing the item of the two numbers. You
select the first number in the left column, move it down the column, and after that choose the 2nd number from the top row. The product will certainly be the square where the numbers fulfill.
Multiplication charts are useful for many reasons, consisting of aiding youngsters learn how to separate as well as streamline fractions. They can likewise aid kids learn just how to select a
reliable common measure. Multiplication charts can likewise be useful as workdesk resources because they function as a consistent tip of the student’s progress. These tools help us establish
independent learners that recognize the basic concepts of multiplication.
Multiplication charts are additionally beneficial for assisting students memorize their times tables. As with any ability, remembering multiplication tables takes time and also technique.
Printable 50×50 Multiplication Chart
Multiplication Chart 50 50 PrintableMultiplication
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Multiplication Chart 50 50 PrintableMultiplication
Printable 50×50 Multiplication Chart
If you’re trying to find Printable 50×50 Multiplication Chart, you’ve concerned the best place. Multiplication charts are offered in various styles, consisting of full dimension, half size, as well
as a range of cute layouts. Some are vertical, while others include a straight layout. You can also find worksheet printables that include multiplication formulas as well as mathematics truths.
Multiplication charts as well as tables are indispensable tools for kids’s education. These charts are terrific for use in homeschool mathematics binders or as classroom posters.
A Printable 50×50 Multiplication Chart is an useful tool to strengthen mathematics truths as well as can assist a youngster learn multiplication rapidly. It’s additionally a wonderful tool for skip
counting as well as finding out the times tables.
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Bra-ket Notation - The Unit Operator
The Unit Operator
Consider a complete orthonormal system (basis), for a Hilbert space H, with respect to the norm from an inner product . From basic functional analysis we know that any ket can be written as
with the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now follows that
must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting its value, for example
where in the last identity Einstein summation convention has been used.
In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary (state) kets is present, while it is possible to say something about the expansion
coefficients and of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to insert the unit operator into the bracket one time or more (for more
information see Resolution of the identity).
Read more about this topic: Bra-ket Notation
Famous quotes containing the word unit:
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men
in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner s jury. In the case of elected bodies the only way of effecting this is by the
Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856 1950)
Related Phrases
Related Words | {"url":"https://www.liquisearch.com/bra-ket_notation/the_unit_operator","timestamp":"2024-11-13T11:43:00Z","content_type":"text/html","content_length":"5494","record_id":"<urn:uuid:6d2abfeb-0c7f-45bb-b9cf-ca8923825bf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00776.warc.gz"} |
How many equations are in the Navier-Stokes equations?
How many equations are in the Navier-Stokes equations?
There is a special simplification of the Navier-Stokes equations that describe boundary layer flows. Notice that all of the dependent variables appear in each equation. To solve a flow problem, you
have to solve all five equations simultaneously; that is why we call this a coupled system of equations.
Can the Navier-Stokes equation be solved?
In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and
engineering. Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven.
What are the forces used in Navier-Stokes equation?
There are three kinds of forces important to fluid mechanics: gravity (body force), pressure forces, and viscous forces (due to friction). Gravity force, Body forces act on the entire element, rather
than merely at its surfaces.
Why is the Navier Stokes problem difficult to solve?
The Navier Stokes equation is so hard to solve because it is non-linear. If the inertial terms were not present (either because of the geometry or because the inertial terms are negligible0, it would
(and can) be much easier to solve.
Why is Navier-Stokes unsolvable?
The Navier-Stokes equation is difficult to solve because it is nonlinear. This word is thrown around quite a bit, but here it means something specific. You can build up a complicated solution to a
linear equation by adding up many simple solutions.
Which forces are not considered in Navier-Stokes equation?
Detailed Solution When all the force is taken in to account the equation of motion is called Newton’s equation of motion. When turbulence and minor forces like surface tension are neglected the
equation of motion is called Navier-stock equation of motion.
What are the difficulties in solving Navier-Stokes equation?
The Navier Stokes equations is a non linear set of partial differental equations describing fluid motion. The problem is twofold: except for very simple configurations and simplified equations there
is no solution in terms of elementary functions.
Why is it difficult to obtain the analytical solution of Navier Stokes NS equations choose the correct reasoning’s from the given options?
There are no methods so far or very highly complex methods to solve these non linearity. N-S equations also show such kind of non linearity hence Analytical solution does not exists.
Has P versus NP been solved?
Although one-way functions have never been formally proven to exist, most mathematicians believe that they do, and a proof of their existence would be a much stronger statement than P ≠ NP. Thus it
is unlikely that natural proofs alone can resolve P = NP.
What is the Navier Stokes equation?
The Navier stokes equation or Navier Stokes theorem is so dynamic in fluid mechanics it explains the motion of every possible fluid existing in the universe. It is always been challenging to solve
million-dollar questions and the solution for the Navier Stokes equation is one among them.
What is the time derivative of the Navier-Stokes equation?
The motion of a non-turbulent, Newtonian fluid is governed by the Navier-Stokes equation: The above equation can also be used to model turbulent flow, where the fluid parameters are interpreted as
time-averaged values. The time-derivative of the fluid velocity in the Navier-Stokes equation is the material derivative , defined as:
Why are Navier-Stokes equations so difficult to solve?
Usually, however, they remain nonlinear, which makes them difficult or impossible to solve; this is what causes the turbulence and unpredictability in their results. The Navier-Stokes equations can
be derived from the basic conservation and continuity equations applied to properties of fluids.
What is computational fluid dynamics?
This area of study is called Computational Fluid Dynamics or CFD . The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass , three time-dependent
conservation of momentum equations and a time-dependent conservation of energy equation. | {"url":"https://www.shakerdesignproject.com/students-advice/how-many-equations-are-in-the-navier-stokes-equations/","timestamp":"2024-11-09T01:25:36Z","content_type":"text/html","content_length":"61136","record_id":"<urn:uuid:88b6699d-4187-40fa-8868-95fdb4c90cd7>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00682.warc.gz"} |
ebert_graph | OR-Tools | Google for Developers
C++ Reference: ebert_graph
Note: This documentation is automatically generated.
A few variations on a theme of the "star" graph representation by Ebert, as described in J. Ebert, "A versatile data structure for edge-oriented graph algorithms." Communications of the ACM 30
(6):513-519 (June 1987).
In this file there are three representations that have much in common. The general one, called simply EbertGraph, contains both forward- and backward-star representations. The other, called
ForwardEbertGraph, contains only the forward-star representation of the graph, and is appropriate for applications where the reverse arcs are not needed.
The point of including all the representations in this one file is to capitalize, where possible, on the commonalities among them, and those commonalities are mostly factored out into base classes as
described below. Despite the commonalities, however, each of the three representations presents a somewhat different interface because of their different underlying semantics. A quintessential
example is that the AddArc() method, very natural for the EbertGraph representation, cannot exist for an inherently static representation like ForwardStaticGraph.
Many clients are expected to use the interfaces to the graph objects directly, but some clients are parameterized by graph type and need a consistent interface for their underlying graph objects. For
such clients, a small library of class templates is provided to give a consistent interface to clients where the underlying graph interfaces differ. Examples are the AnnotatedGraphBuildManager<>
template, which provides a uniform interface for building the various types of graphs; and the TailArrayManager<> template, which provides a uniform interface for applications that need to map from
arc indices to arc tail nodes, accounting for the fact that such a mapping has to be requested explicitly from the ForwardStaticGraph and ForwardStarGraph representations.
There are two base class templates, StarGraphBase, and EbertGraphBase; their purpose is to hold methods and data structures that are in common among their descendants. Only classes that are leaves in
the following hierarchy tree are eligible for free-standing instantiation and use by clients. The parentheses around StarGraphBase and EbertGraphBase indicate that they should not normally be
instantiated by clients:
(StarGraphBase) |
/ \ |
/ \ |
/ \ |
/ \ |
(EbertGraphBase) ForwardStaticGraph |
/ \ |
/ \ |
EbertGraph ForwardEbertGraph |
In the general EbertGraph case, the graph is represented with three arrays. Let n be the number of nodes and m be the number of arcs. Let i be an integer in [0..m-1], denoting the index of an arc.
* head_[i] contains the end-node of arc i,
* head_[-i-1] contains the start-node of arc i. Note that in two's-complement arithmetic, -i-1 = ~i.
* head_[~i] contains the end-node of the arc reverse to arc i,
* head_[i] contains the start-node of the arc reverse to arc i.
Note that if arc (u, v) is defined, then the data structure also stores
(v, u).
Arc ~i thus denotes the arc reverse to arc i.
This is what makes this representation useful for undirected graphs and for
implementing algorithms like bidirectional shortest paths.
Also note that the representation handles multi-graphs. If several arcs
going from node u to node v are added to the graph, they will be handled as
separate arcs.
Now, for an integer u in [0..n-1] denoting the index of a node:
* first_incident_arc_[u] denotes the first arc in the adjacency list of u.
* going from an arc i, the adjacency list can be traversed using
j = next_adjacent_arc_[i].
The EbertGraph implementation has the following benefits:
* It is able to handle both directed or undirected graphs.
* Being based on indices, it is easily serializable. Only the contents
of the head_ array need to be stored. Even so, serialization is
currently not implemented.
* The node indices and arc indices can be stored in 32 bits, while
still allowing to go a bit further than the 4-gigabyte
limitation (48 gigabytes for a pure graph, without capacities or
* The representation can be recomputed if edges have been loaded from
* The representation can be recomputed if edges have been loaded from
external memory or if edges have been re-ordered.
* The memory consumption is: 2 * m * sizeof(NodeIndexType)
+ 2 * m * sizeof(ArcIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant.
The EbertGraph implementation differs from the implementation described in
[Ebert 1987] in the following respects:
* arcs are represented using an (i, ~i) approach, whereas Ebert used
(i, -i). Indices for direct arcs thus start at 0, in a fashion that is
compatible with the index numbering in C and C++. Note that we also tested
a (2*i, 2*i+1) storage pattern, which did not show any speed benefit, and
made the use of the API much more difficult.
* because of this, the 'nil' values for nodes and arcs are not 0, as Ebert
first described. The value for the 'nil' node is set to -1, while the
value for the 'nil' arc is set to the smallest integer representable with
ArcIndexSize bytes.
* it is possible to add arcs to the graph, with AddArc, in a much simpler
way than described by Ebert.
* TODO(user) although it is already possible, using the
GroupForwardArcsByFunctor method, to group all the outgoing (resp.
incoming) arcs of a node, the iterator logic could still be improved to
allow traversing the outgoing (resp. incoming) arcs in O(out_degree(node))
(resp. O(in_degree(node))) instead of O(degree(node)).
* TODO(user) it is possible to implement arc deletion and garbage collection
in an efficient (relatively) manner. For the time being we haven't seen an
application for this.
The ForwardEbertGraph representation is like the EbertGraph case described
above, with the following modifications:
* The part of the head_[] array with negative indices is absent. In its
place is a pointer tail_ which, if assigned, points to an array of tail
nodes indexed by (nonnegative) arc index. In typical usage tail_ is NULL
and the memory for the tail nodes need not be allocated.
* The array of arc tails can be allocated as needed and populated from the
adjacency lists of the graph.
* Representing only the forward star of each node implies that the graph
cannot be serialized directly nor rebuilt from scratch from just the head_
array. Rebuilding from scratch requires constructing the array of arc
tails from the adjacency lists first, and serialization can be done either
by first constructing the array of arc tails from the adjacency lists, or
by serializing directly from the adjacency lists.
* The memory consumption is: m * sizeof(NodeIndexType)
+ m * sizeof(ArcIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant when the array of arc tails is absent. Allocating
the arc tail array adds another m * sizeof(NodeIndexType).
The ForwardStaticGraph representation is restricted yet farther
than ForwardEbertGraph, with the benefit that it provides higher performance to those applications that can use it.
* As with ForwardEbertGraph, the presence of the array of arc
tails is optional.
* The outgoing adjacency list for each node is stored in a
contiguous segment of the head_[] array, obviating the
next_adjacent_arc_ structure entirely and ensuring good locality
of reference for applications that iterate over outgoing
adjacency lists.
* The memory consumption is: m * sizeof(NodeIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant when the array of arc tails is absent. Allocating
the arc tail array adds another m * sizeof(NodeIndexType). | {"url":"https://developers.google.com/optimization/reference/graph/ebert_graph","timestamp":"2024-11-07T03:33:11Z","content_type":"text/html","content_length":"147697","record_id":"<urn:uuid:5d77d0ed-4705-47ae-b867-2ac424e4f20e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00536.warc.gz"} |
Embarrassingly Parallel GFlowNets
Embarrassingly Parallel GFlowNets
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:45406-45431, 2024.
GFlowNets are a promising alternative to MCMC sampling for discrete compositional random variables. Training GFlowNets requires repeated evaluations of the unnormalized target distribution, or reward
function. However, for large-scale posterior sampling, this may be prohibitive since it incurs traversing the data several times. Moreover, if the data are distributed across clients, employing
standard GFlowNets leads to intensive client-server communication. To alleviate both these issues, we propose embarrassingly parallel GFlowNet (EP-GFlowNet). EP-GFlowNet is a provably correct
divide-and-conquer method to sample from product distributions of the form $R(\cdot) \propto R_1(\cdot) ... R_N(\cdot)$ — e.g., in parallel or federated Bayes, where each $R_n$ is a local posterior
defined on a data partition. First, in parallel, we train a local GFlowNet targeting each $R_n$ and send the resulting models to the server. Then, the server learns a global GFlowNet by enforcing our
newly proposed aggregating balance condition, requiring a single communication step. Importantly, EP-GFlowNets can also be applied to multi-objective optimization and model reuse. Our experiments
illustrate the effectiveness of EP-GFlowNets on multiple tasks, including parallel Bayesian phylogenetics, multi-objective multiset and sequence generation, and federated Bayesian structure learning.
Cite this Paper
Related Material | {"url":"https://proceedings.mlr.press/v235/silva24a.html","timestamp":"2024-11-09T14:42:18Z","content_type":"text/html","content_length":"17171","record_id":"<urn:uuid:7af9c203-0358-4acf-88df-27ebb7fcbf89>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00272.warc.gz"} |
On a class of optimal control problems arising in mathematical economics
Aseev, S.M. & Kryazhimskiy, A.V. (2008). On a class of optimal control problems arising in mathematical economics. Proceedings of the Steklov Institute of Mathematics 262 (1) 10-25. 10.1134/
Full text not available from this repository.
This paper is devoted to the study of the properties of the adjoint variable in the relations of the Pontryagin maximum principle for a class of optimal control problems that arise in mathematical
economics. This class is characterized by an infinite time interval on which a control process is considered and by a special goal functional defined by an improper integral with a discounting
factor. Under a dominating discount condition, we discuss a variant of the Pontryagin maximum principle that was obtained recently by the authors and contains a description of the adjoint variable by
a formula analogous to the well-known Cauchy formula for the solutions of linear differential equations. In a number of important cases, this description of the adjoint variable leads to standard
transversality conditions at infinity that are usually applied when solving optimal control problems in economics. As an illustration, we analyze a conventionalized model of optimal investment policy
of an enterprise.
Item Type: Article
Research Programs: Dynamic Systems (DYN)
Bibliographic Reference: Proceedings of the Steklov Institute of Mathematics; 262(1):10-25 (September 2008)
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:40
Last Modified: 27 Aug 2021 17:38
URI: https://pure.iiasa.ac.at/8495
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Which property justifies this statement? x-4=x-4x-4=x-4 Reflexive Property of Equality Substitution Property of Equality Transitive Property of Equality Symmetric Property of Equality
Find an answer to your question 👍 “Which property justifies this statement? x-4=x-4x-4=x-4 Reflexive Property of Equality Substitution Property of Equality Transitive ...” in 📗 Mathematics if the
answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers | {"url":"https://cpep.org/mathematics/2305716-which-property-justifies-this-statement-x-4x-4x-4x-4-reflexive-propert.html","timestamp":"2024-11-02T15:04:58Z","content_type":"text/html","content_length":"23785","record_id":"<urn:uuid:76f5b07c-63f7-4f66-bdee-43675373468e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00076.warc.gz"} |
Predictive Analytics | Atlas.Science
Predictive Analytics refers to the use of knowledge extracted from descriptive analytics to realize what will happen in the future. It goes through techniques such as statistical analysis,
forecasting models, Natural Language Processing (NLP), text mining, and Artificial Neural Networks (ANNs). It allows users to predict future possibilities and discover hidden relationships to make
the most likely patterns.*
Predictive analytics have the ability to anticipate risk and identify relationships that may not be apparent with descriptive analytics. Through statistical modeling, data mining, and other advanced
techniques, predictive analytics can identify hidden relationships or patterns in huge volumes of data. This is integral to group or segment data into meaningful sets for detecting tends or
predicting behavior.*
Organisation that are matured in descriptive analytics move into this level where they look beyond what happened and try to answer the question of β What will happen?β . Prediction essentially is
the process of making intelligent/scientific estimates about the future values of some variables like customer demand, interest rates, stock market movements, etc. If what is being predicted is a
categorical variable, the act of prediction is called classification otherwise it is called regression. If the predicted variable is time-dependent, then the prediction process is often called
time-series forecasting.*
Predictive analytics answers such questions as;
• What is likely to happen?
• What trends are foreseen?
• What are multiple alternatives and scenarios?*
The most common predictive analyzing techniques used are: classification, clustering, regression, association analyzes, graph analyses, and decision tree. These are advanced analytics that provide
self-learning models that can be used in real-time applications and forecasting. The use of data, analytics, predictive modeling, machine learning help perform complex correlations on data to gain
insights and are the future of organizations.*
Predictive analytics is no longer confined to statisticians and mathematicians. Corporate strategic planners utilize it significantly in their decision-making processes. The following are a few of
the most common forecasting options:
• Fraud Detection
• Risk Mitigation
• Effectiveness of Marketing Campaigns
• Operational Enhancement
• Clinical Decision Making * | {"url":"https://www.atlas.science/predictive-analytics","timestamp":"2024-11-08T22:03:48Z","content_type":"text/html","content_length":"172876","record_id":"<urn:uuid:c1ea6538-af9a-4a81-ac76-56be4c0e71f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00229.warc.gz"} |
SPPU Engineering Mechanics (EM) - April 2013 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
Answer any one question from Q1 & Q2
1 (a) The resultant of two forces P and Q is 1200 N vertical. Determine the force Q and the corresponding angle ? for the system of forces as shown in Fig. 1a.
