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Word Problem Solver-Free AI-powered math solutions
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Word Problem Solver-AI-powered math solutions
AI-Powered Solutions for Math Problems
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Introduction to Word Problem Solver
Word Problem Solver is an AI-based tool designed to help users understand and solve mathematical word problems. Its primary function is to provide step-by-step solutions to a variety of math-related
queries, ranging from basic arithmetic to complex algebraic equations. The tool emphasizes explaining the reasoning behind each step, ensuring that users not only get the correct answer but also
understand the process. For example, if a user is given a problem about dividing a number of apples equally among friends, Word Problem Solver will break down the division process, clarify any
necessary mathematical concepts, and guide the user through the solution.
Main Functions of Word Problem Solver
• Step-by-Step Solutions
Solving a quadratic equation like x^2 - 5x + 6 = 0
A student struggling with algebra can input their problem, and Word Problem Solver will provide a detailed breakdown of the solution, explaining each step from factoring the equation to finding
the roots.
• Concept Clarification
Explaining the concept of fractions and their operations
If a user doesn't understand how to add fractions, Word Problem Solver will explain the concept of a common denominator, show how to convert fractions, and guide through the addition process.
• Application in Real-Life Situations
Calculating the total cost of items after applying a discount and tax
A shopper wants to know how much they will pay after a 20% discount and 8% tax on a $50 item. Word Problem Solver will walk through each step, from calculating the discount to adding the tax.
Ideal Users of Word Problem Solver
• Students
Students at various educational levels, from elementary to college, can benefit from using Word Problem Solver. It aids them in understanding complex math problems, offers homework help, and
provides additional learning support to reinforce classroom instruction.
• Educators and Tutors
Teachers and tutors can use Word Problem Solver as a supplementary tool to explain difficult concepts to students. It can serve as a resource for creating lesson plans, offering additional
practice problems, and providing detailed solutions that enhance their teaching methods.
How to Use Word Problem Solver
• 1
Visit aichatonline.org for a free trial without login, also no need for ChatGPT Plus.
• 2
Prepare your word problem or mathematical query in clear, concise language.
• 3
Enter your problem into the chat interface and submit it for analysis.
• 4
Review the detailed step-by-step solution provided by Word Problem Solver.
• 5
Use the explanations to understand the methodology and apply similar techniques to other problems.
• Problem Solving
• Homework Help
• Exam Preparation
• Study Aid
• Concept Review
Word Problem Solver Q&A
• What types of math problems can Word Problem Solver handle?
Word Problem Solver can handle a wide range of mathematical problems, from basic arithmetic to complex algebra, geometry, and even calculus. It is designed to provide clear, step-by-step
solutions to help users understand the underlying concepts.
• Do I need any special software or subscription to use Word Problem Solver?
No special software or subscription is required. You can access Word Problem Solver by visiting aichatonline.org for a free trial without the need for login or ChatGPT Plus.
• Can Word Problem Solver help with word problems in subjects other than math?
Yes, while Word Problem Solver is primarily focused on mathematical problems, it can also assist with word problems in related subjects that require logical reasoning and problem-solving skills.
• How does Word Problem Solver ensure the accuracy of its solutions?
Word Problem Solver uses advanced AI algorithms to analyze and solve problems. It provides detailed explanations and verifies each step to ensure accuracy, helping users understand the solution
process thoroughly.
• Is Word Problem Solver suitable for all age groups and education levels?
Yes, Word Problem Solver is designed to be user-friendly and accessible for all age groups and education levels. It provides explanations that cater to both beginners and advanced learners,
making it a versatile tool for education. | {"url":"https://theee.ai/tools/Word-Problem-Solver-2OTo9yv39U","timestamp":"2024-11-04T04:25:49Z","content_type":"text/html","content_length":"104943","record_id":"<urn:uuid:392bb237-6fae-4274-b484-e98025fbf304>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00248.warc.gz"} |
Joules to volts
Joules to volts calculator
Joules (J) to volts (V) calculator.
Enter the energy in joules, charge in coulombs and press the Calculate button:
Joules to volts calculation
The voltage V in volts (V) is equal to the energy E in joules (J), divided by the charge Q in coulombs (C):
V[(V)] = E[(J)] / Q[(C)]
See also | {"url":"https://jobsvacancy.in/calc/electric/Joule_to_Volt_Calculator.html","timestamp":"2024-11-07T05:55:15Z","content_type":"text/html","content_length":"8423","record_id":"<urn:uuid:e89fb857-4ba0-418e-8084-d0f9e2e83b76>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00362.warc.gz"} |
Radical expressions interactive
radical expressions interactive Related topics: solving addition equation word problems
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Author Message
Cindic_Nad Posted: Sunday 15th of Jun 09:22
Can somebody help me? I am in deep difficulty. It’s about radical expressions interactive. I tried to find somebody in my locality who can lend a hand with decimals, least common
denominator and binomial formula. But I was unsuccessful. I also know that it will be not easy for me to bear the expense . My assignment are coming up very soon . What should I do?
Somebody out there who can help me?
Back to top
Vofj Posted: Monday 16th of Jun 18:37
Timidrov I think I know what you are searching for. Check out Algebrator. This is an excellent product that helps you get your homework done faster and right. It can help out with problems in
radical expressions interactive, roots and more.
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MoonBuggy Posted: Wednesday 18th of Jun 08:50
I discovered a number of software programs that are appropriate. I checked them out . The Algebrator turned out to be the best suited one for powers, percentages and binomials. It was
also uncomplicated to work with . It took me step by step towards the answer rather than only giving the solution. That way I got to learn how to solve the problems too. By the time I was
done with it , I had learnt how to explain the problems. I found them useful for College Algebra, Algebra 2 and College Algebra which helped me in my math classes. May be, this is just
the thing for you . Why not try this out?
Leeds, UK
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Dam-gl Posted: Thursday 19th of Jun 10:30
Thank you very much for your help ! Could you please tell me how to download this program? I don’t have much time since I have to solve this in a few days.
Back to top
Mibxrus Posted: Thursday 19th of Jun 20:46
The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://softmath.com/ordering-algebra.html.