4 M
1 (b) The 4.5 × 10^6 kg tanker is pulled with constant acceleration of 0.001 m/s^2 using cable that makes an angle of 15° with the horizontal as shown in in Fig. 1 b. Determine the force in the cable
using Newton's second law of motion.
4 M
1 (c) During a race the dirt bike was observed to leap up off the small hill at A at an angle of 60° with the horizontal as shown in Fig. 1c. If the point of landing is 6 m away, determine the
approximate speed at which the bike was travelling just before it left the ground.
4 M
1 (d) A woman having a mass of 70 kg stands in an elevator which has a downward acceleration of 4 m/s^2 starting from rest. Determine the work done by her weight and the normal force which the floor
exerts on her when the elevator descends 6 m.
4 M
2 (a) Determine the y coordinate of centroid of the shaded area as shown in Fig. 2a.
4 M
2 (b) A girl having mass of 25kg sits at the edge of the merry go-round so her centre of mass G is at a distance of 1.5 m from the centre of rotation as shown in Fig. 2b. Neglecting tangential
component of acceleration, determine the maximum speed which she can have before she begins to slip off the merry go-round. The coefficient of static friction is ?[s]=0.3 Use Newton's second law of
4 M
2 (c) A baseball is thrown downward from a 15 m tower with an initial speed of 5 m/s. determine the speed at which it hits the ground and the time of travel.
4 M
2 (d) A ball has a mass of 30 kg and is thrown upward with a speed of 15 m/s. Determine the time to attain maximum height using impulse momentum principle. Also find the maximum height.
4 M
Answer any one question from Q3 & Q4
3 (a) The motor at B winds up the cord attached to the 65 N crate with a constant speed as shown in Fig. 3a. Determine the force in cord CD supporting the pulley and the angle ? for equilibrium.
Neglet the size of pulley at C.
6 M
3 (b) The boom supports the two vertical loads P[1]=800 N and P[2]=350 N as shown in Fig. 3b. Determine the tension in cable BC and component of reaction at A.
6 M
3 (c) A concrete foundation mat in the shape of regular hexagon with 3 m side support column loads as shown inFig. 3c. Determine the magnitude of the additional loads P[1] and P[2] that must be
applied at B and F if resultant of all six loads is to pass through the centre of the mat.
5 M
4 (a) The rope BC will fail when the tension becomes 50 kN as shown In Fig. 4A. Determine the greatest load P that can be applied to the beam At B and reaction at A for equilibrium.
6 M
4 (b) The three cables are used to support the 800 N lamp as shown in Fig. 4b. Determine the force developed in each cable for equilibrium.
6 M
4 (c) State and explain active forces, reactive forces and free body diagram with suitable example.
5 M
Answer any one question from Q5 & Q6
5 (a) Determine the magnitude and nature of forces in the members BC, HC and HG of the truss loaded and supported as shown in Fig. 5a.
6 M
5 (b) The 15 m ladder has a uniform weight of 80 N and rest against the smooth wall at B as shown in Fig. 5b. If the coefficient of static friction ?[s]=0.4, determine if the ladder will slip?
6 M
5 (c) Define angle of repose, angle of friction, coefficient of friction and cone of friction with sketches.
5 M
6 (a) Determine the forces in each member of the truss and state if the members are in tension or compression. Refer Fig. 6a.
6 M
6 (b) Two loads are suspended as shown in Fig. 6b from cable ABCD. Knowing that d[c]=0.75 m and d[b]=1.125 m, determine the component of reaction at A maximum tension in the cable.
6 M
6 (c) A 400 N block is resting on a rough horizontal surface as shown in Fig. 6c for which the coefficient of friction is 0.4. Determine the force P required to cause motion if applied to the block
horizontally. What minimum force is required to start motion?
5 M
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The previous example is quite impressive and seems to be complex. In the following sections we are going to see how to deal with such complex problems and how to perform inference on them, whatever
the model is. In practice, we will see that things are not as idyllic as they seem to be and there are a few restrictions. Moreover, as we saw in the first chapter, when one solves the problem of
inference, one has to deal with the NP-hard problem, which leads to algorithms that have an exponential time complexity.
Nevertheless there are dynamic programming algorithms that can be used to achieve a high degree of efficiency in many problems of inference.
We recall that inference means the computing of a posterior distribution of a subset of variables, given observed values of another subset of the variables of the model. Solving this problem in
general means we can choose any disjoint subsets.
Let be the set of all the variables in the graphical model and let Y and E be two... | {"url":"https://subscription.packtpub.com/book/data/9781784392055/2/ch02lvl1sec08/variable-elimination","timestamp":"2024-11-07T07:34:37Z","content_type":"text/html","content_length":"90230","record_id":"<urn:uuid:29363b14-3958-4304-b585-8d1e86a48e5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00500.warc.gz"} |
Gompertz Distribution: Simple Definition, PDF
Probability Distributions > Gompertz Distribution
What is the Gompertz Distribution?
The Gompertz distribution, named after Benjamin Gompertz, is an exponentially increasing, continuous probability distribution. It’s basically a truncated extreme value distribution (Johnson et. al,
1994). Therefore, it is also called an EVD Type I. Although the theoretical range is from zero to positive infinity, most applications for this distribution are for human mortality rates with a range
of 0 to ≅ 100. It is also used in other fields, including biology and demography.
Probability Density Function
Two shape parameters, δ and κ control the shape of the probability density function.
Gompertz distribution showing various values for shape parameters delta and kappa.
Formal definition
The Gompertz can be defined by the following differential equation (Marshall & Olkin, 2007):
When λ = 1, this becomes the exponential distribution.
Gompertz supposed that the hazard rate was your chances of death at time t. He concluded that this rate increased in geometrical progression:
When λ and ξ are both less than zero, this becomes the negative Gompertz distribution.
Relation to Other Distributions
The Gompertz is related to:
• The extreme value distribution: If X has the standard extreme value distribution for minimums, then the conditional distribution of X (given X ≥ 0 ) is a standard Gompertz.
• The exponential distribution: The exponential distribution is made up of limits of sequences of Gompertz distributions (Marshall & Olkin, 2007). Put more simply, a series of exponential
distributions can be combined to make a Gompertz. If X has a basic Gompertz dist. (with shape parameter a), then Y = e^X − 1 has an exponential distribution with rate parameter a.
• The Gompertz is a log-Weibull distribution This is because the log of the hazard rate is linear in t, giving: λ(t) = exp{α + Βt). The log of the hazard rate in the Weibull is also linear in t
Other Forms
Various forms of the Gompertz exist, in part because of its long history. It was first developed by Benjamin Gompertz in 1825, as a way to model age-specific mortality rates. Although it was used
widely in Victorian times, the Gompertz “law” lost popularity at the turn of the 20th century but is recently regaining ground in several different forms, which include:
• The generalized Gompertz with three parameters, introduced by El-Gohary et al. (2013).
• The beta-Gompertz, introduced by Ali et al. (2014), is a generalized version with four parameters.
• The negative Gompertz distribution has an additional negative rate of aging parameter.
• The generalized Gompertz distribution (GGD) differs from the “regular” distribution in that “it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape
parameter” (El-Gohary et. al).
• The Exponentiated Generalized Weibull-Gompertz Distribution generalizes several distributions, including the Gompertz.
El-Gohary, A et. al. (2013). The generalized Gompertz distribution. Applied Mathematical Modelling. Volume 37, Issues 1–2, January 2013, Pages 13–24
Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994), Continuous Univariate Distributions (Vol. I, 2nd ed.), New York: Wiley.
Marshall & Olkin (2007. Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Science & Business Media.
Rodriguez, G. Parametric Survival Models. Available here.
Comments? Need to post a correction? Please Contact Us. | {"url":"https://www.statisticshowto.com/gompertz-distribution/","timestamp":"2024-11-04T14:43:05Z","content_type":"text/html","content_length":"73398","record_id":"<urn:uuid:abd7c3d5-60ce-4aa5-9d52-0e2bcc70fa61>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00811.warc.gz"} |
Da Hood Values – Accurate Calculator Tool
This tool helps you calculate the in-game value of items in Da Hood quickly and accurately.
How to Use the Da Hood Values Calculator
This calculator helps you determine an overall value based on various in-game parameters in the game “Da Hood”. Below is a quick rundown on how to use it:
1. Enter values for each parameter: Weapon, Money, Influence, Inventory, Respect, and Level. Make sure to use numerical input.
2. Click on the “Calculate” button to compute the overall value.
3. The result will be shown in the “Result” field below the button.
How It Calculates the Results
The calculator uses the following formula to calculate the overall value:
Result = (Weapon * 0.3) + (Money * 0.2) + (Influence * 0.15) + (Inventory * 0.2) + (Respect * 0.1) + (Level * 0.05)
Each factor is given a different weight in the final computation to reflect its relative importance.
This calculator does not account for qualitative aspects such as skill or reputation, which may also play significant roles in the game. It strictly computes based on numerical input provided.
Use Cases for This Calculator
Calculating Average Income
Enter the incomes of residents of Da Hood to find the average income in the neighborhood. Simply input the values, click calculate, and get the average income instantly.
Evaluating Total Property Value
Use this calculator to add up the total value of all properties in Da Hood. Input the property values, hit calculate, and see the total property value quickly.
Assessing Crime Rate per Capita
Determine the crime rate per person in Da Hood by inputting the total number of crimes and the population. Click calculate to get the crime rate per capita in seconds.
Estimating Total Annual Tax Revenue
Calculate the total annual tax revenue of Da Hood by entering the tax rate and the total property value. Hit calculate to see how much tax revenue the neighborhood generates per year.
Projecting Population Growth
Forecast the population growth of Da Hood by entering the current population and the growth rate. Click calculate to predict the future population size of the neighborhood.
Determining Average Education Level
Find out the average education level in Da Hood by entering the education levels of residents. Hit calculate to see the average education level of the neighborhood.
Calculating Unemployment Rate
Determine the unemployment rate in Da Hood by inputting the number of unemployed individuals and the total labor force. Click calculate to get the unemployment rate percentage instantly.
Estimating Median Home Price
Estimate the median home price in Da Hood by entering the values of individual home prices. Hit calculate to find the median price of homes in the neighborhood.
Assessing Average Commute Time
Calculate the average commute time for residents of Da Hood by entering individual commute times. Click calculate to see the average time it takes to commute to work in the neighborhood.
Evaluating Percentage of Homeowners
Determine the percentage of homeowners in Da Hood by inputting the total number of homeowners and the population. Click calculate to see the percentage of residents who own their homes. | {"url":"https://madecalculators.com/da-hood-values/","timestamp":"2024-11-06T19:55:49Z","content_type":"text/html","content_length":"144825","record_id":"<urn:uuid:0762f6b2-8029-46cc-98d1-f18c6fda42db>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00480.warc.gz"} |
Transient structural analysis in ansys workbench tutorial pdf
In thissketch,drawacirclecentredattheoriginbyusingthe. Please practice handwashing and social distancing, and. Transient dynamic analysis is a technique used to determine the dynamic response of a
structure under a timevarying load. Understand what it takes to be a structural analysis engineer. Im confused about having 310 steps, each 1 second long, and taking 100 seconds of data to paste into
step 1. Ansys tutorial structural ansys structural tutorial ansys stress analysis tutorial. A transient structural analysis of a simply supported beam with udl has been performed using ansys
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Step by step procedure of how to do transient structural analysis varying load force with time of a bridge ansys workbench. Transient structural that is also termed as time history load is the type
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It is used across the globe in various industries such as aerospace, automotive, manufacturing, nuclear, electronics, biomedical, and so on. Inthissketch,drawacirclecentredattheoriginbyusingthe.
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what is a linear foot - what is a linear footwhat is a linear foot
When it comes to an online quote for a kitchen countertop, most people run into a problem. They do not know or have doubts about how to measure the countertop correctly. That is something that makes
it difficult to ask for different quotes online to compare. Today we will explain what is a linear foot, and how to ensure that we are providing the correct measures.
Most companies that offer the possibility of making estimates online will ask you for the countertop measurements to provide you with a price that will be subject to the review of those same
measurements by a company technician. So in this sense, you should not worry if you have passed any information that is not correct.
Since they will always give the final price after verifying that the kitchen countertop measurements are accurate. If they were not, the company would prepare a final budget with the professional
technician’s modifications.
Today, we will talk about the recommendations to measure a kitchen countertop to request quotes online correctly. If you want to know how not to make mistakes when measuring a countertop, do not
stop reading this article.
What is a linear foot?
The unit for measuring the length and width of an object is a linear foot. You will need to measure your linear foot when you are going to build a house. There is usually no difference between
regular feet and linear foot. This type of measurement typically used to denote the length of a piece of wood. The foot is no longer a commonly use measure. Although it standard in the United States
and is currently used in addition to the UK metric system measurement.
An ancient unit of linear foot measurement has used in one form or another for thousands of years. The foot’s exact size varied from country to country. And even from person to person and was not
officially standardize on international lines until 1959. When it gives a precise metric equivalent of 1 foot to 0.3048 meters. It is still in use today. Previously, it standardized one-foot length
with different lengths. That were about 0.0 meters apart in individual countries. This is possible because the size of the actual human foot varies from person to person. Board yards often use the
term linear foot exchange with feet to describe the length of the board. The term foot measurement also commonly used to measure wood and refers to an inch thick board area.
Now, possibly you would like
to know the exact price before calling any company to go and measure it. Since, if you finally do not buy the countertop from them. Most will charge the visit for incurring travel and handling costs.
Which is the work of the meter technician. Surely you want to compare the price of different companies.
But not pay the visit to each of them. Therefore, the key is to provide the correct measurements to each of these companies online so that the budgets that they offer you online are as tight and
accurate as possible. In this way, you can compare via the Internet and then decide the company you will buy the countertop from.
That will be the one that will perform the measurements at your home, but you will have ensured that the budget does not vary or that this variation is minimal. In addition to that, then they will
not charge you for the visit to measure the countertop.
At first, you have to know what form it will take the counter. It can be in L, in U, linear to the wall, two linear facing each other. It is not the most important thing since we will make a total
sum of linear meters in the end. But it is vital that you understand what to take into account when measuring, so you can know your countertop measurements, whatever their shape.
To know the countertop’s length, we must always measure from wall to wall and from behind.
How many centimeters is afoot?
Discover the equivalence of a foot with centimeters. Unit of measurement is used in Anglo-Saxon countries and is also used to express height. You can also discover this unit of measure’s origin and
other equivalences with the foot’s action.
The foot is a unit of measurement that helps us measure length. It used in the Anglo-Saxon measurement system. So its use is not widespread, although it is still an essential measurement in
One foot equals 12 inches. This measurement given by the estimated length of a human foot.
It causes discrepancies with other measurement systems since there are different foot measurements in ancient civilizations.
As we said, this measure of length used in countries such as the United States, Canada, and the United Kingdom. Especially to locate the height of an airplane or other aerial vehicles. However, it is
also common to use it as a measure to express measurements.
There also the timber foot or block foot used in the wood industry. But in this case, it is also a unit of volume. 1 lumber foot is equal to 2.359.737216 cm3.
The case of linear countertops is the simplest:
Measure from one end to the other, regardless of columns or other elements. Remember always to do it from behind, even if it seems to you that in this case, it does not matter, you will avoid
mistakes, and you will get used to always doing it the right way.
If they two linear facing each other and each of them align to a different wall, you only have to add them.
The longest side is the same as for a linear countertop – measure end to end, wall to wall.
Write down the total meters of side A
The other side of the L, painted in a lighter color in the image. As you can see, you should not measure to the end of the wall since the piece or square. Where the two sides coincide will already
covered by the measure.
Remember to measure from behind. In this case, it is crucial since if we did it from the front. We would let ourselves measure that same square that we talked about before belonging to side A
Write down the total meters of B.
Add A + B, and you will have a total of linear meters.
To measure a U-shaped countertop:
Make and record the measurements of A and B as we have explained in the L-shaped countertop.
The new side C, also indicated in the image in another color for clarity as possible, maybe the same length as A or be different, more or less long.
Wood is a natural material. Therefore, it has different defects and characteristics that need to know and consider for any application that you want to give it. The classification by categories of
the sawn wood allows determining to a large extent. The value and possible potential use of each table.
The grade of wood purchased by the manufacturer determines the cost and residual factor achieved.
Because grades based on the percentage of clean wood in the chart. Many of the hardwoods’ natural aesthetic characteristics not considered when calculating net yield.
Higher grades, including FAS, One-Sided FAS (FAS / 1F), and select best suited for long moldings. The joinery products such as door frames and architectural interiors. And furniture applications,
which require a high percentage broad cuts.
FAS ONE SIDE (FIF)
It derived from the original “First And Seconds” grade. The FAS grade provides the buyer with long, clean cuts best suited for high-quality furniture, interior joinery, and solid wood trim. The
minimum size of the boards is 6″ comprehensive and eight ‘long.
Besides, it includes a range of boards that go from 83 1/3% (10 / 12th) to 100% clean woodcuts on the board’s total surface. Clean cuts must be a minimum size of 3″ wide by seven ‘long, or 4″ wide by
5’ long. The number of reductions allowed depends on the table’s size.
Although most tables will enable one to two cuts. The minimum width and length may vary depending on the table’s species and green or stove.
The best face must meet all the fas requirements, while the poor face must meet Common Mo grade requirements. 1. Thus, the buyer is assured that he will have at least one FAS face.
FOOT TABLE: What is a linear foot?
A board foot is a unit in which wood is measured.
One board foot equals one foot long x one foot wide x one inch thick (one foot = 0.305m, one inch = 25.4mm).
The formula to determine a board foot on a board or plank is:
(Width in inches x lake in feet x thickness in inches) divided by 12.
The percentages of clean wood required for each grade are based on this 12′ measurement.
The table above is 2 “thick, 6¼” wide, and 8 ‘long, so 6¼ “÷ 12 x 2” x 8’.
When preparing a package, the width and length of the boards are recorded. Widths greater or less than a half-inch are rounded to the nearest whole inch, while widths that equal precisely one-half
inch are rounded either up or down randomly. Lengths between full foot unit increments are rounded to the next lower integer. For example, the values for a 5 ¼” wide and 8 ½” long table are set to
5″ and 8 ‘.
FAQ of What is a linear foot?
1. How to calculate the board feet that are in a wooden package?
First, calculate the surface measurement of a layer of boards. The package’s width, minus the spaces between the tables. By the length of the box and divide the result by 12. If there are different
lengths in the package, use an average size. Once the measurement of a layer is calculated, multiply the result by the total number of layers.
2. How long is a linear foot?
It is used in the Anglo-Saxon measurement system, so its use is not widespread, although it is still an essential measurement in aeronautics. One foot equals 12 inches.
3. How to calculate linear feet to square feet?
To calculate a room’s area in square feet, measure the room’s length and width in feet, then multiply these figures together to get a place in square feet. For example, a 12-foot x 15-foot room can
be said to have an area of 180 square feet (12 x 15 = 180).
4. How much is a foot of wood in cm?
In the metric system, a square foot of wood is equivalent to a square piece with a volume of 30.48 cm on a side by 2.54 cm thick. Its equivalent in the Anglo-Saxon system is 12 inches per side by 1
inch thick.
5. How to calculate square feet for tile?
In the slabs, you should always buy more. To calculate the amount of glue needed, you divide the area by 40, which is the approximate square footage that a 50 lb. bag will cover with a 1/2 “x 1/2”
7. How to calculate the amount of wall ceramic?
To calculate how many tiles we need, we must multiply the amount that can fit in a square meter by the total meters of the surface, 12 m2. Knowing the number of units required, we can calculate how
many boxes of each type of ceramic we need.
8. How does the foot abbreviate?
The foot abbreviates with the letters ft from the word feet, feet in English, which gives its name to this measurement unit.
Conclusion of what is a linear foot?
Generally, wood classified based on the size and number of cuts obtained from aboard and use in the manufacture of products. High grades provide the buyer with long clean pieces while cutting common
ones back into smaller clean pieces.
Common grades, mainly standard number 1 (No. 1c) and typical number 2 (No. 2C), seem to be the most appropriate for the kitchen cabinet industry. For most furniture, and flooring. Planked or stave
wood. You should note that you will obtain the same clean high-grade wood once the common grade pieces are trimmed. But in minor cuts (either lengthwise or widthwise). The grade name is only used to
designate the percentage of clean wood in the boards and not its overall appearance. | {"url":"https://www.technewsera.com/what-is-a-linear-foot/","timestamp":"2024-11-10T15:08:43Z","content_type":"text/html","content_length":"95439","record_id":"<urn:uuid:7cd287f9-55eb-47de-8c17-184cd6ac4119>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00715.warc.gz"} |
SS-V: 5070 Pinched Hemispherical Shell
Test No. VNL08 Find displacements of a hemispherical shell loaded with inward and outward concentrated forces.
A hemispherical shell is loaded with inward and outward concentrated forces at point A and point B, respectively. The hemisphere has an 18 degree hole at the top and the quadrant of the hemisphere is
modeled utilizing symmetric boundary conditions (
Figure 1
). Correspondingly, forces P shown in
Figure 1
are acting on the quadrant. Displacement at points A and B are to be determined for force values P = 40, 60, and 100 lbf.
Figure 1.
The material properties are:
Modulus of Elasticity
6.825e+7 psi
Poisson's Ratio
Symmetry conditions were simulated via sliding boundary conditions applied at faces coinciding with symmetry planes (Figure 2). Concentrated forces were applied in points of outer face of the sphere
(Figure 3). In order to eliminate rigid body motion along Z-axis a point on the top of the sphere was constrained in Z-direction (Figure 4).
The following tables summarize the comparison results.
Load [lbf] Ref Solution *, Ux at Point A [in] SimSolid, Displacement Uy at Point A [in] % Difference
40 -3.280 -3.148 -4.02%
60 -4.360 -4.120 -5.50
100 -5.950 -5.482 -7.87
Load [lbf] Ref Solution *, Uy at Point B [in] SimSolid, Displacement Ux at Point B [in] % Difference
40 -2.330 -2.268 -2.66%
60 -2.830 -2.725 -3.71%
100 -3.430 -3.255 -5.10%
* Ref Solution is a thin shell model
Figure 2.
Figure 3.
Figure 4.
^1 Test 3DNLG-9 from NAFEMS Publication R0024 “A Review of Benchmark Problems for Geometric Non-linear Behaviour of 3-D Beams and Shells (SUMMARY).” | {"url":"https://help.altair.com/ss/en_us/topics/simsolid/verification%20manual/nonlinear_analysis_test_problem_vnl08_r.htm","timestamp":"2024-11-05T03:44:59Z","content_type":"application/xhtml+xml","content_length":"58162","record_id":"<urn:uuid:c8cb83cd-5572-4663-9a88-2bbee0d2a86b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00178.warc.gz"} |
Re: st: calculating group means
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Re: st: calculating group means
From David Kantor <[email protected]>
To [email protected]
Subject Re: st: calculating group means
Date Thu, 25 Jul 2002 18:39:53 -0400
At 04:40 PM 7/25/2002 -0400, Gayatri Koolwal. wrote:
I have a data set of individuals who belong to different households, each with a value of a variable x2. For each household, I would like to create a new variable that is the mean of x2 for the
individuals in that household. Would anyone be able to help me with the procedure for this? Do I need to collapse the data set first somehow?
As Marcela Perticara indicated, you can use egen ... mean ... by(householdid)
But note that there are two possibilities. If you want to take a household mean and spread it onto your set of individuals (by household), then use egen ... mean ... by(householdid). But if you want
to reduce to a set of households (i.e., one observation per household), then -collapse- is more convenient. (You could still use egen ... mean, but you would then want to reduce it to one
representative per household.)
I hope this helps.
-- David
David Kantor
Institute for Policy Studies
Johns Hopkins University
[email protected]
* 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/2002-07/msg00464.html","timestamp":"2024-11-10T10:38:10Z","content_type":"text/html","content_length":"7979","record_id":"<urn:uuid:acdc264c-de09-4de8-b2d5-8042dfeac9d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00430.warc.gz"} |
Symmetric Square Puzzle | Math = Love
This blog post contains Amazon affiliate links. As an Amazon Associate, I earn a small commission from qualifying purchases.
Today I finally got around to taking down the last puzzle we tackled before Christmas Break. It is always my aim to switch out my puzzles on a weekly basis, but sometimes the hectic nature of the
classroom prevails. This was a reminder that I haven’t shared this symmetric square puzzle here on the blog yet.
This Symmetric Square puzzle is from Robert Allen’s Mensa All-Color Puzzle Book. It’s an okay puzzle book for personal puzzling, but I struggled to find enough puzzles with classroom applications to
really recommend it for teachers to purchase.
Students are given ten tiles which must be arranged in such a way that a square is formed in which each horizontal line matches up with a vertical line. You should notice quite quickly that this
partial attempted solution does not work.
Only a day after I tweeted a picture of the puzzle on my dry erase board, Tracy Esposito was sending me pictures of student reflections after having her year 6 class tackle this puzzle. How awesome
is that?!?
Puzzle Solutions
I intentionally do not make answers to the printable math puzzles I share on my blog available online because I strive to provide learning experiences for my students that are non-google-able. I
would like other teachers to be able to use these puzzles in their classrooms as well without the solutions being easily found on the Internet.
However, I do recognize that us teachers are busy people and sometimes need to quickly reference an answer key to see if a student has solved a puzzle correctly or to see if they have interpreted the
instructions properly.
If you are a teacher who is using these puzzles in your classroom, please send me an email at sarah@mathequalslove.net with information about what you teach and where you teach. I will be happy to
forward an answer key to you.
Not a teacher? Go ahead and send me an email as well. Just let me know what you are using the puzzles for. I am continually in awe of how many people are using these puzzles with scouting groups,
with senior adults battling dementia, as fun activities in their workplace, or as a birthday party escape room.
More Puzzles with Movable Pieces
4 Comments
1. Your blog has inspired my teaching SO MUCH. I teach grade 7 in Vancouver BC. I hope you know how much we appreciate the effort you put into this blog. THANK YOU!
2. A little confused… The square piece has two 6s, and there is a 2×1 piece with a 0 & 9. Those are both obvious because they're underlined. Then there's a 3×1 piece that doesn't have any
underlining, but based on the direction of the 5 it seems to read 5,8,6. Is the 6 not underlined on purpose, or just because it's unnecessary since the orientation of the 5 gives it away? Or are
they supposed to struggle with if it's a 6 or 9?
1. That was just an oversight… It should be underlined.
3. Hi Sarah.
Thank you so much.
It's a great job you're doing here. | {"url":"https://mathequalslove.net/symmetric-square-puzzle/","timestamp":"2024-11-08T11:37:07Z","content_type":"text/html","content_length":"261282","record_id":"<urn:uuid:96a65ba8-5d97-4471-be30-d80af6eeac83>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00674.warc.gz"} |
Commission Formula
Hope you're all well.
I’m hitting a brick wall inserting a formula into a tracker for my sales teams commission structure which is being redone.
It’s a tiered system whereby the salesperson will get commission in brackets until they reach target, and then a higher percentage on anything over target; (0.1% for the first 20%) of their target,
(0.2% on the next 20%), (0.3% on the next 20%) so on.
Until they reach target, at which point everything over target they are rewarded at 2%.
If they are under target, they will still receive commission, but only within the brackets.
If anyone who’s done a similar structure that would be able to take a look and let me know their thoughts, it would be really appreciated, as I seem to be going around in circles.
I've extracted a section and published it below.
Thanks in advanced, any suggestions welcome :D
Best Answer
• Hi @Paul Newcome @Glen Urquhart
Hope you are fine, i do it using the following formula, please confirm if this ok for you or we must change the factors of calculation.
=IF([Month End Sales Actual]@row > [2021 Target]@row, (0.001 * [First 20%]@row + 0.002 * [Second 20%]@row + 0.003 * [Third 20%]@row + 0.004 * [Last 20%]@row + ([Over Target]@row * 0.02)), (IF(AND
([Month End Sales Actual]@row <= [Last 20%]@row, [Month End Sales Actual]@row > [Third 20%]@row), ((0.001 * [First 20%]@row + 0.002 * [Second 20%]@row + 0.003 * [Third 20%]@row + (([Month End
Sales Actual]@row - [Third 20%]@row) * 0.004))), (IF(AND([Month End Sales Actual]@row > [Second 20%]@row, [Month End Sales Actual]@row <= [Third 20%]@row), ((0.001 * [First 20%]@row + 0.002 *
[Second 20%]@row + ([Month End Sales Actual]@row - [Second 20%]@row) * 0.003)), (IF(AND([Month End Sales Actual]@row > [First 20%]@row, [Month End Sales Actual]@row <= [Second 20%]@row), (0.001 *
[First 20%]@row + ([Month End Sales Actual]@row - [First 20%]@row) * 0.002), (IF(AND([Month End Sales Actual]@row > Divider@row, [Month End Sales Actual]@row <= [First 20%]@row), 0.001 * ([Month
End Sales Actual]@row - Divider@row))))))))))
☑️ Are you satisfied with my answer to your question? Please help the Community by marking it as an ( Accepted Answer), and I will be grateful for your "Vote Up" or "Insightful"
• Hi @Glen Urquhart
Hope you are fine, i designed the following sheet using the information you mentioned in your question to calculate the Actual Commission please check.
☑️ Are you satisfied with my answer to your question? Please help the Community by marking it as an ( Accepted Answer), and I will be grateful for your "Vote Up" or "Insightful"
• I want to make sure I understand this correctly...
Lets assume a target of $10,000.00 for easier math.
If they reach their target, do they get 2% of 10,000, or is it
.1% of 2,000
.2% of 2,000
.3% of 2,000
.4% of 2,000
.5% of 2,000
all added together?
Then if they were to go to maybe $15,000 they would get 2$ of 5,000?
• Hi @Paul Newcome @Glen Urquhart
Hope you are fine, i do it using the following formula, please confirm if this ok for you or we must change the factors of calculation.
=IF([Month End Sales Actual]@row > [2021 Target]@row, (0.001 * [First 20%]@row + 0.002 * [Second 20%]@row + 0.003 * [Third 20%]@row + 0.004 * [Last 20%]@row + ([Over Target]@row * 0.02)), (IF(AND
([Month End Sales Actual]@row <= [Last 20%]@row, [Month End Sales Actual]@row > [Third 20%]@row), ((0.001 * [First 20%]@row + 0.002 * [Second 20%]@row + 0.003 * [Third 20%]@row + (([Month End
Sales Actual]@row - [Third 20%]@row) * 0.004))), (IF(AND([Month End Sales Actual]@row > [Second 20%]@row, [Month End Sales Actual]@row <= [Third 20%]@row), ((0.001 * [First 20%]@row + 0.002 *
[Second 20%]@row + ([Month End Sales Actual]@row - [Second 20%]@row) * 0.003)), (IF(AND([Month End Sales Actual]@row > [First 20%]@row, [Month End Sales Actual]@row <= [Second 20%]@row), (0.001 *
[First 20%]@row + ([Month End Sales Actual]@row - [First 20%]@row) * 0.002), (IF(AND([Month End Sales Actual]@row > Divider@row, [Month End Sales Actual]@row <= [First 20%]@row), 0.001 * ([Month
End Sales Actual]@row - Divider@row))))))))))
☑️ Are you satisfied with my answer to your question? Please help the Community by marking it as an ( Accepted Answer), and I will be grateful for your "Vote Up" or "Insightful"
• After taking another look at the sheet you provided... Where exactly are you wanting to populate the formula(s)?
• Hi both, thank you for your input.
Correct- as they increase through the tiers, the commision increases as you say.
0.2% of the first 2k
0.3% of the next 2k, so on.
And then any sales that surpass the target 2% on the 'over'
i.e target of 10k, and sales of 15k, would result in 2% of the 5k over the target.
If they achieve for example 5k sales however, they should recieve:
0.1% of the first 2k
0.2% of the second 2k
0.3% of the remaining 1k.
= 2 + 4 + 3...
The numbers are actually different per salesperson / area / agreed percentage etc, but i can't figue the basis for it.
I'm looking for a formula to populate the (now) red filled cells.
• The template was inspired by the furthest to the right "Tiered Commision" scheme on the attached, however there are a couple of errors in the original;
• It may be that the first approch was over complicated for what is actually needed, and infact the below would suffice, however im still having trouble with these ifs rules, is the below formula
description logical / possible?
• Hi Both,
Thank you for your help and input.
I have achieved what I was looking to have as an output now.
@Paul Newcome , it was actually a thread that you answered in 2018 that wa the final piece to my puzzle.
The below is an example of what I was going for:
The "Actual sales" field is fed live from another sheet, which is linked to our CRM via Automate.io, collecting sales reps monthly sales figures.
Thanks again for your help and comments.
• Glad you were able to get it working! 👍️
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/75219/commission-formula","timestamp":"2024-11-13T14:57:27Z","content_type":"text/html","content_length":"435377","record_id":"<urn:uuid:9ce39ef2-8584-47ea-bb37-fbf915b9bce1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00885.warc.gz"} |
So You Like Your Food Hot?
Problem F
So You Like Your Food Hot?
Peter is co-owner of the incredibly successful Pete and Pat’s Pitas and Pizzas and his sales are on fire! But unfortunately, so is his building, due to carelessly laid delivery boxes placed too close
to Pete’s famous wood burning pizza oven. After sifting though the remnants, one of the few things Pete is able to salvage is a ledger book, but the only thing he can make out on the charred pages is
the profit he made during the last month. The insurance company would like to know how many pitas and how many pizzas Pete actually delivered over that period. Pete does recall how much profit he
makes on each of these products, so he’s pretty confident that he can determine how many of each were sold during the last month given the total profit. Well perhaps “confident” is not the exact word
Peter is looking for – it’s more like clueless. Can you help Pete out? I’m sure there are some discount coupons in it for you, or at least a really cheap price on a used pizza oven.
Input consists of a single line containing $3$ values $p_ t$ $p_1$ $p_2$, where $0 \leq p_ t \leq 10\, 000.00$ is the profit for the month and $0 < p_1, p_2 \leq 100.00$ are the profits Pete makes on
a pita ($p_1$) and on a pizza ($p_2$). All values are in dollars and cents.
Display two integers: the number of pitas sold and the number of pizzas sold so that the total profit equals the value given. If there is more than one combination of pitas and pizzas that give the
specified profit, list them all, one combination per line, listing the combination with the smallest number of pitas first, then the combination with the second smallest number of pitas, and so on.
If there are no combinations of pizza and pita sales that realize the profit, output none.
Sample Input 1 Sample Output 1
725.85 1.71 2.38 199 162
Sample Input 2 Sample Output 2
100.00 20.00 10.00 2 6 | {"url":"https://nus.kattis.com/courses/CS3233/CS3233_S2_AY1920/assignments/ok5kbf/problems/soyoulikeyourfoodhot","timestamp":"2024-11-07T22:30:45Z","content_type":"text/html","content_length":"29174","record_id":"<urn:uuid:2145768b-f15c-45e2-b9d8-de6d1c9703be>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00543.warc.gz"} |
My formula is broken, not sure house to fix | Microsoft Community Hub
Forum Discussion
My formula is broken, not sure house to fix
Here is my formula =IFS(D22-D18>D18,{" "},D22-D18<D18,{"0"},D22-D18=D17,{"5000000"}) It is broken at the first part, I need it to come to a number that is between $3M and $5M, how do I get a
I don't know what you are trying to do / why it is 'broken'. The formula itself 'works' but not sure why you did a few things:
a) why is each answer formatted as a set of 1 (i.e. {" "} ) you don't need the set around those.
b) why is the 3rd condition =D17 instead of D18? you realize that if D22-D18 = D18 and D18 <> D17 then there is no correct response excel will return #N/A
c) are you sure you want those numbers 0 and 5000000 formatted as TEXT?
so maybe you want:
=IFS(D22-D18>D18,"", D22-D18<D18,0, D22-D18=D18, 5000000)
this is what I am attempting to do is take the d22 less d18 if larger then the difference between d22, d18 and d17 if smaller then 0, or if it higher the d17 make it the amount in d17.
I get the 500,000 but I can't seem to find the formula for the difference between d22-d18>d17,'"
can you help?
• The now information is useful, but I still do not really understand exactly what it is you hope to see. My formula returned 500,000 but I wouldn't know whether that is good or bad.
= IFS(
paidLoss-retention > limit, limit,
paidLoss-retention < retention, 0,
TRUE, 500000
□ I am attempting to make a figure out how to write a formal to determine if the paid loss is below retention, for a zero to enter, if not what is the difference between paid loss, retention
and limit, and if the loss is higher then the limit to have the limit entered.
Does that makes sense?