Back to top | {"url":"https://softmath.com/algebra-software/subtracting-exponents/radical-expressions.html","timestamp":"2024-11-04T13:56:38Z","content_type":"text/html","content_length":"41335","record_id":"<urn:uuid:42154c6d-1f96-47e8-ae71-bf4886ef87e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00820.warc.gz"} |
Bagging with decision trees
A base learning algorithm (e.g., decision tree, neural network) is trained independently on each bootstrap sample. Therefore, each decision tree is likely to be slightly different due to the
variations introduced by bootstrapping. One popular implementation of bagging with decision trees is the Random Forest algorithm. In Random Forest, each tree is trained on a random subset of features
as well as a random subset of data, adding an extra layer of diversity to the ensemble.
Figure 3739 shows bagging in decision trees. In this example, the "Original" dataset is generated with 100 data points. For each "Bootstrap Sample," the same number of data points as the original
dataset is used. The bootstrap sampling is done with replacement, so some data points may be repeated in a particular bootstrap sample. In the current case, there are 5 bootstrap samples, and each
bootstrap sample has 100 data points. Therefore, the total number of data points used across all bootstrap samples is 5 * 100 = 500. In the figure, each bootstrap sample is used to train a single
decision tree. That is, there are 5 bootstrap samples, and each bootstrap sample is used to train one decision tree.
Figure 3739. Bagging in decision trees (Code). | {"url":"https://www.globalsino.com/ICs/page3739.html","timestamp":"2024-11-12T06:37:07Z","content_type":"text/html","content_length":"13646","record_id":"<urn:uuid:f2a273f6-294e-48d2-b37b-a2e1e43bc327>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00159.warc.gz"} |
How much does an air conditioner weigh? - CleanCrispAir
How much does an air conditioner weigh?
During the summer, it is common to find most homeowners installing air conditioners. The one challenge they often face is the units are cumbersome.
Although there are six different types of air conditioners, it is hard to find a window unit that weighs less than 46 pounds or about 21 kilograms. You’ll need a lot of heavy lifting to install
bulkier types.
What is the average weight of an air conditioner?
On average, an air conditioner weighs between 60 and 180 pounds or 27 and 82 kilograms. But since there are different types of ACs, the weight can vary significantly.
For example, window air conditioners tend to be light. They weigh around 30 kilograms on average. You can also buy portable units that also weigh 30 kilograms.
Ductless air conditioners or indoor ACs weigh 12 kilograms. The heaviest air cons are the whole-house ACs weighing up to 80 kilograms.
The weight of an air conditioner increases functionally with its power, measured in tons or BTU. The BTU is a unit of energy. It is an acronym that stands for British Thermal Unit.
One BTU is the amount of energy required to increase the temperature of a pound of water by 1^0 F. But BTU measures how much heat it can remove in an air conditioner.
A higher rating means you have a powerful AC. The BTU value tells you the power of your air con.
What makes an air conditioner heavy?
Various factors determine the weight of an AC. These include:
1. Copper
Copper is a metal responsible for the heavy nature of air conditioners. It weighs around 253 kilograms (558 pounds) per cubic foot.
All copper components account for about 60% of the weight of smaller air conditioners. However, the weight declines significantly in heavier units.
2. Steel
If copper accounts for 60% of smaller air conditioners’ weight, steel covers the rest.
The primary function of the metal is to provide structural support. It also covers or houses the compressor.
3. Aluminum
The weight of aluminum is about 77 kilograms (169 pounds) per cubic foot. That means you can use it in place of copper.
However, it is not the best conductor of heat compared to the former.
Image credit: ramustagram
How much does a 5000 BTU air conditioner weigh?
Most air conditioners with a rating of 5000 BTU are not heavy. On average, they weigh approximately 45 pounds or 20 kilograms.
These air cons can cover a room of about 150 square feet. Manufacturers design the machines for window placement. Also, their rating is usually the lowest you will find in most home air conditioners.
How much does a 10000 BTU air conditioner weigh?
A 10000 BTU window air conditioner is best for spaces of up to 450 square feet. It is also ideal for areas with high ceilings, open floor plans, and lots of traffic.
On average, a 10000 BTU-rated AC will weigh around 65 pounds or 30 kilograms.
How much does a 12 000 BTU air conditioner weigh?
A 12000 BTU window air conditioner can cover spaces of up to 550 square feet. The AC has a dehumidifying capacity of around 3.6 pints per hour or 1.7 liters per hour.
On average, this unit will weigh about 72 pounds or 33 kilograms.
How much does a 14000 BTU air conditioner weigh?
A 14000 BTU air conditioner weighs around 80 pounds or 36 kilograms. Many of these units are powerful enough to cover spaces of up to 700 square feet.
Furthermore, they have a dehumidification capacity of about 90 pints per day.
How much does a 2-ton AC unit weigh?
When somebody says they require a 2-ton AC unit, it does not imply they are talking about the machine’s weight.
In this context, it refers to the amount of heat or British Thermal Units the AC can remove in an hour. One ton of AC capacity is equivalent to 12000 BTU per hour.
Therefore, it is safe to assume that a 2-ton AC unit has a rating of 24000 BTU. Most air conditioners with a 24000 BTU rating weigh around 160 pounds or 73 kilograms.
However, you can find some that are as heavy as 220 pounds or 100 kilograms.
How much does a 5-ton AC unit weigh?
A 5-ton AC unit is the most powerful of all ACs. It has a BTU rating of 60000. On average, the units weigh around 240 pounds or 109 kilograms.
But you can also find one that weighs more than 250 pounds.
This table summarizes the average weight of air conditioners
Air con Rating Coverage Weight Weight
(BTU) (Sq. ft.) (Pounds) (Kgs)
On average, an air conditioner with a BTU rating of 5000 weighs around 20 kilograms. It is the smallest unit you can find for window ACs.
If you buy one with a rating of 10000 BTU, it will weigh around 30 kilograms. A 2-ton AC weighs between 150 and 220 pounds.