☆ This tries to follow your written criteria as I followed them
= IFS(
paidLoss-retention < 0, 0,
paidLoss-retention > limit, limit,
TRUE, paidLoss-retention
If that is correct, an alternative formula might be
= MEDIAN(0, limit, paidLoss- retention)
[p.s. I hope that the use of names makes the formula more meaningful written here, evenif you doen't use them in your workbooks] | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/my-formula-is-broken-not-sure-house-to-fix/4093231/replies/4093289","timestamp":"2024-11-10T18:14:51Z","content_type":"text/html","content_length":"311161","record_id":"<urn:uuid:197d535a-275e-46ae-b0f2-864915118c54>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00788.warc.gz"} |
Deep Learning with R - The Best Book to Start With - reason.townDeep Learning with R – The Best Book to Start With
Deep Learning with R – The Best Book to Start With
Deep Learning with R is the best book to start your journey with deep learning. This book provides a comprehensive introduction to the subject with clear and concise explanations.
Checkout this video:
Why deep learning with R is the best book to start with?
If you’re looking for a book that will teach you the basics of deep learning with R, then “Deep Learning with R” is the best choice. Written by Francois Chollet, the creator of the Keras deep
learning library, this book is a great way to get started with deep learning.
Chollet begins by explaining what deep learning is and why it is such a powerful tool for data analysis. He then walks readers through the steps of building a simple neural network using the Keras
library. Throughout the book, Chollet provides clear and concise explanations of the concepts behind deep learning.
“Deep Learning with R” is an excellent resource for anyone who wants to learn more about this exciting field.
What makes deep learning with R so special?
There are a lot of great things about the deep learning with R book. Firstly, it provides a very gentle introduction to the subject matter. Secondly, it covers all of the major concepts and
techniques in a very concise and straightforward manner. Thirdly, it includes a lot of code examples to help you better understand the techniques. Finally, it also has a very good bibliography and
index so you can easily find more information on the topics that interest you.
How deep learning with R can help you achieve success?
Deep learning is a subset of machine learning that is concerned with algorithms inspired by the structure and function of the brain. Deep learning networks are also called artificial neural networks
R is a programming language and free software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing. The R language is widely used among
statisticians and data miners for developing statistical software and data analysis.
There are many books that talk about deep learning, but how do you know which one is the best to start with? If you’re looking for a book that will help you get started with deep learning using R,
then look no further than Deep Learning with R by François Chollet. This book provides a comprehensive and easy-to-understand introduction to the fundamentals of deep learning. It also covers how to
develop, train, and deploy neural network models using the R programming language.
If you’re new to deep learning, or if you’re looking for a book that will help you get started with using R for deep learning, then Deep Learning with R is the perfect book for you!
The benefits of deep learning with R
deeper understanding of how to get started with deep learning using R. The book will guide you through the basics of deep learning, starting with an introduction to the basic concepts and working
your way up to more advanced topics such as convolutional neural networks and sequence modeling. There is also a section on practical applications of deep learning, with worked examples in R that
show you how to build models for tasks such as image classification, text classification, and time series forecasting.
The key features of deep learning with R
There are many key features of deep learning with R that make it an excellent choice for those looking to get started with the subject. For starters, deep learning with R is designed to be beginner
friendly, and does an excellent job of explaining all the concepts in a clear and concise manner. Additionally, deep learning with R comes with a number of code examples which helps to consolidate
the knowledge learned in the book. Finally, deep learning with R is also packed with information on how to apply deep learning in real world scenarios.
How to get started with deep learning with R
Deep learning is a subset of machine learning that is inspired by how the brain works. Deep learning algorithms are able to learn from data in a way that is similar to how humans learn. This book
will teach you how to get started with deep learning using the R programming language.
The different types of deep learning with R
Deep learning is a subset of machine learning that is concerned with models inspired by the structure and function of the brain called artificial neural networks. Neural networks are a set of
algorithms that are designed to recognize patterns. They interpret sensory data such as images, sound, and text into predetermined classes or interpretations.
Deep learning is used to power applications like computer vision, natural language processing, and speech recognition. It is also used in healthcare, finance, agriculture, and many other industries.
There are different types of deep learning: supervised, unsupervised, and semi-supervised. Supervised deep learning requires that the data be labeled in order for the algorithm to learn from it.
Unsupervised deep learning does not require labels but instead relies on algorithms to find structure in data. Semi-supervised deep learning is a mix of both supervised and unsupervised deep learning
where some of the data is labeled and some is not.
R is a programming language that is widely used for statistical computing and graphics. It is also a popular language for machine learning and deep learning. R has many packages that make it easy to
develop deep learning models. In this book, you will learn how to use R to develop different types of deep learning models for applications such as computer vision, natural language processing, and
time series forecasting.
The challenges of deep learning with R
Deep learning is a branch of machine learning that is concerned with teaching computers to learn from data in a way that is similar to the way humans learn. It is a very powerful tool that has been
used to create some of the most successful machine learning applications, such as Google Translate and self-driving cars.
However, deep learning is also a very challenging field, and there are not many resources available for people who want to learn about it. This is why we have written Deep Learning with R – The Best
Book to Start With.
This book is aimed at people who are new to deep learning and want to get started with using R to build deep learning models. It covers the basics of deep learning, including how to train and
evaluate models, as well as how to avoid overfitting.
We also provide practical guidance on how to apply deep learning to real-world problems, such as image classification and natural language processing.
Deep Learning with R – The Best Book to Start With is available now from Amazon.
The future of deep learning with R
Deep learning has been gaining a lot of popularity lately, and it is no surprise that the R programming language has been one of the main driving forces behind this popularity. R is a perfect
language for data analysis and machine learning, and it offers a wide range of deep learning libraries that can be used to develop powerful models.
In this book, we will take a look at the basics of deep learning with R. We will start by taking a look at the fundamental concepts of deep learning, and then we will move on to actual implementation
using the popular TensorFlow library. By the end of this book, you will have a good understanding of how deep learning works, and you will be able to build your own deep learning models using R.
Why you should read deep learning with R
If you want to know everything about deep learning, this is the book for you. It starts with the basics, covering all the important concepts in a simple and easy-to-understand way. The book then
gradually moves on to more advanced topics, such as convolutional neural networks and recurrent neural networks.
But this book is not only about deep learning. It is also about R. R is a programming language that is widely used in the data science community. This book shows you how to use R to build deep
learning models. You will learn how to install R and all the necessary packages, how to clean and prepare data for modeling, and how to evaluate the performance of your models.
Deep Learning with R is the perfect book for anyone who wants to get started with deep learning. It is also a great reference for experienced data scientists who want to use R to build deep learning | {"url":"https://reason.town/deep-learning-r-book/","timestamp":"2024-11-14T18:57:51Z","content_type":"text/html","content_length":"99139","record_id":"<urn:uuid:7784a242-4fc7-432e-9d03-847e87d7e5e2>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00550.warc.gz"} |
Deficiency subspace
From Encyclopedia of Mathematics
defect subspace, defective subspace, of an operator
The orthogonal complement resolvent of
Deficiency subspaces play an important role in constructing the extensions of a symmetric operator to a maximal operator or to a self-adjoint (hyper-maximal) operator.
[1] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian)
[2] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in a Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[3] N. Dunford, J.T. Schwartz, "Linear operators" , 1–2 , Interscience (1958–1963)
[4] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
The definition of a regular value of an operator as given above is not quite correct and should read as follows. The value
How to Cite This Entry:
Deficiency subspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deficiency_subspace&oldid=15718
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Deficiency_subspace&oldid=15718","timestamp":"2024-11-12T09:12:16Z","content_type":"text/html","content_length":"20390","record_id":"<urn:uuid:25617172-f4ba-439a-986b-5c3d569dff9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00355.warc.gz"} |
: Having trouble using template problems from Wiki
Im sure Im doing something wrong, just cant figure out what the issue is this morning.
I am trying to create essay problems. I started by using the problems posted at the wiki. For example, the following problem is found doing a search on the Wiki for essay problems.
I tried copying this problem into a text editor and then uploaded to a course to access via "local problems" using the browser.
I have tried several problems and I keep getting the same error:
Problem1 ERROR caught by Translator while processing problem file:WWprobs/Essay2.pg **************** Unrecognized character \xEF; marked by <-- HERE after ExPr__; <-- HERE near column 17 at line 1 of (eval 1915)
Here is the file I made that produces the above error:
#Load the Essay Macros
$m = random(2,5);
$b = random(20,30);
Context()->variables->add(D => 'Real');
$F = Formula("$m D + $b");
$y1 = $F->eval(D=>1);
$y2 = $F->eval(D=>2);
The cost to download data using your phone is a linear function of the amount of data downloaded. Suppose it costs $y1 dollars to download 1 gigabyte of data and $y2 dollars to download 2 gigabytes
of data.
Find a formula for the cost C in terms of the amount downloaded D. $BR
What do the slope and y-intercept of this function represent? $BR
\{#Put an Essay Box where ever you want a essay type answer\}
#Essay Boxs use the essay_cmp evaluator.
I found how to find the Wiki template problems in the NPL using the browser. The location is posted at the top of the example (not sure how I missed that)
However, I would still like to know why the above pg file wont work. I copied and pasted from the wiki to a text editor and then saved as a text file. Then uploaded to course and viewed in the local
area using the browser. I think the problem has to do with how I am saving the text file, but I dont recall having issues before. It looks fine when viewing with the editor in file manager. The above
file was copied to this forum directly from the file manager editor.
I forgot to put the header in the file. Moderator, please feel free to delete this thread. | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=3105","timestamp":"2024-11-13T01:39:40Z","content_type":"text/html","content_length":"83324","record_id":"<urn:uuid:f751de98-ffa7-4170-8fd2-6c1945028331>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00262.warc.gz"} |
How does geometry apply to photographs? Read more on the RetouchMe blog.
At school during math and geometry classes, students usually yawn out of boredom looking at these geometry pictures of different shapes. And this is mainly due to the fact that if a person does not
understand why he needs this knowledge and how it can be used, the subject will simply not be interesting to him. And this is normal. In our today's article, we will show you how your knowledge of
geometry can be useful in photography. Whether you are a beginner or an advanced photographer, theoretical knowledge of how to apply geometry together with the magic of numbers in your profession or
hobby will definitely not be superfluous, and on the contrary, will bring more charm and creativity to your photos.
How is geometry applicable to photos?
Before we go deep into geometric photography ideas, let's start with the basics. Why do we need geometry in photography at all, don't we just take pictures of whatever we are interested in? Geometry
is the basis of many fundamental rules in nature, which apply just as well to photography. In particular, knowing rules such as the rule of the golden ratio or the rule of thirds can help you frame
your composition in a way that looks harmonious and creative. Even people who know nothing about geometry will notice the difference. And all because the harmony in the image is visible even to the
naked eye, and we notice it on some unconscious, intuitive level.
Geometric pattern images
Have you ever admired flowers for example? What is so fascinating about plants? And all the matter is in proportion, which was noticed even by the ancients. This can be obtained by dividing a line
into two parts so that the long part corresponds with the short one in the same proportion as the whole line corresponds with the long one. It is the rule of the golden ratio, and the proportion is
calculated by the number Phi which is equal to 1,618. At the level of numbers, the rule of the golden ratio is traced in such sequence 1-2-3-5-8-8-13-21. In this sequence, each following number is
equal to the sum of two previous numbers and if we take two numbers following each other in this sequence and then divide them on each other, then we receive the number Phi. This is called the
Fibonacci sequence. Let's see how the rule of the golden ratio will look on the graph.
Even without delving deep into fundamentals, we can see the pattern to which many natural phenomena can correspond. The way flowers grow, spider net developed, the way hurricanes and cyclones are
formed are all images of geometry following one of the fundamental rules. Even if we go beyond our planet and out of the solar system, we will see that the rule of the golden ratio applies to
galaxies formation with the same geometric aesthetic. It turns out that in math and geometry, there is a kind of magic of numbers and shapes that are constantly interacting with each other. And when
we are actively interested in such facts, we will start noticing them everywhere, including photography. It is enough just to think at the level of subconsciousness, and we will be able to say which
frame we like more and which one we like less with high probability. The photo which will have more harmony in it would be connected with numbers and geometry to some extent and please our eyes.
Rule of Thirds
So, what other geometry pics can we obtain besides the Fibonacci sequence and the golden ratio to our composition in photography? We mentioned the rule of thirds earlier, let's break down what it is.
If you have ever used a camera, you have probably noticed that looking through the viewfinder or your camera's display you may see the grid. This grid is there for a reason, it represents the rule of
thirds. The rule of thirds on this grid is made as two vertical and two horizontal lines crossing each other at 90 degrees and thus forming 9 equal sections with 4 cross-sections. These intersections
of the lines are the power points. If we place the main subject of our photograph in a certain way above or below the lines, or on the crossing point, it will give different effects of perception of
the overall composition.
Geometry and composition rules
Let us go through a few examples of how you can easily apply the rule of thirds to your geometric form photography. First, if you do not have the rule of thirds grid turned on by default, find it and
turn it on. Once enabled, we need to think about what we want to photograph. We can start with the most available and easiest to find and go to photographing landscapes. Get out with our camera away
from the hustle and bustle of the city, to a place where we can see the horizon line and large open spaces. If we align the horizon line with one of the grid lines, we will get a more harmonious shot
than exposing the horizon line in the center of the frame.
But which line is better to set the horizon line on? You need to evaluate which part of the frame is more interesting for your specific geometry photos. What looks more attractive, the sky or the
ground? For example, there is an airplane flying in the sky, or beautiful clouds, lots of birds, and so on, then we level the horizon line with the bottom line, thus allowing the sky to occupy most
of our frame. If the horizon stretches along the sea line, you have a beautiful sunset that is reflected in the water and stretches across the entire width of the frame, it will be better to capture
less sky and more land or the water surface. So we even out the top line with the horizon line. These were the compositional basics for landscape photography.
Rule of thirds in portraits
Let us now consider how the rule of thirds is used in portrait or more subjective photography. We mentioned earlier the power points of intersections. These are where our subjects of interest should
be placed in the frame. Here everything works on the same principle that we applied in landscape photography: we try to avoid rigid centering of the subject in the frame, and line intersection points
help us in this by setting the right coordinates. You can point your camera lens at a person's face so that one of their eyes is at the intersection of the lines, for example in the upper right
intersection. Make sure that you do not leave any empty space towards the subject, otherwise, the shot can appear as intrusive as if we positioned our model just in the center of the frame. Empty
spaces can be distracting and ruin our narrative. But why does this happen? That's where the rule of symmetry comes in.
Symmetry in geometric design images
Symmetry in photography can also form harmony at the expense of order. It can be direct or indirect depending on our tasks. As we found out in the example above when working with the rule of thirds,
we do not want to have a lot of empty spaces in the frame. This is passive symmetry, which allows us to emphasize our main subject by filling in the blanks with something less important.
Symmetry can also be active or direct. Returning to landscape photography, we can use this compositional rule if we have an expanse of rivers or lakes and mountains in front of us. We can find the
reflection of our hill or mountain in the river, thus mirroring the image, which has a little more meaning than just a beautiful view of nature. There is nothing to stop you from applying the rule of
thirds and the golden ratio.
Rules interactions
The rule of thirds is a simplified version of the golden ratio and symmetry rules. If we display the rule of thirds grid, we can superimpose the golden section graph on top and see that the
concentric formula will be going to infinity exactly at the point where the lines of the rule of thirds cross each other forming power points.
Application in architecture
Symmetry is perfectly applied in photography of architecture, buildings, and landscapes which allows us to bring a variety of spatial perspectives. You just need to learn to notice it and adjust the
composition of your shot accordingly. For example, you are in the center of your city, where business centers grow like mushrooms and skyscrapers rise high into the sky. Most of these buildings are
proportioned and may provide simple geometric images. The exterior is usually made up of solid windows or reflective surfaces in one way or another. We have already learned how to use reflections in
nature, so try to find an interesting angle that will also allow you to capture geometric shapes pictures seasoned with symmetry in the frame.
Symmetry has good use in repetition. Imagine you have perfectly lined bicycles parked nearby some KFC or McDonald's. You can frame through that on to the brand logo telling the viewer how convenient
it is to get to your restaurant or any other service. And we can find those shapes and repetitions everywhere to shoot through, thus framing our main subject and accentuating the viewer's attention
where we need it.
Symmetry is the same harmony that is closely intertwined with other rules of photography. Imagine that you have an architecturally beautiful building in front of you, with a water channel or swimming
pool in the center and plants and towers on the sides of the pool. We can place the building exactly in the center of the frame, thus following the symmetry due to the towers and plants on the sides
and the water channel will be the leading line to the subject of our interest in the frame being symmetrical at the same time. Now we have introduced the concept of leading lines on photo, let us
dwell on it in detail!
Leading lines
Leading lines inherently speak for themselves. These are any geometric shapes that somehow lead us to the object of interest in the frame. They can be straight lines, such as buildings or a road
leading into the distance. It can also be curves or curved lines of rivers also leading to the point of interest.
Leading lines are everywhere, we just have to start looking for them. For example, the same railing that a person holds on to, with the right approach to the composition in the frame can directly
point to the main subject in the frame manipulating the viewer's attention and as if hinting to look here. That is why the composition in the frame is so important, knowing how to manipulate the
attention can cause more interest to the image. Thus, it will be evaluated as more creative and will allow you to stand out against other photographers with your work by the right approach to
creating the composition.
Depth of Field
Have you ever heard about the Bokeh effect? This effect allows us to emphasize the viewer's attention on our subject by blurring the background, thus cutting off the unnecessary but preserving the
composition of the frame. The aperture setting is responsible for the depth of field, which is a part of the lens in the camera. Aperture works like our eye, the narrower our pupil is the more we
see, the wider it is, the more blurred everything is in the distance and the better we see up close. The camera and the iris are made on exactly the same principle. The diaphragm has a hole in the
center, which is formed by its expansion or contraction. This hole in the center allows a certain amount of light to pass through, which is directly responsible for the depth of field. Depth of field
is a plan in the frame within which the subject remains equally sharp, and everything outside goes into a blur.
The aperture parameter is measured in F-stops and the smaller the number, for example, F/1.4, the greater the depth of field and the clearer the entire image. Such depth of field values are usually
used in landscape photography where a uniform level of detail is important. This allows our camera to focus on the most distant points, making the entire frame sharp.
By increasing the aperture value, at F/14 which is 10 times bigger, we will get a shallow depth of field and the very same bokeh blur effect. That way we can now only focus on a nearby subject and
the depth of field will only allow this subject to stay in focus and make the background blurred. This has common use in portrait photography and night urban photography. You can get interesting
light effects with bokeh. Imagine you are sitting in a restaurant and looking behind the glass or window onto the street where traffic is going, glowing headlights and traffic lights. You can focus
on your dish placed on the table while keeping the widow in the frame. Open your aperture while keep focusing on the dish, and you will see how nicely all these lights will blur in the background.