More helpful guides | {"url":"https://cleancrispair.com/air-conditioner-weight/","timestamp":"2024-11-10T06:13:28Z","content_type":"text/html","content_length":"170639","record_id":"<urn:uuid:782f52fc-399e-4904-b934-60c2cc311292>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00459.warc.gz"} |
Year 7 Scheme of Learning
Solving problems with multiplication and division
Term 2 starting in week 3 :: Estimated time: 3 weeks
• Properties of multiplication and division
• Understand and use factors
• Understand and use multiples
• Multiply and divide integers and decimals by powers of 10
• Convert metric units
• Use formal methods to multiply integers
• Use formal methods to multiply decimals
• Use formal methods to divide integers
• Use formal methods to divide decimals
• Understand and use order of operations
• Solve problems using the area of rectangles and parallelograms
• Solve problems using the area of triangles
• Solve problems using the mean
For higher-attaining pupils:
• Multiply by 0.1 and 0.01
• Solve problems using the area of trapezia
• Explore multiplication and division in algebraic expressions
This page should remember your ticks from one visit to the next for a period of time. It does this by using Local Storage so the information is saved only on the computer you are working on right | {"url":"https://transum.org/Maths/National_Curriculum/SOLblock.asp?ID_SOL=7","timestamp":"2024-11-07T12:58:11Z","content_type":"text/html","content_length":"58170","record_id":"<urn:uuid:c866561f-12e0-4ca5-b3f1-ccb5a31bfff2>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00536.warc.gz"} |
The 26.2 Blog—Wolfram Blog
The 26.2 Blog
It’s four months into the new year. Spring is here. Well, so they say. And if the temperatures do not convince you, the influx of the number of runners on our roads definitely should. I have always
loved running. Despite the fact that during each mile I complain about various combinations of the weather, the mileage, and my general state of mind, I met up with 37,000 other runners for the
Chicago Marathon on October 11, 2015. As it turns out, this single event makes for a great example to explore what the Wolfram Language can do with larger datasets. The data we are using below is
available on the Chicago Marathon results website.
This marathon is one of the six Abbott World Marathon Majors: the Tokyo, Boston, Virgin Money London, BMW Berlin, Bank of America Chicago, and TCS New York City marathons. If you are looking for
things to add to your bucket list, I believe these are great candidates. Given the international appeal, let’s have a look at the runners’ nationalities and their travel paths. Our GeoGraphics
functionality easily enables us to do so. Clearly many people traveled very far to participate:
The vast majority, of course, came from the US:
Let’s create a heat map to see the distribution of all US runners. As expected, most of them are from Chicago and the Midwest:
What did the race look like in Chicago? Recreating the map in the Wolfram Language, taking every runner’s running times, and utilizing my coworker’s mad programming skills, we can produce the
following movie:
As you can see, the green dot is the winning runner. I am red, and the median is shown in blue. This movie made me realize that while the fastest runner was already approaching the most northern
point of the course, I was still trying to meet up with my running partner! The purple bars indicate the density of runners at any given time along the race course. You might wonder what the gold
curve is. That would be the center of gravity given the distribution of the runners.
The dataset also includes age division and placement within age group, gender and placement within gender group, all split times, and overall placement. The split times were taken every 5 km, at the
half-marathon distance, and, of course, at the finish line. The following image illustrates the interpolated split times for all participants after deducting the starting time of the winning runner:
The graphic reflects several things about this race: runners were grouped into two waves, A and B, depending on their expected finishing time. This is illustrated by the split around 2,500 seconds at
the starting line. Within each wave, runners were then grouped into corrals. Again, faster runners started in earlier corrals. Thus the later runners got started, the slower they were overall. The
resulting slower split times are expressed in a much faster rise of the corresponding lines. It also took 4,503 seconds, a little over 75 minutes, for all runners to get started. In contrast, the
last person crossed the finish line 19,949 seconds after the winner of this race. I was neither…
Let’s take a more detailed look at everyone’s start and finish in absolute time. We’re letting the first runner start at 0 seconds by subtracting his time from all participants’. The red dots
indicate the mean of the finish time for runners with the exact same starting time:
Again, the two waves are clearly visible. The smaller breaks within each wave indicate the corral changes. But what caught my eye was the handful of people preceding the first wave. Because the
dataset provides us with the names of the participants, I was able to drill down and find out whose data I was looking at: it is the “Athletes with Disabilities” (AWD), as the group is named by the
Chicago Marathon administration. Checking back with the schedule of events, I was able to confirm that this group started eight minutes ahead of the first wave.
Let’s investigate a bit more and see what we can learn about this group. Of course, the very first person to cross the starting line is part of this group. Everyone else started very closely around
him. We can query for the AWD subgroup by looking for everyone who started within a generous 200 seconds of the first person. We find that there were 49 members in this group:
Here is the plot of their start and finish times. It is equivalent to a zoom on the 0-second start line in the above plot:
Due to their physical disabilities, many of these runners were joined by one or two guides who helped them navigate the course. With our Nearest functionality, we can try to identify such groups. We
just need to gather everyone’s time stamps, convert them to UnixTime, and define our Nearest function:
Let’s find the group of nearest people for all 49 runners by limiting the variations of their time stamps to 10 seconds over the course of the race:
Out of the 49 runners, we find that 35 ran in 15 groups of 2 or more people:
These are the groups we could identify:
I tip my hat to everyone who participated in this race. But I am in awe of people running a marathon with a physical disability. I would like to give them, as well their guides, a special shoutout!
Did I run with someone? As mentioned above, I sure did. I am lucky to have my next-door neighbor Michael as my running partner. Cursing and whining during a long run is a lot easier if you have
someone on your side. Otherwise you just look crazy while mumbling to yourself. Let’s build the Nearest function:
Then we can apply it to the entire dataset. Any result of length greater than 1 indicates a running group. We find that 2,784 runners ran in 1,394 groups:
There were 1,329 groups of 2, 62 groups of 3, and 3 groups of 4. The latter were:
By the way, you will not find my and Michael’s names in any of these groups. Why? Because there was nothing in this world that could keep Michael from his tenth attempt to finish the marathon in
under four hours—whereas halfway through the race I had to give in to that nagging voice telling me to take a break and walk. Just taking the first half of the race into account, here we are:
We finished only three minutes apart, but that can be a whole lot of time during a marathon. Michael came in just under four hours; I barely missed that time.
Now let’s take a look at how the race progressed split by split. The following histograms show how participants’ split times compared to the mean time at each split distance:
Interestingly, for each split the curve shows a little bump just before the 0 marker, which indicates the mean split time. To find out which runners these might be, we have to consider who the
participants are. The vast majority are recreational marathon runners. We hope to stay injury free and maybe achieve a personal record, but our goal is to have a great experience and a rush of
endorphins. We are not there to win and collect prize money. But, as Michael did above, one thing that people might attempt is to break the illusive four-hour mark. To beat four hours, a runner—let’s
call her “Molly”—has to average 341.517 s/km, or 9 minutes and 9 seconds per mile:
To make sure Molly comes in under four hours, let’s assume she runs at a pace five seconds faster per kilometer, 336.517 s/km. By not allowing any change of pace, we are basically turning Molly into
a robot. But let’s see where her split times (indicated in red) fall compared to the mean at each of the kilometer markers. Indeed, Molly’s split times match the “hump,” and thus are a representation
of all runners trying to finish the marathon in less than four hours:
As can be seen in the above histograms, with each split we plot more bins representing fewer runners, while the variations from the mean steadily increase. Here is another look at the same fact, just
from a different angle. Again taking the differences of the runners’ split times to the mean, and then sorting them from smallest to largest, we can see how the differences between the fastest and
slowest runners steadily increase over the course of the race:
Again, the group of people trying to finish in under four hours is nicely visible in the small hump to the left of the y axis. How many people did make it in under four hours? We could not make this
number up: it was exactly 11,111 people, or 29.7% of all participants:
As mentioned above, I could not keep pace with Michael after about halfway through the race. But let’s look at “keeping pace” and how consistent people ran their race. The dataset provides all the
information we need to look at everyone’s average pace and absolute variations from it at each split. Adding up those variations per person gives us the following picture:
The maximum of accumulated variations from the average pace is around 10 minutes. I averaged 9 minutes and 16 seconds:
My variations from that average added up to almost three minutes:
In the charts below, we are looking at the distribution of those variations versus a runner’s finishing time. Since a slower runner takes more time between splits and thus automatically accumulates
more minutes and more variations, we additionally normalized the pace variation by the corresponding finishing time:
Of course, these pace variations cause people to pass each other. Let’s have a quick look at how often this happened. We counted an amazing 276,121,258 occurrences of runners’ position changes. Below
is an illustration. Inside the attached notebook, please hover over the data points to see the number of takeovers at a given distance:
To explain the numerous peaks, we should have another look at the race. Every mile or two, aid stations were providing runners with fluids, medical assistance, and other necessities. These aid
stations were about two city blocks long, giving runners plenty of opportunities to move through and to avoid crowds. Consider the aid stations on the map:
Also consider their locations along the course by using our new GeoDistanceList function:
We can nicely match the peaks with the locations of the aid stations. At each of these points, a huge number of runners change their paces, resulting in the jump in takeovers. While taking in fluids,
one runner might choose to walk while another just slows down but continues to run. A third runner might not utilize the station at all and run through it. Turns out I am not very gifted when it
comes to drinking while running, so I walk whenever necessary.
Interestingly, a Histogram3D of time versus distance versus the number of takeovers looks like the city of Chicago itself:
Running a marathon does not just take a good number of months of training, battles with injuries, and bouts of laziness (as well as a good sense of the craziness of this endeavor). It also takes a
financial commitment. Race registration and travel costs can add up to an intimidating sum of money. This made me wonder if there is a correlation between travel distance and finishing time, i.e. can
I assume that the farther you have to travel and the more money you have to spend on the event, the better you are as a runner? The following plot shows the finishing time versus travel distance to
the US. Upon hovering inside the notebook, you can see the runners’ countries, their finishing times, and their overall placement in the race:
Clearly my assumption is incorrect. We do see a small number of runners from Kenya and Ethiopia who traveled thousands of miles and came in first. But we also see runners who traveled all the way
from India, New Zealand, Indonesia, Swaziland, and Singapore who finished in more than six or seven hours. The means for these countries are all around six hours.
Let’s see if another assumption can be proven wrong, e.g. if the travel expense is not as prohibitive as thought, does the number of runners from a country decrease with increasing travel distance?
And could it be true that the more runners a country has in the race, the higher its GDP per capita is? In the notebook, hover over each data point in the charts below to see the country, number of
runners from that country, and travel distance or GDP per capita:
The data is not as obvious as one might think. More than 28,000 participants came from the US, whereas only a single person came from countries such as Réunion and Mauritius. We do have a number of
countries with less wealth and only single-runner representation. But the single-runner representation also holds true for Qatar and Luxembourg—both known for their financial muscle.
I’ll admit that the country of origin might not be as much of a statement about the size of one’s wallet or someone’s performance as I might have thought. What about age?
Marathons seem to appeal mainly to people in their mid-twenties to mid-forties. And, of course, the higher your age, the better your chances of winning your division. But what is interesting to see
is that this is not actually a sport favoring the younger athletes. The fastest times were achieved by the 40–44 age division. So I might still have my Olympic years ahead of me!
To add a note of obscurity: have you ever considered if your name is any indication of your performance? Or if there are other runners by your name in this exact race? There are many shared first and
last names. If you were a “Cabada” or a “Zac” in this race, you did awfully well:
You may have guessed the most common first name: there were 641 Michaels. The leading last name was, also not very surprising, “Smith” with a count of 157. Of course, these numbers decrease
considerably when we look at shared full names:
And the most common full names and their counts are:
The combination of my family watching on the sidelines, including my mother visiting from Germany, the outstanding work of all the volunteers, and the huge crowds of spectators and the entertainment
they provided, all made for a memorable race. Plus the weather, which is usually a liability in Illinois, was just impeccable. Both Michael and I had a blast, which I think is visible using
But as it turns out, not just the event itself was fun. This was a great dataset for me to play around with and learn a lot more about the capabilities of the Wolfram Language. I am not a developer,
but I greatly enjoyed this opportunity to combine my professional and personal lives. If you are interested in more scientific approaches to the topic of marathon running, you might find this article
and this article intriguing.
But most importantly, registration is now open for the 2016 event!
Download this post as a Computable Document Format (CDF) file. New to CDF? Get your copy for free with this one-time download.
Join the discussion
10 comments
1. Wow! Much more in marathoning stats than I expected. Yes, see you in Tokyo 2020!
2. It would be great if the code that produced the movie can be made available.
3. What a wonderful analysis of running statistics. Let’s not forget to take some extra time on the roads and watch out for those runners. Happy Running!
4. You realise that people across the world (i.e. both hemispheres) use Mathematica? The seasons are different in different hemispheres.
5. this godd blog thank
6. Awesome , Thank You Eile For this Amazing Article .
7. Lots of data summed up nicely! Appreciate the infographics!
8. Nice article. Wonderful analysis of running statistics and I really like the infographics.
9. Thank You Eile For this godd blog and Amazing Article …. Love it. | {"url":"https://blog.wolfram.com/2016/04/15/the-26-2-blog/","timestamp":"2024-11-06T14:04:33Z","content_type":"text/html","content_length":"210216","record_id":"<urn:uuid:8688b81b-b838-4071-b852-1dc766473717>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00477.warc.gz"} |
no graph, but i labeled steps to draw the graph
As shown in the graph, the area of regular hexagon ABCDEF is 72, P is the midpoint of EF. What is the area of the shadow?