All the mentioned rules can be applied both separately and in combination to compose geometric images. You can spend hours looking for a landscape in which symmetry, the rule of thirds, and the rule
of the golden ratio will give the maximum of composition. But it is better to start with the basics, and you will gradually begin to notice more and more geometry and harmony around you, which you
can capture. Photography is first and foremost an art, and understanding how numbers and geometry can affect the perception of your work should be considered but not blindly followed. | {"url":"https://retouchme.com/blog/geometric-photography-ideas","timestamp":"2024-11-14T22:19:45Z","content_type":"text/html","content_length":"126485","record_id":"<urn:uuid:289dd929-5839-4f80-8922-43c7c8d671cb>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00475.warc.gz"} |
Best T-Test Calculator Online
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• Our T test calculator is the most sophisticated and comprehensive T-test calculator online.
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• Results page will give you t-Score, standard error of difference, degrees of freedom, two-tailed p-value, mean difference and confidence range. You can choose the confidence interval. | {"url":"https://www.meta-calculator.com/t-test-calculator.php?panel-406-t-tests-input","timestamp":"2024-11-08T14:08:01Z","content_type":"text/html","content_length":"56005","record_id":"<urn:uuid:808a125f-d178-4d89-8572-0d907176866d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00663.warc.gz"} |
Art Academy V: Ruin
Submit solution
Points: 12 (partial)
Time limit: 0.5s
Memory limit: 512M
Long live the Queen!
After a heavy bombardment by the forces of Queen Alice, the Art Academy has started to collapse. Since , the faithful leader of the Art Academy, was so fanatical in his obsession for art, pieces have
been hung everywhere; including on the ceiling. has already managed to remove all his pieces on the walls, so he turns his salvation efforts to the sky, and prepares to catch the pieces as they fall.
Fortunately for him, he is not alone. His loyal aide has come back to help him, and together, they will save the art!
The Academy will be represented on a 2D plane where artworks will fall sequentially, given as a point . Either one of them must collect a given piece of art when it falls. Determine the minimum sum
of the Manhattan distances that and his aide will have to travel.
For example, if there is a single painting that falls, starts at , his aide starts at , and the painting falls at , the optimal Manhattan distance would be for his aide to move a distance of .
Input Specification
The first line will contain the integer , representing the number of artworks that will fall.
The next line will have the integers , , , and , representing the coordinates of where and his aide will start.
The following lines will have two integers and , representing the coordinate of where the artwork will drop down to.
For all subtasks:
Subtask 1 [10%]
Subtask 2 [20%]
Subtask 3 [70%]
No additional constraints.
Output Specification
Save the Academy! Save the artwork! Output the minimum total distance and his aide will have to travel in order to catch all the artwork.
Sample Input 1
Sample Output 1
• I'm very confused. Why is my solution clearing the last subset but not the other two? Shouldn't the last one be harder? | {"url":"https://dmoj.ca/problem/art5","timestamp":"2024-11-12T00:51:02Z","content_type":"text/html","content_length":"27848","record_id":"<urn:uuid:f7de29c9-13b4-457c-b525-7e7aba60ae24>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00441.warc.gz"} |
Library prosa.analysis.facts.transform.swaps
In this file, we establish invariants about schedules in which two allocations have been swapped, as for instance it is done in the classic EDF optimality proof.
For any given type of jobs...
... any given type of processor states:
...consider any given reference schedule.
First, we note that the trivial case where t1 == t2 is not interesting because then the two schedules are identical.
In this trivial case, the amount of service received hence is obviously always identical.
In any case, the two schedules do not differ at non-swapped times.
By definition, if a job is scheduled at t2 in the original schedule, then it is found at t1 in the new schedule.
Similarly, a job scheduled at t1 in the original schedule is scheduled at t2 after the swap.
If a job is scheduled at any time not involved in the swap, then it remains scheduled at that time in the new schedule.
To make case analysis more convenient, we summarize the preceding three lemmas as a disjunction.
From this, we can easily conclude that no jobs have appeared out of thin air: if a job scheduled at some time in the new schedule, then it was also scheduled at some time in the original schedule.
Mirroring swap_job_scheduled_cases above, we also state a disjunction for case analysis under the premise that a job is scheduled in the original schedule.
Thus, we can conclude that no jobs are lost: if a job is scheduled at some point in the original schedule, then it is also scheduled at some point in the new schedule.
As another trivial invariant, we observe that nothing has changed before time t1.
Similarly, nothing has changed after time t2.
Thus, we observe that, before t1, the two schedules are identical with regard to the service received by any job because they are identical.
Likewise, we observe that, *after* t2, the swapped schedule again does not differ with regard to the service received by any job.
Finally, we note that, trivially, jobs that are not involved in the swap receive invariant service.
For any given type of jobs...
... any given type of processor states:
...consider any given reference schedule.
First, we observe that if jobs never accumulate more service than required, then that's still the case after the swap.
From the above service bound, we conclude that, if if completed jobs don't execute in the original schedule, then that's still the case after the swap, assuming an ideal unit-service model (i.e.,
scheduled jobs receive exactly one unit of service).
Suppose all jobs in the original schedule come from some arrival sequence...
...then that's still the case in the new schedule.
In the next section, we consider a special case of swaps: namely, when the job moved earlier has an earlier deadline than the job being moved to a later allocation, assuming that the former has not
yet missed its deadline, which is the core transformation of the classic EDF optimality proof.
For any given type of jobs with costs and deadlines...
... any given type of processor states...
...consider a given reference schedule...
...in which complete jobs don't execute...
...and scheduled jobs always receive service.
...that are ordered.
Further, assume that, if there are jobs scheduled at times t1 and t2, then they either have the same deadline or violate EDF, ...
...and that we don't move idle times or deadline misses earlier, i.e., if t1 is not an idle time, then neither is t2 and whatever job is scheduled at time t2 has not yet missed its deadline.
Consider the schedule obtained from swapping the allocations at times t1 and t2.
The key property of this transformation is that, for any job that meets its deadline in the original schedule, we have not introduced any deadline misses, which we establish by considering a number
of different cases.
Consider any job...
... that meets its deadline in the original schedule.
First we observe that jobs that are not involved in the swap still meet their deadlines.
Next, we observe that a swap is unproblematic for the job scheduled at time t2.
Finally, we observe is also unproblematic for the job that is moved to a later allocation.
From the above case analysis, we conclude that no deadline misses are introduced in the schedule obtained from swapping the allocations at times t1 and t2. | {"url":"https://prosa.mpi-sws.org/branches/master/pretty/prosa.analysis.facts.transform.swaps.html","timestamp":"2024-11-07T15:13:53Z","content_type":"application/xhtml+xml","content_length":"141929","record_id":"<urn:uuid:a1755875-8856-47bc-ae1e-92e6fc3d40a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00206.warc.gz"} |
On the herbrandised interpretation for nonstandard arithmetic
Instituto Superior Técnico, Universidade de Lisboa, 2016
Functional interpretations are useful tools of proof theory. After Godel described his dialectica interpretation for Heyting arithmetic in 1941, many other interpretations have been proposed, each
focusing on different goals. We start with an overview of the interpretations of Godel and Shoenfield. We propose a functional interpretation for nonstandard Heyting arithmetic based on previous work
by Van den Berg, Briseid and Safarik. This interpretation enables the transformation of proofs in nonstandard arithmetic of internal statements into proofs in standard arithmetic of those same
statements. The witnesses for external, existential statements of the interpreting formulas are functions whose output is a finite sequence. Syntactically, the terms representing these functions are
called end-star terms. It is possible to define a preorder of end-star terms. Our interpretation is monotone over this preorder: if a certain end-star term is a witness for an existential statement,
then any "bigger" term also is. Using this property, we are able to prove a soundness theorem for our interpretation, which eliminates principles recognisable from nonstandard analysis. From this
theorem, we get as corollary the conservativity of nonstandard arithmetic over standard arithmetic, as well as a term extraction theorem. It is also possible to prove a characterization theorem for
our interpretation. As corollary, we show that the countable saturation principle does not add proof theoretical strength to our intuitionistic nonstandard system. Finally, we give a short
description of Weihrauch reducibility and comment on an application of Godel’s dialectica interpretation to, in certain circumstances, prove that a ∀∃-formula Weihrauch reduces to another one.
Functional interpretations, nonstandard arithmetic, Weihrauch reducibility
de Almeida Borges, A. (2016). On the herbrandised interpretation for nonstandard arithmetic (Master's thesis). Instituto Superior Técnico, Universidade de Lisboa.
author = "{de Almeida Borges}, Ana",
title = "On the herbrandised interpretation for nonstandard arithmetic",
school = "{I}nstituto {S}uperior {T}écnico, {U}niversidade de {L}isboa",
year = "2016",
url = "https://fenix.tecnico.ulisboa.pt/cursos/mma/dissertacao/1691203502342670",
keywords = "Functional interpretations, nonstandard arithmetic, Weihrauch reducibility",
abstract = {Functional interpretations are useful tools of proof theory. After Godel described his dialectica interpretation for Heyting arithmetic in 1941, many other interpretations have been proposed, each focusing on different goals. We start with an overview of the interpretations of Godel and Shoenfield. We propose a functional interpretation for nonstandard Heyting arithmetic based on previous work by Van den Berg, Briseid and Safarik. This interpretation enables the transformation of proofs in nonstandard arithmetic of internal statements into proofs in standard arithmetic of those same statements. The witnesses for external, existential statements of the interpreting formulas are functions whose output is a finite sequence. Syntactically, the terms representing these functions are called end-star terms. It is possible to define a preorder of end-star terms. Our interpretation is monotone over this preorder: if a certain end-star term is a witness for an existential statement, then any "bigger" term also is. Using this property, we are able to prove a soundness theorem for our interpretation, which eliminates principles recognisable from nonstandard analysis. From this theorem, we get as corollary the conservativity of nonstandard arithmetic over standard arithmetic, as well as a term extraction theorem. It is also possible to prove a characterization theorem for our interpretation. As corollary, we show that the countable saturation principle does not add proof theoretical strength to our intuitionistic nonstandard system. Finally, we give a short description of Weihrauch reducibility and comment on an application of Godel’s dialectica interpretation to, in certain circumstances, prove that a ∀∃-formula Weihrauch reduces to another one.} | {"url":"https://aborges.eu/publication/2016-01-01-On-the-herbrandised-interpretation-for-nonstandard-arithmetic","timestamp":"2024-11-07T01:26:35Z","content_type":"text/html","content_length":"14935","record_id":"<urn:uuid:39624f86-f44e-4083-920d-a325802043ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00077.warc.gz"} |
logistic regression multiclass
In its vanilla form logistic regression is used to do binary classification. But there you have it. Fortunately, this simplifies to computing (in vectorised form), which updates all the values of
simultaneously, where is a learning rate and is the index of the iterations (and not a power superscript!). Applications. Multiclass Classification Out there are algorithms that can deal by
themselves with predicting multiple classes, like Random Forest classifiers or the Naive Bayes Classifier. The typical cost function usually used in logistic regression is based on cross entropy
computations (which helps in faster convergence in relation to the well known least squares); this cost function is estimated during each learning iteration for the current values of , and in
vectorised form is formulated as. Usually learning about these methods starts off with the general categorisation of problems into regression and classification, the first tackling the issue of
learning a model (usually also called a hypothesis) that fits the data and the second focusing on learning a model that categorises the data into classes. One has to keep in mind that one logistic
regression classifier is enough for two classes but three are needed for three classes and so on. The hypothesis in logistic regression can be defined as Sigmoid function. Apparently this piece of
code is what happens within each learning iteration. In this module, we introduce the notion of classification, the cost function for logistic regression, and the application of logistic regression
to multi-class classification. You consider output as 1 if it is class 1 and as zero if it is any other class. It is used when the outcome involves more than two classes. Everything seems perfectly
fine in cases in which binary classification is the proper task. you train one model each for different class. Logistic regression algorithm can also use to solve the multi-classification problems.
In logistic regression, instead of computing a prediction of an output by simply summing the multiplications of the model (hypothesis) parameters with the data (which is practically what linear
regression does), the predictions are the result of a more complex operation as defined by the logistic function, where is the hypothesis formed by the parameters on the data , all in vector
representations, in which for data samples and data dimensions. L1 regularization weight, L2 regularization weight: Type a value to use for the regularization parameters L1 and L2. The way it works
is based on an iterative minimisation of a kind of an error of the predictions of the current model to the actual solution (which is known during training). Notify me of follow-up comments by email.
This can be compactly expressed in vector form: Thus, the logistic link function can be used to cast logistic regression into the Generalized Linear Model. A more complex case is the case of
multi-class classification, in which data are to be assigned to more than two classes. Logistic Regression method and Multi-classifiers has been proposed to predict the breast cancer. In statistics,
multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. Your email address will not be published. Next step in the study of
machine learning is typically the logistic regression. Regression, and particularly linear regression is where everyone starts off. Linear regression focuses on learning a line that fits the data.
The change in this case is really spectacular. multioutput regression is also supported.. Multiclass classification: classification task with more than two classes.Each sample can only be labelled as
one class. This is called as Logistic function as well. \(C=2\)). Active today. About multiclass logistic regression. Logistic Regression (aka logit, MaxEnt) classifier. Numpy: Numpy for performing
the numerical calculation. For consistency in the computations the data dimensions are supposed to have been augmented by a first ‘virtual’ dimension (column in the data matrix) having one (1) as a
value for all samples due to the fact that there is a first parameter, which is a kind of an ‘offset’. The model has a 92% accuracy score. Logistic regression for multi-class classification problems
– a vectorized MATLAB/Octave approach sepdek February 2, 2018 Machine learning is a research domain that is becoming the holy grail of data science towards the modelling and solution of science and
engineering problems. Multiclass Logistic Regression: How does sklearn model.coef_ return K well-identified sets of coefficients for K classes? Use multiclass logistic regression for this task.
Unlike linear regression which outputs continuous number values, logistic regression transforms its output using the logistic sigmoid function to return a probability value which can then be mapped
to two or more discrete classes. (Currently, the ‘multinomial’ option is supported only by the ‘lbfgs’, ‘sag’ and ‘newton-cg’ solvers.) We use logistic regression when the dependent variable is
categorical. See below: The idea in logistic regression is to cast the problem in the form of a generalized linear regression model. @whuber Actually, I am confused related to multiclass logistic
regression not binary one. Sklearn: Sklearn is the python machine learning algorithm toolkit. model = LogisticRegression(solver = 'lbfgs'), # use the model to make predictions with the test data,
count_misclassified = (test_lbl != y_pred).sum(), print('Misclassified samples: {}'.format(count_misclassified)), accuracy = metrics.accuracy_score(test_lbl, y_pred), print('Accuracy: {:.2f}'.format
(accuracy)). Load your favorite data set and give it a try! Enter your email address to subscribe to this blog and receive notifications of new posts by email. In this chapter, we’ll show you how to
compute multinomial logistic regression in R. Ltd. 2020, All Rights Reserved. While prediction, you test the input using all the 10 models and which ever model gives the highest value between zero
and one considering you are using sigmoid transfer function, the input belongs to that particular class. Since this is a very simplistic dataset with distinctly separable classes. To produce deep
predictions in a new environment on the breast cancer data. In this post, I will demonstrate how to use BigQuery ML for multi class classification. Next, you train another model where you consider
output to be 1 if it class 2 and zero for any other class. The occupational choices will be the outcome variable whichconsists of categories of occupations.Example 2. Why we are not using dummies in
target data ? We will mainly focus on learning to build a logistic regression model for doing a multi-class classification. Classify a handwritten image of a digit into a label from 0-9. I have
already witnessed researchers proposing solutions to problems out of their area of expertise using machine learning methods, basing their approach on the success of modern machine learning algorithm
on any kinds of data. Taken that there are only first-order data (linear terms only, ) the result of this algorithm is shown in the following figure. It is a good database for,
train-images-idx3-ubyte.gz: training set images (9912422 bytes), train-labels-idx1-ubyte.gz: training set labels (28881 bytes), t10k-images-idx3-ubyte.gz: test set images (1648877 bytes),
t10k-labels-idx1-ubyte.gz: test set labels (4542 bytes). The model has a 92% accuracy score. That’s how to implement multi-class classification with logistic regression using scikit-learn. $\
begingroup$ I have edited the equation. It is essentially a binary classification method that identifies and classifies into two and only two classes. Logistic function is expected to output 0 or 1.
? The digits have been size-normalized and centered in a fixed-size image. Logistic regression is usually among the first few topics which people pick while learning predictive modeling. Which is not
true. Use multiclass logistic regression for this task. But linear function can output less than 0 o more than 1. After this code (and still inside the loop of the training iterations) some kind of
convergence criterion should be included, like an estimation of the change in the cost function or the change in the parameters in relation to some arbitrary convergence limit. In the multi class
logistic regression python Logistic Regression class, multi-class classification can be enabled/disabled by passing values to the argument called ‘‘multi_class’ in the constructor of the algorithm.
handwritten image of a digit into a label from 0-9. I am assuming that you already know how to implement a binary classification with Logistic Regression. it is a multi-class classification problem)
then logistic regression needs an upgrade. The simpler case in classification is what is called binary (or binomial) classification, in which the task is to identify and assign data into two classes.
In this video, learn how to create a logistic regression model for multiclass classification using the Python library scikit-learn. I wrote this kernel to first start with the easiest method to
classify the handwritten digits. In the multi class logistic regression python Logistic Regression class, multi-class classification can be enabled/disabled by passing values to the argument called
‘‘multi_class’ in the constructor of the algorithm. Logistic regression is a very popular machine learning technique. Logistic regression is used for classification problems in machine learning. The
first (which we don’t actually use) shows a simple implementation of the softmax function. Logistic regression is based on the use of the logistic function, the well known. The digits have been
size-normalized and centered in a fixed-size image. Logistic regression uses a more complex formula for hypothesis. So in this article, your are going to implement the logistic regression model in
python for the multi-classification problem in 2 different ways. So, we cannot use the linear regression hypothesis. Logistic regression. Logistic regression, although termed ‘regression’ is not a
regression method. The MNIST database of handwritten digits, available from this page, has a training set of 60,000 examples, and a test set of 10,000 examples. Although nothing has changed in the
algorithm and the code given above, now the classes are successfully separated by curves. Yes, we can do it. Note that the levels of prog are defined as: 1=general 2=academic (referenc… Load your
favorite data set and give it a try! This is part of my serie of posts (www.marcelojo.org) where I compare the results here with an implementation in Octave/Matlab. This site uses Akismet to reduce
spam. Use multiclass logistic regression for this task. Classification in Machine Learning is a technique of learning, where an instance is mapped to one of many labels. บทที่ 17-Multiclass Logistic
Regression. It is a good database for people who want to try learning techniques and pattern recognition methods on real-world data while spending minimal efforts on preprocessing and formatting. How
to Do Multi-Class Logistic Regression Using C#. Data Science Bootcamp with NIT KKRData Science MastersData AnalyticsUX & Visual Design. machine-learning neural-network numpy jupyter-notebook
regression python3 classification expectation-maximization vae logistic-regression bayesian polynomial-regression support-vector-machines gaussian-processes svm-classifier ica
independent-component-analysis multiclass-logistic-regression baysian-inference vae-pytorch Learn how your comment data is processed. Multiclass logistic regression for classification; Hands on Multi
class classification. It's called as one-vs-all Classification or Multi class classification. For example you have 10 different classes, first you train model for classifying whether it is class 1 or
any other class. Following is the graph for the sigmoidal function: The equation for the sigmoid function is: It ensures that the generated number is always between 0 and 1 since the numerator is
always smaller than the denominator by 1. # Apply transform to both the training set and the test set. logistic regression is used for binary classification . Similarly you train one model per every
class. The outcome prog and the predictor ses are bothcategorical variables and should be indicated as such on the class statement. About the Dataset. From here on, all you need is practice.