Graph: Step 1: Draw regular hexagon ABCDEF. Step 2: Connect AD. Step 3: Mark the midpoint of EF as P, and connect BP. Name the intersection between these two lines as M. Find [AFPM]
celestial Oct 20, 2024
Analyzing the Problem
Regular hexagon ABCDEF
Area of hexagon ABCDEF = 72
P is the midpoint of EF
We need to find the area of triangle APF (the shadow)
Find the area of triangle ADF.
Find the ratio of the area of triangle APF to triangle ADF.
Calculate the area of triangle APF.
Step 1: Find the area of triangle ADF.
Since a regular hexagon can be divided into six equilateral triangles, the area of one equilateral triangle is 72/6 = 12.
Triangle ADF is made up of two equilateral triangles.
Area of triangle ADF = 2 * 12 = 24
Step 2: Find the ratio of the area of triangle APF to triangle ADF.
Triangle APF is similar to triangle ADF (both right triangles with the same angle at A).
The ratio of their areas is the square of the ratio of their corresponding sides.
Since P is the midpoint of EF, AP = AF/2.
The ratio of the areas of triangle APF to triangle ADF is (AF/2)^2 / AF^2 = 1/4.
Step 3: Calculate the area of triangle APF.
Area of triangle APF = (1/4) * Area of triangle ADF
Area of triangle APF = (1/4) * 24 = 6
Therefore, the area of the shadow (triangle APF) is 6 square units.
AUnVerifedTaxPayer Oct 20, 2024 | {"url":"https://web2.0calc.com/questions/no-graph-but-i-labeled-steps-to-draw-the-graph","timestamp":"2024-11-12T23:57:11Z","content_type":"text/html","content_length":"22450","record_id":"<urn:uuid:a22ba981-9cb4-47c9-a8cf-9652740f7820>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00449.warc.gz"} |
Kinetics of the Thermal Decomposition of Alum Sourced from Kankara Kaolin
Volume 03, Issue 02 (February 2014)
Kinetics of the Thermal Decomposition of Alum Sourced from Kankara Kaolin
DOI : 10.17577/IJERTV3IS20732
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Olubajo O. O., S. M. Waziri, B. O. Aderemi, 2014, Kinetics of the Thermal Decomposition of Alum Sourced from Kankara Kaolin, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume
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Kinetics of the Thermal Decomposition of Alum Sourced from Kankara Kaolin
1Olubajo O. O., 2S. M. Waziri, 2B. O. Aderemi
1Department of Chemical Engineering, Abubakar Tafawa Balewa University Bauchi, Nigeria
2Department of Chemical Engineering, Ahmadu Bello University, Zaria, Nigeria
ABSTRACT- This study investigated the kinetics of the thermal decomposition of alum produced from Kankara kaolin. The work involved preparation of alum as well as calcination of the prepared alum.
The Coats-Redfern kinetic model was adopted in estimating the kinetic parameters for the decomposition reactions. The experiment was conducted using a muffle furnace and the rate data were obtained
using thermogravimetric (TG) and Xray fluorescence (XRF) analyses. From the TGA experimental data, an average order of reaction, n of 1.3 with corresponding activation energy value of 321.03 kJ/molK
and frequency factor 2.57E+14min-1 gave the smallest residual deviation values and best fitting curves compared to other reaction orders tested within the range of 0 to 2, while equivalent values
from XRF residual sulphate data are n = 1, activation energy = 337.22 kJ/molK and frequency factor of 5.27E+14 min-1. The discripances in the corresponding values from the two modes of data
generation is attributed to the differences in the reaction mechanisms captured by the different methods of analysis. While, the XRF solely followed the sulfate decomposition phenomenon, TGA was more
encompassing including dehydroxylation and impurity decomposition reactions.
(KEYWORDS: Kankara Kaolin, decomposition, Coats-Redfern model, TGA, XRF, kinetics)
1. INTRODUCTION
The great interest in alumina is mostly due to its ever increasing applications as adsorbents, catalysts support, refractory, filler and as constituents of various household products. Invariably,
bauxite has been the dominant source alumina, through the popular Bayers process [1]. However, notwithstanding the acclaimed bauxites relative economic advantages over all other alternative
sources, its rapid depletion in quantity and quality globally is an impetus promoting intensive research activities on alumina extraction from clay materials worldwide [2]. In fact, this is a
welcome development to many countries such as Nigeria that are not naturaly endowed with bauxite but having kaolin in abundance.
It must be admitted that in the last decade, a lot of research efforts have been expended in producing alums, alumina and silica for various purposes from Kankara kaolin clay [3-8]. However, it
is of note that while much of the efforts were geared towards establishing favourable process conditions,
apparently, no such attention has been paid to the prevailing kinetics.
The work of Moselhy et al. (1994) [9] , on thermal treatment of aluminium sulfate hydrate made a fundamental contribution in addressing the peculiarity of alum decomposition including the
testfitting of the Coat-Redfern kinetic model. However, they limited their consideration to only the ideal solid-state reaction cases of order n= ½, 2/3 and 1, thus ignoring the obvious
peculiarity of alum. At the onset, it is obvious that a well crystalline alum behaves as a non porous material, in which the shrinking core reaction model ought to describe, howbeit, as reaction
progresses and it is evacuated of the occluded free water and sulfate ions, it becomes a porous material to mimic a diffusion-progressive phenomenon. Even at that, decomposition of alum to
release the bonded waters (water of crystallization) requires different energy level (activation energy) to that of sulfate decomposition, with no clear cut demarcation in time or space of the
occurence of these duo phenomena.
The strenght of the Coat-Redfern model resides in its development, which is free from the majority of the idealized models assumptions. It equally merged the kinetic constants with the
thermodynamic parameters in a single equation, thus affording the evaluation of reaction order, activation energy and reaction frequency factor from same set of data. Its major shortcoming being
that the order of reaction determination, inherently depends on the trial and error principle, whether by numerical or graphical approach.