Apparently this is not a good choice and I have also witnessed failures, since those modern methods in many cases rely on an intuition on the data at hand. The second applies the softmax function to
each row of a matrix. The MNIST database of handwritten digits, available from this page, has a training set of 60,000 examples, and a test set of 10,000 examples. In multi-class classification
applications using logistic regression, similar to binary classification, the response of each of the classifiers (the prediction) represents the probability of each unknown input to be in the ‘Class
1’ of each classifier. The algorithm successfully ‘draws’ a line separating the space for each of the classes. Logistic regression for multi-class classification problems – a vectorized MATLAB/Octave
approach, Expectation Maximization for gaussian mixtures – a vectorized MATLAB/Octave approach, Matrix-based implementation of neural network back-propagation training – a MATLAB/Octave approach,
Computational Methods in Heritage Science. Explained with examples, Mastering Big Data Hadoop With Real World Projects, Using Decision Trees for Regression Problems >>, How to Access Hive Tables
using Spark SQL. In its vanilla form logistic regression is used to do binary classification. is usually among the first few topics which people pick while learning predictive modeling. The machine
learns patterns from data in such a way that the learned representation successfully maps the original dimension to the suggested label/class without any intervention from a human expert. Multiclass
classification with logistic regression can be done either through the one-vs-rest scheme in which for each class a binary classification problem of data belonging or not to that class is done, or
changing the loss function to. But there you have it. Nevertheless, the particular field of deep learning with artificial neural networks has already successfully proposed significant solutions to
highly complex problems in a diverse range of domains and applications. Logistic regression is used for classification problems in machine learning. Copyright © AeonLearning Pvt. Since this is a very
simplistic dataset with distinctly separable classes. n the multiclass case, the training algorithm uses the one-vs-rest (OvR) scheme if the ‘multi_class’ option is set to ‘ovr’ and uses the
cross-entropy loss if the ‘multi_class’ option is set to ‘multinomial’. where ŷ =predicted value, x= independent variables and the β are coefficients to be learned. Multiclass logistic regression
•Suppose the class-conditional densities दध༞गis normal दध༞ग༞द|ථ,༞ Յ Ն/ഈ expᐎ༘ Յ Ն द༘ථ ഈ ᐏ •Then एථ≔lnदध༞गध༞ग ༞༘ Յ Ն दद༗थථ … Gradient descent intuitively tries to find the lower limits of the cost
function (thus the optimum solution) by, step-by-step, looking for the direction of lower and lower values, using estimates of the first (partial) derivatives of the cost function. We can study
therelationship of one’s occupation choice with education level and father’soccupation. For example, the Trauma and Injury Severity Score (), which is widely used to predict mortality in injured
patients, was originally developed by Boyd et al. linear_model: Is for modeling the logistic regression model metrics: Is for calculating the accuracies of the trained logistic regression model.
Choose Your Course (required) What is Data Analytics - Decoded in 60 Seconds | Data Analytics Explained | Acadgild, Acadgild Reviews | Acadgild Data Science Reviews - Student Feedback | Data Science
Course Review, Introduction to Full Stack Developer | Full Stack Web Development Course 2018 | Acadgild. This site uses Akismet to reduce spam. Logistic regression, despite its name, is a
classification algorithm rather than regression algorithm. Note that you could also use Logistic Regression for multiclass classification, which will be discussed in the next section. Apparently this
operation applies on all input data at once, or in batches, and this is why this is usually termed as batch training. Learn how your comment data is processed. Below we use proc logistic to estimate
a multinomial logisticregression model. The multinomial logistic regression is an extension of the logistic regression (Chapter @ref(logistic-regression)) for multiclass classification tasks. Before
fitting our multiclass logistic regression model, let’s again define some helper functions. People’s occupational choices might be influencedby their parents’ occupations and their own education
level. Abhay Kumar, lead Data Scientist – Computer Vision in a startup, is an experienced data scientist specializing in Deep Learning in Computer vision and has worked with a variety of programming
languages like Python, Java, Pig, Hive, R, Shell, Javascript and with frameworks like Tensorflow, MXNet, Hadoop, Spark, MapReduce, Numpy, Scikit-learn, and pandas. Post was not sent - check your
email addresses! Of course, in this case, as the dimensionality of the training data increases so does the parameter space and the parameters are now 5-dimensional vectors. To find the optimal
decision boundary, we must minimize this cost function, which we can do with an... Training our model. Logistic regression is used in various fields, including machine learning, most medical fields,
and social sciences. Using Logistic Regression to Create a Binary and Multiclass Classifier from Basics Minimizing the cost. Logistic regression is not a regression algorithm but a probabilistic
classification model. Let’s see a similar but even more complicated example of a 5-class classification training, in which the following features for the logistic regression are being used . To show
that multinomial logistic regression is a generalization of binary logistic regression, we will consider the case where there are 2 classes (ie. (Currently, the ‘multinomial’ option is supported only
by the ‘lbfgs’, ‘sag’ and ‘newton-cg’ solvers.) Multiclass logistic regression¶ In the linear regression tutorial, we performed regression, so we had just one output \(\hat{y}\) and tried to push
this value as close as possible to the true target \(y\). Apparently, this is a completely different picture. That is, it is a model that is used to predict the probabilities of the different
possible outcomes of a categorically distributed dependent variable, given a set of independent variables. This tutorial will show you how to use sklearn logisticregression class to solve multiclass
classification problem … Modeling multiclass classifications are common in data science. The sklearn.multiclass module implements meta-estimators to solve multiclass and multilabel classification
problems by decomposing such problems into binary classification problems. Gradient descent is usually the very first optimisation algorithm presented that can be used to optimise a cost function,
which is arbitrarily defined to measure the cost of using specific parameters for a hypothesis (model) in relation to the correct choice. Machine learning is a research domain that is becoming the
holy grail of data science towards the modelling and solution of science and engineering problems. •The multiclass logistic regression model is •For maximum likelihood we will need the derivatives
ofy kwrtall of the activations a j •These are given by –where I kjare the elements of the identity matrix Machine Learning Srihari 8 By default. This article will focus on the implementation of
logistic regression for multiclass classification problems. It is a subset of a larger set available from NIST. Ask Question Asked today. Logistic regression is not a regression algorithm but.
Required fields are marked *. It is a subset of a larger set available from NIST. which has a very convenient range of values and a really handy differentiation property. The MNIST database of
handwritten digits, available from this page, has a training set of 60,000 examples, and a test set of 10,000 examples. วิธีการ Classification คุณภาพของไวน์ด้วยโมเดล Multiclass Logistic Regression โดย
AzureML Just subscribe to our blog and we will send you this step-by-step guide absolutely FREE! Logistic regression is one of the most fundamental and widely used Machine Learning Algorithms. A
biologist may be interested in food choices that alligators make.Adult alligators might ha… Based on a given set of independent variables, it is used to estimate discrete value (0 or 1, yes/no, true/
false). Let’s see what happens when this algorithm is applied in a typical binary classification problem. So how it can be used for multiclass classification without using any parameter (multi_class)
Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regres Practically, the above operation may result in computations with infinity,
so one might implement it in a slightly tricky way, During the main algorithm in logistic regression, each iteration updates the parameters to gradually minimise this error (of course if everything
works smoothly, which means that a proper learning rate has been chosen–this will appear a little later). Logistic regression is a classification algorithm used to assign observations to a discrete
set of classes. The following figure presents a simple example of a classification training for a 3-class problem, again using gaussian data for better illustration and only linear terms for
classification. An example of this is shown for the matrix Your email address will not be published. The digits have been size-normalized and centered in a fixed-size image. Logistic regression is a
method for classifying data into discrete outcomes. Example 1. In this case, we have predictions ... Multiclass classification; Scalable Machine Learning (UC Davis) Deep Learning with Logistic
Regression. If there are more than two classes (i.e. In the multiclass case, the training algorithm uses the one-vs-rest (OvR) scheme if the ‘multi_class’ option is set to ‘ovr’, and uses the
cross-entropy loss if the ‘multi_class’ option is set to ‘multinomial’. Multivariate Multilabel Classification with Logistic Regression, Logistic regression is one of the most fundamental and widely
used Machine Learning Algorithms. Let’s examine a case of 4 classes, in which only linear terms have been used as features for the classification. In the multiclass case, the training algorithm uses
the one-vs-rest (OvR) scheme if the ‘multi_class’ option is set to ‘ovr’ and uses the cross-entropy loss if the ‘multi_class’ option is set to ‘multinomial’. The algorithm predicts the probability of
occurrence of an event by fitting data to a logistic function. The situation gets significantly more complicated for cases of, say, four (4) classes. with more than two possible discrete outcomes.
Logistic regression is a well-known method in statistics that is used to predict the probability of an outcome, and is popular for classification tasks. It is also called logit or MaxEnt Classifier.
Suppose there are two sets of 1000 2D training samples following gaussian distributions (for simplicity and illustration). The predictions resulting from this vectorised operation are all stored in a
vector, which ideally should match the training ground truth vector that conveys the correct class. Multiclass Logistic Regression - MNIST. Save my name, email, and website in this browser for the
next time I comment. After this training process, the prediction formula actually represents the probability of a new (unknown) data sample being classified in ‘Class 1’ (). For example, we might use
logistic regression to classify an email as spam or not spam. Sorry, your blog cannot share posts by email. A simple practical implementation of this is straight-forward. The MNIST database of
handwritten digits is available on the following website: MNIST Dataset. * in this figure only the first 3 of the 5 θ values are shown due to space limitations. So, the one-vs-one or one-vs-all is
better approach towards multi-class classification using logistic regression. Free Step-by-step Guide To Become A Data Scientist, Subscribe and get this detailed guide absolutely FREE. Logistic
regression has a sigmoidal curve. Wecan specify the baseline category for prog using (ref = “2”) andthe reference group for ses using (ref = “1”). Multiclass classification with logistic regression
can be done either through the one-vs-rest scheme in which for each class a binary classification problem of data belonging or not to that class is done, or changing the loss function to cross-
entropy loss. The MNIST database of handwritten digits is available on the following website: from sklearn.datasets import fetch_mldata, from sklearn.preprocessing import StandardScaler, from
sklearn.model_selection import train_test_split, from sklearn.linear_model import LogisticRegression, # You can add the parameter data_home to wherever to where you want to download your data, #
test_size: what proportion of original data is used for test set, train_img, test_img, train_lbl, test_lbl = train_test_split(, mnist.data, mnist.target, test_size=1/7.0, random_state=122). Of
particular interest is also the ‘probability map’ shown in the middle lower diagram in pseudo-colour representation, where the solution of the prediction formula is shown for every possible
combination of the data dimensions. By default, multi_class is set to ’ovr’. Pandas: Pandas is for data analysis, In our case the tabular data analysis. The goal of this blog post is to show you how
logistic regression can be applied to do multi-class classification. Here, instead of regression, we are performing classification, where we want to … It is a subset of a larger set available from
NIST. In the figure that follows it is evident that the decision boundaries are not at all optimum for the data and the training accuracy drops significantly, as there is no way to linearly separate
each of the classes. This upgrade is not any sophisticated algorithmic update but rather a naive approach towards a typical multiple classifier system, in which many binary classifiers are being
applied to recognise each class versus all others (one-vs-all scheme). The param=ref optiononthe class statement tells SAS to use dummy coding rather than effect codingfor the variable ses. The way
to get through with situations like this is to use higher order features for the classification, say second order features like . The Data Science Lab. That’s how to implement multi-class
classification with logistic regression using scikit-learn. Complete information on what skills are required to become a Data Scientist and how to acquire those skills, Comprehensive information on
various roles in Analytics industry and what responsibilities do they have, Simple explanations on various Machine Learning algorithms and when to use them.
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Hard Rock Cafe Cocktails | {"url":"http://www.asesis.com/blood-feud-xwccjri/16rtv.php?id=logistic-regression-multiclass-46913b","timestamp":"2024-11-12T00:39:44Z","content_type":"text/html","content_length":"35997","record_id":"<urn:uuid:977252e8-f8d5-4a61-bf69-2471d8bda57c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00307.warc.gz"} |
How can I verify spec satisfies the properties for infinite states?
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How can I verify spec satisfies the properties for infinite states?
I wonder how can I verify spec satisfies the properties for infinite states.
In my spec, there are infinitely states so TLC is executed infinitely.
In this case, I wonder how can I verify spec satisfies the properties.
For example,
I want to prove "Theorem Spec => []TypeInv".
When i check by the bound length constraint because of infinite states, TypeInV is satisfied.
But, I think it can be not proved that spec satisfies the properties for infinitely states by this bounded checking. | {"url":"https://discuss.tlapl.us/msg02441.html","timestamp":"2024-11-01T23:21:46Z","content_type":"text/html","content_length":"3711","record_id":"<urn:uuid:7fcceec4-4db7-443a-b249-977d17bfc6dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00057.warc.gz"} |
Aberration Sentence Examples
• She is normally calm and level headed, so this outburst is an aberration.
• We considered our defeat to be an aberration, since we'd easily beaten this team last season.
• The discovery of the aberration of light in 1725, due to James Bradley, is one of the most important in the whole domain of astronomy.
• The residents considered the recent surge in burglaries to be an aberration, since their neighborhood was known for its safety.
• The use of mirrors instead of lenses was another way to avoid chromatic aberration.
• The problem with many low priced cameras, is color aberration in the final image.
• Orthodox evangelicalism is tempted to view it as an apostasy or an aberration.
• Newton was led by this reasoning to the erroneous conclusion that telescopes using refracting lenses would always suffer chromatic aberration.
• The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half.
• By one, and likewise by several, and even by an infinite number of thin lenses in contact, no more than two axis points can be reproduced without aberration of the third order.
• The most important is the chromatic difference of aberration of the axis point, which is still present to disturb the image, after par-axial rays of different colours are united by an appropriate
combination of glasses.
• His researches in optics, continued until his death, appear to have been begun about the year 1814, when he prepared a paper on the aberration of light, which, however, was not published.
• This implies that the whole of Western theology has been an aberration or an exoteric veiling of the truth.'
• James Bradley discovered in 1728 the annual shifting of the stars due to the aberration of light, and in 1748, the complicating effects upon precession of the "nutation" of the earth's axis.
• Again, since the constant of aberration defines the ratio between the velocity of light and the earth's orbital speed, the span of the terrestrial circuit, in other words, the distance of the
sun, is immediately deducible from known values of the first two quantities.
• The spherical aberration of a diamond lens can be brought down to one-ninth of a glass lens of equal focus.
• With this mineral also spherical and chromatic aberration are a fraction of that of a glass lens, but double refraction, which involves a doubling of the image, is fatal to its use.
• When well made such constructions are almost free from spherical aberration, and the chromatic errors are very small.
• Axial aberration is reduced by distributing the refraction between two lenses; and by placing the two lenses farther apart the errors of the pencils of rays proceeding from points lying outside
the axis are reduced.
• As shown in Lens and Aberration, for reproduction through a single lens with spherical surfaces, a combination of the rays is only possible for an extremely small angular aperture.
• The aberration of rays in which the outer rays intersect the axis at a shorter distance than the central rays is known as " undercorrection."
• Correction of the spherical aberration in strong systems with very large aperture can not be brought about by means of a single combination of two lenses, but several partial systems are
• If, by these methods, a point in the optic axis has been freed from aberration, it does not follow that a point situated only a very small distance from the optic axis can also be represented
without spherical aberration.
• The representation, free from aberration, of a small surface-element, is only possible, as Abbe has shown, if the objective simultaneously fulfils the " sine-condition," i.e.
• The removal of the spherical aberration and the sine-condition can be accomplished only for two conjugate points.
• By using these glasses and employing minerals with special optical properties, it is possible to correct objectives so that three colours can be combined, leaving only a quite slight tertiary
spectrum, and removing the spherical aberration for two colours.
• A further aberration which can only be overcome with difficulty, and even then only partially, is the " curvature of the field, " i.e.
• A second method for diminishing the spherical aberration was to alter the distances of the single systems, a method still used.
• The lower surface of the slip causes undercorrection on being traversed by the pencil, with over-correction when it leaves it; and since the aberration of the surface lying farthest from the
object, i.e.
• In order to counteract this aberration the whole objective must be correspondingly under-corrected.
• In the apochromats the chromatic difference of the spherical aberrations is eliminated, for the spherical aberration is completely avoided for three colours.
• In consequence of these residual aberrations, every object-point is not reproduced in an ideal image-point, but as a small circle of aberration.
• In the case of a suitable ocular magnification, the details will be well seen, while the aberration circles remain invisible.
• I also looked into the immense size of chromatic aberration in the human eye.
• The image, despite the spherical aberration, was by far superior to any existing microscope made by his contemporaries.
• The transformation from apparent to topocentric consists of allowing for diurnal aberration.
• I knew the cause to be lateral chromatic aberration.
• The annual aberration is the aberration correction for an imaginary observer at the Earth's center.
• Three SLD glass elements are employed for effective compensation of color aberration, which is a common problem with super-wide angle lenses.
• Overall, the COM concluded that the results from the in vitro chromosomal aberration assays were likely to represent a cytotoxic response.
• The reduction of almost all visible chromatic aberration at 60x magnification on the base model ES 80 GA SD is a real breakthrough.
• The spherical aberration correction is in this region too.