The use of thermogravimetric data to evaluate kinetic parameters for solid state reactions involving weight loss has been investigated by a number of workers as noted by Coats and Redfern (1964)
[10] , but sad enough, the single sample, rapid, and continous kinetics calculation over the entire temperature range offered by dynamic thermogravimetry is still not executable in Nigeria, five
decades after, due to lack of requisite thermo-balance facility in the neighborhood.
Hence, this paper attempts to estimate the kinetic parameters such as activation energy, pre-exponential factor and the order of reaction using Coats-Redfern kinetic model. Therefore, the present
work attempts to explore the traditional cumbersome isothermal gravimetry under varying holding time and
temperature. The study also involves the preparation of single alum from local clay and the calcination of the produced alum at various temperatures and holding times. Gravimetric determination
was employed in monitoring the overall rate data, while X-ray Fluorescence analyses of the residual sulfate
dX = dX . dT = dX . 9
dt dT dt dT
Substituting Equation 9 into 1 and /, the subject
of the formula
concentration in the alumina were obtained to model sulphate decomposition specifically.
dX = dT
dX . 1
X 10
Derivation of Coat-Redfern Model
In general, for the reaction
dX = X dT 11
+ the disappearance of component A can be described by the formal kinetic expression:
g X = () = X dT X
dX g X = =
dt = kf X 1a
dX = f X 1b
From Arrhenius,
From calculus:
Therefore, if
dX = 1 a
= 1
Where X is the fractional conversion; t is the time; A is the pre-exponential factor; E is the activation energy; R is the gas constant; T is the temperature in Kelvin and f (X) is the kinetic
function which takes different forms depending on the particular reaction rate equation. In isothermal kinetic studies, the rate equation used to calculate the rate constant has the
Differentiating the above with respect to temperature T gives
dx dt = T 2 dT = T2dx
Substituting dT = T 2dx into Equation13:
= 2 =
g X = t 3
Differentiating with respect to time t
d g X = 4
Substituting the LHS term of Equation 4 into Equation 1a to
Since dx = 1 a
Then using integration by part
= 15
d g X f X =
Let = 1 2
= , then from Equation 14
d g X X = 6
Making dg( X ) subject of the formula
= 1 2
+ 2 1 3 16a
g X =
= + 2 16b
Integrating with respect to X
g X = ()
Taken the integration by part a step furtherand rewrite Equation 16
However, non-isothermal methods are becoming more widely
used because they are more realistic than the classical isothermal methods. In non-isothermal kinetics the time
+ 2 .
2 +
3 =
dependence on the left hand side of Equation 1 is eliminated
using constant heating rate = dT so that T = To + , where To
0 +
is the starting temperature and t is the time of heating. Using integral methods of analysis, from Equation 1:
Its obvious that the integration continues endlessly, however, from the generic definition of x = 1/T, x4 and above goes to
zero, rendering the equation undefined. Therefore, it becomes expedient to stop at the second term on the right hand bracket.
The equation has been written in the form:
1 1 X 1n
T2 1 n
= In R 1 2RT
E RT (27)
a 0
3 .
E E +
Substituting a = and x = 1 T
gives Equation 19
into Equation 18
Equation 27 satisfies for n< 1or n > 1 but for n=1, Equation 27 becomes
= 1 X
RT2 T
= e RT
eE RT
= In R 1 2RT E
E E +
And then factorize Equation 19 to give Equation 20
= RT2 eE RT 1 2RT 20
For Equations 27 and 28 to be amenable to graphical solution, the quantity 1 2RT is assumed to be close to 1, hence the need to verify the reasonability of this assumption [10].
E E
E E
= RT2eE RT 1 2RT 21
Incorporating the power law Equation 21 as follows:
n dX
2. MATERIALS
Raw kaolin used in this investigation was obtained from Kankara village, Katsina State, Nigeria, while beneficiated and metakaolin were obtained from processing of the raw clay
X =
1 X , then g X =
and calcination of the beneficiated clay respectively. Fresh alum produced in this work as described in the following
dX 1 X n
= 1
dX (22)
subsection meets general requirement of 6-9% wt of Al2O3/ wt of fresh alum. All other chemicals used were laboratory grade.
If Equation 22 was integrated to give:
1 1 X 1n
Preparation of Metakaolin from raw Kankara kaolin clay
1 n (23)
Substituting Equation 23 into Equation 21
1 1 X 1n
The raw clay was crushed using a mortar and pestle. The resultant product was then beneficiated by soaking in water and intermitent vigorious stirring for 3 days, after each day the spent water
was replaced while sand particles were removed
1 n
RT2 1 2RT E E
= E e
and discarded. The significance of removal of spent water is to faciliate the removal of soluble impurties. The kaolin suspension was then centrifuged and dried overnight at 120oC to remove free
water [11]. The dried lump was crushed and screened with 315 microns sieve. The sieved clay powder was
Dividing Equation 24 through by T2
then weighed into crucibles and calcined in a muffle furnace at a temperature of 750oC for 2 hrs [2].
1 1 X 1n
T2 1 n
= R E 1 2RT E eE RT
Dealumination of metakaolin using sulphuric acid
The dealumination of metakaolin (Al2Si2O7) was performed by reacting 50 grammes of metakaolin with 168.03cm3 of
Taking the negative natural logarithm of both sides
sulphuric acid of 96wt% (H2SO4) to give a 60 wt% acid solution [12]. A simplified chemical reaction for dealumination process is presented as Equation 29:
1 1 X 1n
T2 1 n
+ 3 + 24
2 2 7 ()
2 4 ( ) 2
= In E 1
E + E RT (26)
2 4 3 .272() + 22 (29)
Distilled water (184.27cm3) was added to quench the reaction and to enhance the seperability of the alum ladden aliquot
1 1 X 1n
T2 1 n
1 T
from the residual silica. The crystal yield obtained from 50ml of filtrate was observed to increase with decrease in temperature. This is in agreement with the established fact that the
solubility of alum decreases at reduced temperatures [13,14]. The crystallization stage was achieved by cooling at – 10oC for 3 hours to initate the growth of alum crystals and resultant content
is then filtered [2]. The hydrated alum crystals (Al2(SO4)3.27H2O) were gradually dried in an oven between 40oC and 160oC for 12 hrs. to enhance size reduction and seperation, after which the
resultant sample is heated at 350oC for 5 hours to remove the occluded acid and the chemically bonded water from the alum [9]. The dried alum was then ground and sieved with 315 microns [2].
Table 2 presents the compositional analysis for fresh alum and pretreated alums at 160oC and 350oC.
Calcination of the Pretreated Alum
40 grammes of ground alum in a crucible was calcined in a muffle furnace at 750oC, 800oC, 810o C, 850o C and 900oC and for holding time intervals of 20,60, 120,150 and 180 minutes respectively.
The residues were weighed and also subjected to X-ray flurorescence analysis after cooling to room temperature in a desiccator. A simplified decomposition reaction for aluminum sulphate is as
shown in Equation 30.
2 4 3 () 23() + 3() (30)
Kinetic parameter estimation using thermogravimetric data
The weight losses obtained were converted to conversion at different temperatures and time interval as shown in Table 5. The calculated conversion and temperature values were inputted into the
Coats-Redfern model. Plot of
T2 1n T
1 1X 1n vs 1 were made. Regressing the left side of the
above equation using the least square criteria for different values of n, the slope is E/R and the intercept is equal to
R E; from these various plots, activation energy, E and the pre-exponential term, A were evaluated [9].
Discriminatory test for X ray Fluorescence results
The samples were then characterize using a Energy Dispersive
-Xray Fluorescence machine to obtain the elemental composition by monitoring the extent of conversion of sulphate decomposition.
RESULTS AND DISCUSSION
The raw clay, beneficiated clay and metakaolin
component analysis
From the results shown in Table 1, it was observed that the alumina content of the beneficiated clay increased significantly over the raw clay from 41.74 wt % to 43.01 wt.
% while the silica content slightly decreased from 54.56 wt % to 53.95 wt %.
TABLE 1: CHEMICAL COMPOSITION OF RAW CLAY, BENEFICIATED CLAY AND METAKAOLIN
Raw Clay Beneficiated Metakaolin
% clay % %
Al2O3 41.74 43.01 44.87
SiO2 54.56 53.95 52.41
CaO 0.44 0.34 0.32
Fe2O3 0.53 0.58 0.53
MgO 1.74 1.31 1.22
K2O 0.71 0.56 0.47
Na2O 0.04 0.05 0.04
TiO2 0.10 0.07 0.07
Cr2O3 0.01 0.01 0.01
Mn2O3 0.02 0.02 0.01
NiO 0.01 0.03 0.02
CuO 0.08 0.06 0.03
ZnO 0.00 0.01 0.00
This could be attributed to the loss of organic material and free silicia respectively during the process of beneficiation. The color of the calcined kaolin changed from white to brick red
indicating the presence of Fe3+ [15].
1 1 X 1n
2RT E
2 =
T2 1 n = In E 1
E +
RT (26)
Coats-Redfern model was adopted to estimate the kinetic parameters such as activation energy, pre-exponential factor and the order of the reaction were obtained by plotting a graph [9]
Effect of drying on alum
Drying at 160oC evidently was able to drive off the adsorbed free water on the alum and possibly part of the occluded water, while heating to 350oC was believed to be potent enough not only to
eliminate the remaining occluded water but also the occluded excess acid as well. Table 2 showed these effects resulting in increase in concentration of alumina in the two heating steps; while
SO2 increased slightly in the first step, evidence of loss of free water and dropped partially in the second step, indicating loss due to occluded acid evacuation.
TABLE 2: ELEMENTAL COMPOSITION OF FRESH ALUM, THERMAL PRETREATED ALUMS AT 160OC AND 350OC
Fresh Alum Dried at 160oC Dried at 350oC
Al2O3 8.41 8.7 16.02
SO3 60.4 61.4 60.81
CaO 0.46 0.38 0.2
Fe2O3 0.18 0.24 0.14
MgO 0.17 0.02 0.31
K2O – 0.15 0.01
Na2O 0.03 0.09 0.11
TiO2 0.08 – 0.06
NiO – – 0.01
This observation is consistent with Moselhy et al. (1994) [9] differential thermal analysis (DTA) on a hydrated aluminium sulfate. The pretreated alum at 160oC comprised of about 8.41 wt/wt % of
alumina which agrees with Alan et al, (2000) [16] that liquid alum contains about 8 wt/wt%.
Calcination of the partially dried alum (at 350oC)
The residual weight of the alum samples (in percentage of the initial quantity) obtained after calcination at different
temperatures and time intervals are shown in Figure 1. Figure 2 shows the percentage of sulphate decomposed at various temperatures from thermogravimetric analysis. From Figures
1 and 2, it could be observed that the minimum residual weight of alum samples stood at 12.67g, while the maximum sulfate decomposition stood at 97.37%. The two figures clearly illustrated that
as the temperature increased from 750 900oC, the sensitivity of the decomposition reaction increased, this is evident by steeper initial rates which flattens out as the reaction progressed from
20 mins. to 180 mins.
Kinetic parameter estimation from thermogravimetric data
The TGA data obtained from the decomposition of the pretreated alum were used to generate values to plot -In g(X) against 1/T, by varying the order n, between 0 and 2. Figure 3 is representative
of such Coats-Redfern plots for estimating the kinetic and thermodynamic parameters.
The summary of the regression analysis and the obtained kinetic parameters are shown in Table 3, indicating that reaction order of 1.3 for TGA gave the best average regression value of 0.9446 as
compared with other orders within the range of 0 to 2 tested. The estimated average activation energy and pre-exponential factor corresponding to the reaction order
1.3 were 321.03 kJ/mol K and 2.57E+14 min-1 respectively.
Kinetic parameters
Order n Equation y Regression Activation Pre-
value R2 energy kJ/molK exponential factor / mins
0.0 y = 2.386x 6.241 0.884 198.33 1.20E+08
0.5 y = 2.811x 10.412 0.9178 233.69 8.28E+09
0.67 y = 2.992x 12.168 0.9274 248.77 5.00E+10
1.0 y = 3.408x 16.163 0.9407 283.13 3.01E+12
1.2 y = 3.702x 18.983 0.9442 307.81 5.41E+13
1.3 y = 3.861x 20.504 0.9446 321.03 2.57E+14
1.4 y = 3.863x 22.097 0.94416 334.90 1.31E+15
1.5 y= 4.202x-23.7587 0.9272 349.37 7.18E+15
1.6 y= 4.383x-24.321 0.9409 364.43 4.2E+16
1.7 y=4.5709x-27.274 0.9384 380.03 2.61E+17
1.8 y=4.765x-29.118 0.9353 396.13 1.71E+18
1.9 y= 4.964x-31.02 0.9317 412.71 1.19E+19
2.0 y= 5.169x-32.97 0.9278 429.71 8.68E+19
Effect of reaction time on thermodynamic parameters values
Table 4 revealed that as decomposition reaction progresses as a function of time while the order of the reaction was held constant at n= 1.3, the thermodynamic parameters: activation energy and
pre-exponential factor experiences decrease in their values. This may be attributed to decrease in the barrier to heat and mass transfer due to increase in voidage in the sample matrix as the
reaction progresses. The higher activation energy at the early stage of calcinations could also be attributed to the energy required for hydroxylation process coupled with sulfate decomposition,
while at later times, sulfate decomposition being the sole reaction, which requires lesser energy for its transformation.