• Rudolf guides Santa's sleigh with the biological aberration of a red, glowing nose capable of penetrating thick fog?
• If you are using an achromatic refractor, the focus errors will be larger due to chromatic aberration of the telescope.
• But the difficulties interposed by spherical and chromatic aberration had arrested progress in that direction until, in 1655, Huygens, working with his brother Constantijn, hit upon a new method
of grinding and polishing lenses.
• In Bode's Jahrbuch (1776-1780) he discusses nutation, aberration of light, Saturn's rings and comets; in the Nova acta Helvetica (1787) he has a long paper "Sur le son des corps elastiques," in
Bernoulli and Hindenburg's Magazin (1787-1788) he treats of the roots of equation and of parallel lines; and in Hindenburg's Archiv (1798-1799) he writes on optics and perspective.
• This expression for the intensity becomes rigorously applicable when f is indefinitely great, so that ordinary optical aberration disappears.
• Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of,, a regular aberration or
dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual
wavesurface from its intended position.
• The effect of aberration may be considered in two ways.
• The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 3), but it increases
unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a
quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should
accompany the increase of aperture.
• A system fulfilling this condition and free from spherical aberration is called " aplanatic " (Greek a-, privative, irXavrl, a wandering).
• This aberration can, however, be successfully controlled by a suitable eyepiece (see below).
• I can only hope these recent comments are an aberration or better yet, taken completely out of context..."
• Some polyamorists argue that, since many other cultures embrace the idea of multiple partners, monogamy is an aberration.
• The indirect method is based upon the observed constant of aberration or the displacement of the stars due to the earth's motion.
• If the surrounding aether is thereby disturbed, the waves of light arriving from the stars will partake of its movement; the ascertained phenomena of the astronomical aberration of light show
that the rays travel to the observer, across this disturbed aether near the earth, in straight lines.
• At Pulkowa he redetermined the " constant of aberration...
• Aberration Of Light This astronomical phenomenon may be defined as an apparent motion of the heavenly bodies; the stars describing annually orbits more or less elliptical, according to the
latitude of the star; consequently at any moment the star appears to be displaced from its true position.
• The application of this observation to the phenomenon which had so long perplexed him was not difficult, and, in 1727, he published his theory of the aberration of light - a corner-stone of the
edifice of astronomical science.
• James Gregory, in his Optica Promota (1663), discusses the forms of images and objects produced by lenses and mirrors, and shows that when the surfaces of the lenses or mirrors are portions of
spheres the images are curves concave towards the objective, but if the curves of the surfaces are conic sections the spherical aberration is corrected.
• He did not attempt the formation of a parabolic figure on account of the probable mechanical difficulties, and he had besides satisfied himself that the chromatic and not the spherical aberration
formed the chief faults of previous telescopes.
• When all is taken into consideration it is scarcely possible to reduce the secondary colour aberration at the focus of such a double object-glass to less than a fourth part of that prevailing at
the focus of a double objective of the same aperture and focus, but made of the ordinary crown and flint glasses.
• The chromatic aberration of the object-glass of one of these telescopes is corrected for photographic rays, and the image formed by it is received on a highly sensitive photographic plate.
• It requires the middle of the aperture stop to be reproduced in the centres of the entrance and exit pupils without spherical aberration.
• The total aberration of two or more very thin lenses in contact, being the sum of the individual aberrations, can be zero.
• In most cases, two thin lenses are combined, one of which has just so strong a positive aberration (" under-correction," vide supra) as the other a negative; the first must be a positive lens and
the second a negative lens; the powers, however, may differ, so that the desired effect of the lens is maintained.
• Freedom from aberration for two axis points, one of which is infinitely distant, is known as " Herschel's condition."
• Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image
plane of the axis point are represented as functions of the angles of the field of view.
• In a plane containing the image point of one colour, another colour produces a disk of confusion; this is similar to the confusion caused by two " zones " in spherical aberration.
• For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of
the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is
proportional to the ratio of the aperture to the focal length, i.e.
• A second method of correcting the spherical aberration depends FIG.
• The definition is better according as the chromatic and spherical aberrations are removed; there always remains in even the best constructions some slight aberration.
• Object details will only be well seen if the aberration circles are small in comparison.
• I realize that this extends the tube length and will cause some spherical aberration.
• It incorporates aspherical lens elements in the front, as well as rear lens groups, to correct spherical aberration.
• Its design employs three (3) aspherical lens elements to minimize spherical aberration, astigmatism and sagittal comma flare.
• Because almost all reflecting telescopes produce diffraction spikes, many people are used to seeing them and don't consider them an aberration.
• In addition to the doubt thrown on this result by the discrepancy between various determinations of the constant of aberration, it is sometimes doubted whether the latter constant necessarily
expresses with entire precision the ratio of the velocity of the earth to the velocity of light.
• In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light
is practically reached.
• If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e.
• If we inquire what is the greatest admissible longitudinal aberration (Sf) in an object-glass according to the above rule, we find Sf =Xa 2 (2), a being the angular semi-aperture.
• If Q be on the circle described upon OA as diameter, so that u = a cos 4,, then Q' lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration
(depending upon a) vanishes.
• They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ' A, and the outstanding aberration is of the fourth order.
• If it were desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a
remedy by making v slightly variable towards the edges.
• This Makuzu faience, produced by the now justly celebrated Miyagawa ShOzan of Ota (near Yokohama), survives in the form of vases and pots having birds, reptiles, flowers, crustacea and so forth
plastered over the surfacespecimens that disgrace the period of their manufacture, and represent probably the worst aberration of Japanese ceramic conception.
• If the path is to be unaltered by the motion of the aether, as the law of astronomical aberration suggests, this must differ from fds/V by terms not depending on the path - that is, by terms
involving only the beginning and end of it.
• But the aether at a great distance must in any case be at rest; while the facts of astronomical aberration require that the motion of that medium must be irrotational.
• He concluded that there could be no refraction without dispersion, and hence that achromatism was impossible of attainment (see Aberration).
• When the earth is at A, in consequence of aberration, the star is displaced to a point a, its displacement sa being parallel to the earth's motion at A; when the earth is at B, the star appears
at b; and so on throughout an orbital revolution of the earth.
• This constant length subtends an angle of about 40" at the earth; the " constant of aberration " is half this angle.
• For relatively short focal lengths a triple construction such as this is almost necessary in order to obtain an objective free from aberration of the 3rd order, and it might be thought at first
that, given the closest attainable degree of rationality between the colour dispersions of the two glasses employed, which we will call crown and flint, it would be impossible to devise another
form of triple objective, by retaining the same flint glass, but adopting two sorts of crown instead of only one, which would have its secondary spectrum very much further reduced.
• The extension of the image away from the axis or size of field available for covering a photographic plate with fair definition is a function in the first place of the ratio between focal length
and aperture, the longer focus having the greater relative or angular covering power, and in the second a function of the curvatures of the lenses, in the sense that the objective must be free
from coma at the foci of oblique pencils or must fulfil the sine condition (see Aberration).
• If the angle u l be very small, O', is the Gaussian image; and 0', 0' 2 is termed the " longitudinal aberration," and 0'1R the " lateral aberration " of the pencils with aperture u 2.
• If the pencil with the angle u 2 be that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O' 1 there is a circular " disk of confusion" of
radius 0' 1 R, and in a parallel plane at 0'2 another one of radius 0' 2 R 2; between these two is situated the " disk of least confusion."
• The component S 1 of the system, situated between the aperture stop and the object 0, projects an image of the diaphragm, termed by Abbe the " entrance pupil "; the " exit pupil " is the image
formed by the component S2 j which is placed behind the aperture stop. All rays which issue from 0 and pass through the aperture stop also pass through the entrance and exit pupils, since these
are images of the aperture stop. Since the maximum aperture of the pencils issuing from 0 is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be
determined by the position and diameter of the entrance pupil.
• This aberration is quite distinct from that of the sharpness of reproduction; in unsharp reproduction, FIG.
• In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see Lens, and above, " Monochromatic Aberration
• The second aberration which must be removed from microscope objectives are the chromatic. To diminish these a collective lens of crown-glass is combined with a dispersing lens of flint; in such a
system the red and the blue rays intersect at a point (see Aberration).
• Chromatic Aberration, where different frequencies of light refract at different angles causing blurred images.
• For still images, the Nikon D300S has enhanced chromatic aberration editing, which removes color fringing and artifacts before the photo is even downloaded.
• Compared to previous Rebel lenses, the T2i lens has increased sharpness and decreased chromatic aberration.
• Not very long after the disappearance of serfdom in the most advanced communities comes into sight the new system of colonial slavery, which, instead of being the spontaneous outgrowth of social
necessities and subserving a temporary need of human development, was politically as well as morally a monstrous aberration.
• In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wavelength (X).
• Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding 4X
can have but little effect.
• Its original use was the determination of geographical latitudes in the field work of geodetic operations; more recently it has been extensively employed for the determination S of variation of
latitude, at fixed stations, under the auspices of the International Geodetic Bureau, and for the astronomical determination of the constant of aberration.
• Of thin positive lenses with n= 1-5, four are necessary to correct spherical aberration of the third order.
• A lens system comprising two elements, used to reduce chromatic aberration.
• Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration.
• A review of the simplest cases of aberration will now be given.
• Systems free of this aberration are called " ortho scopic " (ipgos, right, r ' 0-K071-6 1/, to look) .
• Further, James Bradley discovered in 1728 the annual shifting of the stars due to the aberration of light (see Aberration), and in 1748, the complicating effects upon precession of the " nutation
" of the earth's axis.
• The swing to the right in the voter survey were considered to be a short-term aberration and of no consequence to the election.
• Hence for the rain to centrally traverse the tube, this must be inclined at an angle BAD to the vertical; this angle is conveniently termed the aberration due to these two motions.
• Both the aberration of axis points, and the deviation from the sine condition, rapidly increase in most (uncorrected) systems with the aperture.
• When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of equal focal length and
• Assured that his explanation was true, Bradley corrected his observations for aberration, but he found that there still remained a residuum which was evidently not a parallax, for it did not
exhibit an annual cycle.
• It has been in the past a source of much perplexity to observers of transits, but is now understood to be a result of irradiation, produced by the atmosphere or by the aberration of the
• Bradley recognized the fact that the experimental determination of the aberration constant gave the ratio of the velocities of light and of the earth; hence, if the velocity of the earth be
known, the velocity of light is determined. | {"url":"https://sentence.yourdictionary.com/aberration","timestamp":"2024-11-08T11:30:50Z","content_type":"text/html","content_length":"539965","record_id":"<urn:uuid:3631e287-3fa6-4912-9f39-8908a7be0cde>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00265.warc.gz"} |
SEN Popup 4 Results
SEN Popup 4 took place at 8pm UTC on 6th April 2024. Here are the results and decks for the top 4 players. Each participant earned SEN Points.
Winner: TJ Inevitable (Serena)
Hero: Serena Thoughtripper (41 cards)
Ally (16):
4x Wisened Ironmelter
2x Silent Assassin
3x Aldmor Artisan
3x Nitish, Darklight Mystic
4x Noxious Acquisitioner
Ability (8):
2x Sinkhole
4x Kraken Attack
2x Stop, Thief!
Item (16):
4x Hit List
3x Black Garb
4x Gyre and Gimble
2x Ill-Gotten Gains
1x Diffusion Gate
2x Thoughtripper's Cutlass
Deck Code: 1154113B
Runner-up: RS Cauten (Zaladar)
Hero: Zaladar (45 cards)
Ally (14):
4x Spitfire Hound
4x Hellfire Besieger
2x Scourge Colossus
2x Infernus, Tyrant of the Damned
2x Forgotten Horror
Ability (18):
2x Word of the Prophet
4x Urgent Business
4x Bad Santa
4x Shoreline Skirmish
4x Mimic
Item (8):
4x Coastal Barrage Frigate
4x The Horizon's Reach
Location (4):
4x Ravenscrest: Valley of Secrets
Deck Code: 1025668B
Semi-finalist: BT Khellus (She-Lah)
Hero: She-Lah the Unifier (43 cards)
Ally (22):
2x Treetop Spider
2x Killer Bee
4x Dockside Coercer
4x Seafaring Shipwright
2x Seaworthy Saboteur
4x Buccaneer Captain
4x Commodore Latimer
Ability (14):
4x Blood Moon
4x Kraken Attack
4x Captured Prey
2x Call of the Wild
Item (6):
2x The Tempest Queen
4x The Captor's Crest
Deck Code: 3185347B
Semi-finalist: BT Will Turner (Serena)
Hero: Serena Thoughtripper (45 cards)
Ally (28):
4x Layarian Knight
4x Stunning Swordmaiden
4x Priest of the Light
4x Night Owl
4x Nitish, Darklight Mystic
4x Slithering Viper
4x Noxious Acquisitioner
Ability (8):
2x Kraken Attack
2x Road Less Traveled
4x Stop, Thief!
Item (8):
4x Gyre and Gimble
2x Thoughtripper's Cutlass
2x Barbarian's Bludgeon
Deck Code: 4143082B | {"url":"https://shadowera.net/home/sen-popup-4-results/","timestamp":"2024-11-09T23:45:08Z","content_type":"text/html","content_length":"79037","record_id":"<urn:uuid:eba9b080-57e6-4bfc-9813-e393fda15424>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00536.warc.gz"} |
In this vignette we examine and model the Fujita2023 data in more detail.
Processing the data cube
The data cube in Fujita2023$data contains unprocessed counts. The function processDataCube() performs the processing of these counts with the following steps:
• It performs feature selection based on the sparsityThreshold setting. Sparsity is here defined as the fraction of samples where a microbial abundance (ASV/OTU or otherwise) is zero.
• It performs a centered log-ratio transformation of each sample using with a pseudo-count of one (on all features, prior to selection based on sparsity).
• It centers and scales the three-way array. This is a complex topic that is elaborated upon in our accompanying paper. By centering across the subject mode, we make the subjects comparable to each
other within each time point. Scaling within the feature mode avoids the PARAFAC model focusing on features with abnormally high variation.
The outcome of processing is a new version of the dataset. Please refer to the documentation of processDataCube() for more information.
Determining the correct number of components
A critical aspect of PARAFAC modelling is to determine the correct number of components. We have developed the functions assessModelQuality() and assessModelStability() for this purpose. First, we
will assess the model quality and specify the minimum and maximum number of components to investigate and the number of randomly initialized models to try for each number of components.
Note: this vignette reflects a minimum working example for analyzing this dataset due to computational limitations in automatic vignette rendering. Hence, we only look at 1-3 components with 5 random
initializations each. These settings are not ideal for real datasets. Please refer to the documentation of assessModelQuality() for more information.
# Setup
minNumComponents = 1
maxNumComponents = 3
numRepetitions = 5 # number of randomly initialized models
numFolds = 8 # number of jack-knifed models
ctol = 1e-6
maxit = 200
numCores= 1
# Plot settings
colourCols = c("", "Genus", "")
legendTitles = c("", "Genus", "")
xLabels = c("Replicate", "Feature index", "Time point")
legendColNums = c(0,5,0)
arrangeModes = c(FALSE, TRUE, FALSE)
continuousModes = c(FALSE,FALSE,TRUE)
# Assess the metrics to determine the correct number of components
qualityAssessment = assessModelQuality(processedFujita$data, minNumComponents, maxNumComponents, numRepetitions, ctol=ctol, maxit=maxit, numCores=numCores)
We will now inspect the output plots of interest for Fujita2023.
Jack-knifed models
Next, we investigate the stability of the models when jack-knifing out samples using assessModelStability(). This will give us more information to choose between 2 or 3 components.
stabilityAssessment = assessModelStability(processedFujita, minNumComponents=1, maxNumComponents=3, numFolds=numFolds, considerGroups=FALSE,
groupVariable="", colourCols, legendTitles, xLabels, legendColNums, arrangeModes,
ctol=ctol, maxit=maxit, numCores=numCores)
Model selection
We have decided that a three-component model is the most appropriate for the Fujita2023 dataset. We can now select one of the random initializations from the assessModelQuality() output as our final
model. We’re going to select the random initialization that corresponded the maximum amount of variation explained for three components.
numComponents = 3
modelChoice = which(qualityAssessment$metrics$varExp[,numComponents] == max(qualityAssessment$metrics$varExp[,numComponents]))
finalModel = qualityAssessment$models[[numComponents]][[modelChoice]]
Finally, we visualize the model using plotPARAFACmodel().
plotPARAFACmodel(finalModel$Fac, processedFujita, 3, colourCols, legendTitles, xLabels, legendColNums, arrangeModes,
continuousModes = c(FALSE,FALSE,TRUE),
overallTitle = "Fujita PARAFAC model")
You will observe that the loadings for some modes in some components are negative. This is due to sign flipping: two modes having negative loadings cancel out but describe the same subspace as two
positive loadings. We can manually sign flip these loadings to obtain a more interpretable plot.
finalModel$Fac[[1]][,2] = -1 * finalModel$Fac[[1]][,2] # mode 1 component 2
finalModel$Fac[[1]][,3] = -1 * finalModel$Fac[[1]][,3] # mode 1 component 3
finalModel$Fac[[2]][,3] = -1 * finalModel$Fac[[2]][,3] # mode 2 component 3
finalModel$Fac[[3]][,2] = -1 * finalModel$Fac[[3]][,2] # mode 3 component 2
plotPARAFACmodel(finalModel$Fac, processedFujita, 3, colourCols, legendTitles, xLabels, legendColNums, arrangeModes,
continuousModes = c(FALSE,FALSE,TRUE),
overallTitle = "Fujita PARAFAC model") | {"url":"https://cloud.r-project.org/web/packages/parafac4microbiome/vignettes/Fujita2023_analysis.html","timestamp":"2024-11-04T05:29:53Z","content_type":"text/html","content_length":"175074","record_id":"<urn:uuid:cf751b36-a22f-4f2e-bdea-e539b613e6bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00406.warc.gz"} |
Water Hammer
Water Hammer. Maximum Pressure Change due to Sudden Valve Closure using Joukowski Equation
Register to enable "Calculate" button.
Pressure Wave Travel Time Calculator (free)
Copy and paste the celerity from the above water hammer calculation into the calculation below to compute wave travel time (may need to use keyboard control-c and control-v instead of
Units in water hammer calculators: cm=centimeter, ft=foot, g=gram, G=Giga, k=kilo, kg=kilogram, lb=pound, m=meter, M=Mega, mm=millimeter, N=Newton, Pa=Pascal, psf=pound per square foot, psi=pound per
square inch, s=second.
Prior to water hammer, liquid initially flows at a constant velocity through a pipe. A downstream valve closes instantaneously, and the liquid slams against the closed valve causing a pressure spike
ΔP also called water hammer.
If a valve is closed faster than the wave travel time, then it is considered an instantaneous valve closure for the water hammer.
t[w] = 2L/c
The water hammer instantaneous valve closure calculation predicts the maximum increase in pressure that will occur due to a sudden valve closure. The valve closure time in water hammer is considered
to be instantaneous if the valve closes faster than (or equal to) the time required for a pressure wave to travel two pipe lengths (i.e. the time for the wave to travel upstream from the valve,
reflect off the upstream boundary and return to the valve). The pressure predicted by the water hammer instantaneous valve closure calculation provides the engineer with the expected maximum pressure
increase. The water hammer calculation can also be used in reverse - to compute the pipe velocity - if a maximum pressure rise due to water hammer is input. For the "Click to Calculate" button to
function, the water hammer pressure calculation requires registration, but the wave travel time calculation does not.
Joukowski Equation
One-dimensional momentum conservation for frictionless flow is used to derive the Joukowski equation for water hammer. The equation was developed for a liquid flowing steadily through a pipe and then
instantly the velocity drops to zero due to a sudden valve closure in a water hammer event. The water hammer equation assumes that liquid compression and pipe friction are negligible. Though the
Joukowski equation's primary applicability is for a liquid velocity that drops to zero upon contacting a closed valve causing water hammer, the equation is valid for any instantaneous drop in
velocity, not necessarily a drop to zero velocity. The Joukowski equation is seen with and without a negative sign on the right hand side depending on whether the pressure wave is traveling upstream
or downstream in the water hammer event. In either case, the pressure increase due to water hammer is a positive number. The Joukowski equation is (Wylie, 1993, p. 4; Chaudhry, 1987, p. 8; Hwang and
Houghtalen, 1996, p. 118):
ΔP = ρ c ΔV
The equation for wave speed, c, during water hammer is based on mass conservation and allows the pipe wall material to expand (Wylie, 1993, p. 6; Chaudhry, 1987, p. 34; Hwang and Houghtalen, 1996, p.
The ΔP equation for water hammer was derived for liquid upstream of the valve and does not include effects downstream of the valve. The D/(wE[p]) portion was derived using a thin-walled pipe
Water hammer Wave Travel Time Equation
Instantaneous valve closure due to water hammer is defined to occur if the valve is closed faster than the wave travel time. The wave travel time is (Hwang and Houghtalen, 1996, p. 119):
t[w] = 2L/c
In water hammer, the wave travel time, t[w], is the time for a pressure wave to propagate from the valve, upstream to the reservoir, and back down to the valve. If a valve takes longer than t[w] to
close, then our other water hammer calculation should be used, where the user can enter the valve closure time.
Dimensions: F=Force, L=Length, M=Mass, T=Time
c = Celerity (wave speed) [L/T].
D = Inside diameter of pipe [L].
E = Composite elastic modulus [F/L^2].
E[f] = Elastic modulus of fluid [F/L^2].
E[p] = Elastic modulus of pipe material [F/L^2].
L = Pipe length [L].
t[w] = Wave travel time [T].
w = Pipe wall thickness [L].
ΔP = Maximum pipe pressure increase in water hammer event due to sudden valve closure [F/L^2].
ΔV = Change in velocity in water hammer [L/T].
ρ = Fluid density [M/L^3].
FLUID PROPERTIES, PIPE PROPERTIES, and PIPE WALL THICKNESS
Fluid Properties
Fluid density, viscosity, and elastic modulus provided by the drop-down menus in the water hammer calculation have been compiled from the closed conduit pipe flow references shown on our literature
web page.
Pipe Properties
The pipe material elastic moduli built into our water hammer calculation have been compiled from the references shown below.
Pressure calculation
"Need Density > 0". "Need E[f] > 0". "Need E[p] > 0". "Need Diameter > 0". "Need Thickness > 0". Density, elastic modulus of fluid and pipe, diameter, and wall thickness must be entered as positive
numbers in water hammer calculation.
"Need ΔV > 0". "Need ΔP > 0". If ΔV or ΔP is selected as an input, it must be positive in water hammer calculator.
Wave travel time calculation
"Need Length>0". "Need Celerity>0". Length and celerity (wave speed) must be entered as positive numbers.
Chaudhry, M. Hanif. 1987. Applied Hydraulic Transients. Van Nostrand Reinhold Co. 2ed.
Hwang, Ned H .C. and Robert J. Houghtalen. 1996. Fundamentals of Hydraulic Engineering Systems. Prentice Hall, Inc. 3ed.
LMNO Engineering, Research, and Software. 2009. Newsletter comparing water hammer calculations.
Mays, Larry W. 1999. Hydraulic Design Handbook. McGraw-Hill.
Wylie, E. Benjamin and Victor L. Streeter. 1993. Fluid Transients in Systems. Prentice-Hall, Inc.
© 2009-2024 LMNO Engineering, Research, and Software, Ltd. All rights reserved.
Please contact us for consulting or questions about water hammer.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 707-2614
LMNO@LMNOeng.com https://www.LMNOeng.com | {"url":"https://www.lmnoeng.com/WaterHammer/impulse.php","timestamp":"2024-11-11T17:22:01Z","content_type":"text/html","content_length":"24392","record_id":"<urn:uuid:eb4fb7e1-490d-4df4-8145-c576c1cf80c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00494.warc.gz"} |
and 15 hours respectively, if both are
is. Rape A can fill an e... | Filo
Question asked by Filo student
and 15 hours respectively, if both are is. Rape A can fill an empty tank in 5 hours while pipe B can empty the full tank in 6 hours. If both are opened at the same time in the empty tank, how much
time will they take to fill it up completely? 16. Three taps and can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would the three taps take to fill the empty tank. if
all of them are opened together? 17. A cistern has two inlets and which can fill it in 12 minutes and 15 minutes respectively. together in the empty tank, how much time will they take to fill the
tank completely? 18. A pipe can a cistern in 9 hours. Due to a leak in its bottom. the cistern fills up in I0 hours. If the cistern is full. in how much time will it be emptied by the leak? Hiat.
Wark done by the leak in 1 hour 19. Pipe A can fll a cistern in 6 hours and pipe B can fill it in 8 hours. Both the pipes are opened and after two hours, pipe A is closed. How much time will B take
to fill the remaining part of the tank? Fint. Work tonte by in 1 hour Work done by both in 2 fours Pemaining part Nore. 1 part is fulled by B tn 1 hour. Find how much time B will take to fill part.
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and 15 hours respectively, if both are is. Rape A can fill an empty tank in 5 hours while pipe B can empty the full tank in 6 hours. If both are opened at the same time in the empty tank,
how much time will they take to fill it up completely? 16. Three taps and can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would the three taps take to fill
the empty tank. if all of them are opened together? 17. A cistern has two inlets and which can fill it in 12 minutes and 15 minutes respectively. together in the empty tank, how much time
Question will they take to fill the tank completely? 18. A pipe can a cistern in 9 hours. Due to a leak in its bottom. the cistern fills up in I0 hours. If the cistern is full. in how much time will
Text it be emptied by the leak? Hiat. Wark done by the leak in 1 hour 19. Pipe A can fll a cistern in 6 hours and pipe B can fill it in 8 hours. Both the pipes are opened and after two hours,
pipe A is closed. How much time will B take to fill the remaining part of the tank? Fint. Work tonte by in 1 hour Work done by both in 2 fours Pemaining part Nore. 1 part is fulled by B tn 1
hour. Find how much time B will take to fill part.
Updated Feb 6, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Video solution: 1
Upvotes 134
Video 17 min | {"url":"https://askfilo.com/user-question-answers-mathematics/and-15-hours-respectively-if-both-are-is-rape-a-can-fill-an-34313239323734","timestamp":"2024-11-08T09:08:04Z","content_type":"text/html","content_length":"292626","record_id":"<urn:uuid:bdc286f5-19a0-4289-b1f2-96d5c91f895c>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00234.warc.gz"} |
Developing new iron-nitrogen steels with ab initio thermodynamics
Developing new iron-nitrogen steels with ab initio thermodynamics
18MAT03 / Solid-state physics
Promotor(en): S. Cottenier, L. Duprez /
Developing new steels is one of the most important driving forces for advancements in construction, medicine and the automotive industry. A promising approach is to add nitrogen. In conventional
steel processing, nitrogen alloying is challenging due to its limited solubility during casting and solidification. Alternatively, nitriding as thermochemical treatment on the final material can
be used to significantly improve surface and bulk properties. By studying the Fe-N metallurgy with quantum physical simulation software, the aim is to gain a good understanding of the mechanisms
of nitrogen diffusion, precipitation and the interactions with other alloying elements. The translation of this fundamental knowledge could help in the development of new breakthrough steel
metallurgies based on the Fe-N system. You will be closely involved in a PhD project on iron-nitrogen steels. This project is funded and guided by OCAS NV, a market-oriented research center that
produces the experimental material.
Figure 1: Left: Nitrogen is precipitated out of the Fe matrix in the form of Fe[4]N (slower cooling rate). Right: Nitrogen is in solid solution in the Fe matrix (faster cooling rate).
a) Computational
Calculations on the electronic level require a quantum mechanical treatment. Density-functional theory (DFT) enables us to model any Fe-N material without experimental input (ab initio). This
yields the materials properties at zero kelvin. To extend these to finite temperatures, we use the quasiharmonic approach. Phonon spectra are derived from DFT calculations at different volumes,
which accounts for thermal expansion, from which the vibrational contribution to the free energy can be derived. Because steels are magnetic materials, we need to take into account the magnetic
entropy as well. This can also be done with DFT, by calculating the magnetic exchange interactions between atoms. All of these elements will add up to a fully ab-initio thermodynamic picture of
compounds relevant for the iron-nitrogen steels. These can be used to obtain relative phase equilibria[Figure 2 (left)], a solvus, or simply the heat capacities [Figure 2 (right)].
Figure 2: Left: Fe[16]N[2] dissociation in Fe[4]N and Fe[N] above 400K (blue line), this coincides with a loss of crystallinity in Fe[16]N[2] (red symbols). Right: Heat capacity of bulk Fe[4]N,
with all relevant contributions to finite temperature. The peak in magnetic heat capacity shows the second-order phase transition from the ferromagnetic to the paramagnetic state.
At this point, we have succeeded in calculating the thermodynamic properties, such as free energy and heat capacity, for Fe with N in solid solution, Fe[4]N and Fe[16]N[2]. Now, we wish to expand
the ab initio modeling to more complex alloys. Three possible thesis work plans are listed below, but your input is welcome as well.
• The interactions between the interstitial N and substitutional ternary elements will determine much of the alloying effect. Calculating the defect-defect interaction can be done with
supercells. These supercells can subsequently be used to extend the interaction energy to finite temperatures. By comparing different alloying elements, we may understand the properties of the
Fe-N steels and optimize them for use in applications.
• Taking into account the disorder that will be present when a second alloying element is added is quite a challenge. Both the vibrational and magnetic entropy become much more difficult to
calculate because of it. Two methods that have gained notable interest in the last couple of years to tackle this problem are the itinerant coherent potential approximation (ICPA) and the band
unfolding method. The goal is to select one or both of these methods and apply it to a well-known ternary compound, for example Fe[4-x]Ni[x]N, and investigate the phonon spectra, magnetisation
and free energy. Comparison to a traditional supercell approach will learn to what extent disorder plays a role in Fe-N alloys.
• Fe[3]N[1+y] is often precipitated in non-equilibrium crystallographic form. Y has a very wide range (-0.40 < y < 0.48) depending on the nitrogen concentration and cooling procedure. The
energetic cost of this off-stoichiometry can be investigated by calculating the vacancy and interstitial formation energy with DFT at finite temperatures. Off-stoichiometry of precipitates is a
very important topic in metallurgy. A successful ab initio treatment of Fe[3]N[1+y] would be a significant advancement.
b) Experimental
If it interests you, access to iron-nitrogen steel samples provides you with the opportunity to include an experimental section. The simulation of the cooling process can be investigated with
salt-bath cooling, subjecting the samples to different heat treatments. Afterwards, the precipitates present in the iron matrix can be investigated with optical microscopy, electron backscatter
diffraction or transmission electron microscopy. These characterization experiments, guided by professor Roumen Petrov and Vitaliy Bliznyuk, can help uncover the orientation relationship, habit
planes and morphology of the precipitate phases.
Physics: Electronic structure theory, quasiharmonic approach, Curie temperature
Engineering: Fe-X-N ternary alloys. Phase diagram, experimental materials characterization (optional)
1. Study programme
Master of Science in Engineering Physics [EMPHYS], Master of Science in Sustainable Materials Engineering [EMMAEN], Master of Science in Physics and Astronomy [CMFYST]
For Engineering Physics students, this thesis is closely related to the cluster(s) MODELING, MATERIALS, NANO
density-functional theory, steel nitriding, quasiharmonic approximation | {"url":"https://molmod.ugent.be/subject/developing-new-iron-nitrogen-steels-ab-initio-thermodynamics-0","timestamp":"2024-11-05T10:23:26Z","content_type":"text/html","content_length":"24702","record_id":"<urn:uuid:2e8ba753-872f-4813-af1b-8dd00745acbd>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00567.warc.gz"} |
Homeruns Are King
It’s a Tuesday night, the Cubs game is postponed, and I should either be working or doing homework. Instead, here I am exploring homeruns in baseball because it is way more fun. I’d argue that aside
from seeing something historical (no hitter/perfect game), most fans want to see home runs when they go to games.
In this analysis, I perform some data cleanup to calculate raw numbers from the %’s that fangraphs provides and shift around some data to build a predictive model to predict homerun totals. The
inputs to this model will be the players previous season statistics. Homeruns are becoming increasingly important in baseball, and if you read my analysis on walks, you’d remember that they are being
hit at an alarming rate. Teams are building their teams around power hitters, but want to be fiscally conscious. So if a team can identify a future homerun hero and save some cash, it’s a major win.
Data Support
To begin, let’s take a look at the league leader and league average in homeruns over the last 18 seasons.
Take out Barry Bond’s 73 home run season in 2001 (which is crazy!) and 2017 had the highest single season HR total by a player over the entire time frame analyzed. The season high total has risen
each year since 2014. Perhaps coincidentally, the league average home run total has risen each year since 2014 and 2017 had the highest league average total.
Whether this increase is driven by worse pitching, better hitting, or juiced baseballs, the increase in home runs is real and needs to be taken seriously. The game is shifting towards power now more
than ever. Teams that are in the Top 5 in hit home runs average 87 wins while teams in the bottom 5 average 72 wins. In a sport where every win and loss matters, this could be the difference between
a wild card play-in game and winning your division, or missing the playoffs entirely.
The plot below illustrates the average win total by season for the best and worst of the league at hitting home runs.
Next, let’s actually see how accurately we can predict a players home runs. Since it’s a Tuesday night and I should be doing homework, let’s use a modeling technique that is great at quickly giving
answers, but is somewhat of a black box. Random Forests are essentially decision trees, but tons of them all put together and then the average answer is the output. Using a random forest is typically
not a great solution, but instead a great first step. Random Forests are a quick, dirty solution that you can use to back up your initial findings from exploratory data analysis.
Typically, once you get answers from a model like this, you would build more in depth models by analyzing what variables are most important, but we’re not going to do that tonight.
The inputs to our model are going to be ~20 offensive statistics from the prior year. Since this is typically an early stage in the analysis and data science is an art, I’m going to leave Team and
Season in the inputs. Logically thinking, I think using Team and Season as a predictor make sense. As we saw above, with the change in season came an increasing average homerun total and this could
be important in predicting what a player does next year (think of it as inflation). Similarly, there is a case to be made for including team. Teams play half of their games in one park and some of
these parks are more hitter friendly or pitcher friendly. We want our model to pick up the fact that more homeruns are hit in the Red’s stadium than the Padres stadium each year.
First, I’ll split the dataset into 2 datasets – 1 to build the model and one to test the model on. Then after building the model we can see how accurate the model is by comparing how many homeruns
were hit vs. how many were predicted on the test set. In this case about 3500 players + season combinations were randomly selected from 2000-2016 as a test, and about 1500 were used in the test
Now, I’ll quickly gloss over the model call: we will use 500 trees in our random forest. Loosely put, this means we’ll be building 500 models and then average them all together for a final solution.
In each one of these models, I specified that 7 variables be used. This will help prevent our model from overfitting (or focusing too much on one situtaion). I choose 7 for analytical reasons,
typically in random forests standard practice calls for using about #predictors / 3, which in our case was 22 / 3 = 7 (roughly).
The graph below will help visualize the results of the model.
Interpreting this graph looks intimitading (and it should be because we just used machine learning!), however it is actually very simple. You can think of these graphs as listing the most important
variables in the model once it has been built. For example, previous seasons homeruns and a batters ISO (isolated power) are strong predictors of how many homeruns a player will hit next year. On the
flip size, things listed as the bottom (playerID, season, and baseruns) are not hugely important in predicting homeruns.
Armed with this information, typically we could build more predictive models, but let’s stick with this for now.
Accuracy and Results
So how accurate was the model? Of the 1500 players in our test set, they went on to hit 19,829 home runs. Our model predicted 19,522 home runs, or 98% of the actual! Now this is not a typical measure
of model accuracy, but I think it makes sense given the environment we are considering.
Looking at the predictions, there are some cases where our model was dead accurate and some were it was very very wrong. Consider one observation: in 2017 the model predicted 22 home runs for
Giancarlo Stanton…and he went on to hit 59. Stanton had one of the best offensive seasons in recent memory. Typically models are not very good at picking up outliers like this, so it is
With that said, what can we expect for 2018 given this model?
Our model predicts that the league average home run will continue to rise to about 14.5 and that the top ten players in home runs will be:
1. Aaron Judge – 36
2. Giancarlo Stanton – 33
3. Josh Donaldson – 30
4. Justin Smoak – 30
5. Nelson Cruz – 29
6. Charlie Blackmon – 29
7. Paul Goldschmidt – 29
8. Nolan Arenado – 29
9. Marcell Ozuna – 28
10. Joey Votto – 27
11. Jonathan Schoop – 27
12. Jose Abreu – 27
I suspect these totals to be a tad low, given the rise in home runs over the last few years, and this is certainly a drawback of using a quick and dirty model. Had we continued, we could tune our
model to be more sensitive to higher home run seasons and naturally inflate the totals. Also, this is certainly where the art in data science comes in. We know that home runs will be high, and we
have to make that reflection in our modeling approaches.
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