TABLE 4: TYPICAL VARIATION OF THE THERMODYNAMIC PARAMETERS ESTIMATED WITH REACTION HOLDING TIME FROM TGA DATA AT CONSTANT REACTION ORDER OF 1.3
Activation Pre-
Time/ mins Equation y Regressio n value R2
energy kJ/molK exponential factor / mins
20 0.9410 333.50 5.79E+14
60 y = 3.7744x 19.406 0.9432 313.80 8.09E+13
120 y = 3.6326x 18.535 0.9461 302.01 3.26E+13
150 y = 3.5632x 18.047 0.9513 296.24 1.96E+13
180 y = 3.5296x 17.817 0.9415 293.45 1.54E+13
Kinetic parameter estimation from X-ray Fluorescence data Figure 4 is a representative of the Coats-Redfern plots for estimating kinetic parameters for reaction orders of 0, 0.5, 1.0,
1.3 and 1.5 respectively from XRF conversion data on sulfate decomposition.
The summary of the regression analysis and the obtained reaction parameters are shown in Table 5, where order of 1.0 for XRF gave the best average regression value of 0.9818.
Kinetic parameters
Order N Activation Pre-
Equation y Regression value R2
energy kJ/molK exponential factor / mins
0.0 y = 2.431x 6.400 0.8619 202.14 1.38E+08
0.5 y = 3.246x 13.468 0.9698 269.85 2.48E+11
0.67 y = 3.494x 15.744 0.9758 290.50 2.61E+12
1.0 y = 4.056x 20.834 0.9818 337.22 5.27E+14
1.3 y = 4.509x 25.266 0.9238 374.86 5.47E+16
1.4 y = 4.739x 27.322 0.9244 394.01 4.79E+17
1.5 y = 5.114x 30.281 0.9780 425.19 1.08E+19
1.6 y= 5.229x- 31.680 0.9241 434.70 4.78E+19
1.7 y=5.487x-33.976 0.9235 456.17 5.41E+20
1.8 y=5.753x-36.342 0.9225 478.34 6.59E+21
1.9 y=6.028x-38.776 0.9213 501.16 8.64E+22
2.0 y=6.592x-43.492 0.9731 548.04 1.28E+25
The estimated activation energy and pre-exponential factor for the best average regression analysis of order 1.0 were 337.22 kJ/mol K and 5.27E+14 min-1 respectively. Table 6 clearly shows that
as the calcinations progressed, the activation energy and pre-exponential factor decreased as the reaction order was held at n=1. This reduction in the activation energy and frequency factor
could be attributed to decrease in the diffusion barrier to heat and mass transfer resulting from increase in voidage as the reaction progresses.
TABLE 6: TYPICAL KINETIC PARAMETERS VARIATION WITH REACTION TIME FROM XRF DATA
Kinetic parameters
Equation y Pre- exponential
/ mins Regression value R2 Activation energy kJ/molK
factor / mins
60 y = 4.270x 16.42 0.9726 355.01 3.76E+15
120 y = 4.144x 22.344 0.9977 344.52 1.68E+15
180 y = 3.754x 19.192 0.9752 312.12 6.50E+13
From Table 7, the kinetic and thermodynamic parameters obtained from XRF data were slightly higher than those from TGA data. This can be attributed to the fact that X ray fluorescence analysis
monitors the sulfate decomposition process only, while that of TGA is more or less average of several reactions including the effect of other metallic sulfates acting as impurities (Table 2)
present in the prepared alum which could decrease the overall observed activation energy. It worth mentioning that the kinetic parameters obtained from the XRF data were in good agreement with
that of Moselhy et al. from derivatograph data using the standard equipment, dynamic thermogravimetric method for data generation (n=1, E
= 348 kJmol-1 and A = 1.3E+16 s-1)
Validation of Coats-Redfern model
To validate the Coats-Redfern model, the assumption of the term 1 2 1. It was observed that after substituing
typical activation energy , gas constant R, and temperature T.
The fraction 2RT/E 0.05, was slightly insignificant resulting in a value of the term in the bracket approximately equal to 1. Thus validating the assumption.
3. CONCLUSION
From this study, it can be concluded that reaction orders of 1.0 and 1.3 were most acceptable for XRF data and TGA data respectively; having activation energy and pre-exponential factor of 337.22 kJ/
molK (5.27E+14 min-1) and 321.03 kJ/molK (2.57E+14 min-1) respectively. The large activation energy indicates that the reaction is temperature sensitive. It was observed that as the calcination
progressed while the reaction order was held constant there was a reduction in the activation energy and pre-exponential factor for both TGA and XRF results. This can be attributed to decrease in the
diffusion barrier to heat and mass transfer most likely due to increase invoidage as the reaction progresses. Based on the assumption considered for Coats-Redfern model, the model was found to fit
the rate data obtained satisfactorily.
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Edomwonyi-Otu, L.C; Aderemi, B.O. and Ahmed A.S. (2010): Beneficiation of Kankara kaolin for alum production. Nig. J. of Eng. 16(2) 27-35
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J. of Eng. 16(2) 36-43
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5. Edomwonyi-Otu L.C.; Aderemi B.O.; Simo A. and Maaza M. (2012): Alum production from Nigerian kaolinite deposits. Int. J. Eng Res. in Africa 7 pp 13 19 Switzerland.
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7. Edomwonyi-Otu- L.C., Aderemi B.O., Ahmed A.S., Coville N.J. and Maaza M. Influence of thermal treatment on Kankara kaolinite Opticon1826 15(5) 1-5 (2013)
8. Moselhy H.J.; Madarasz, G; Pokol, S; Pungor, E.(1994). Aluminum Sulphate hydrates: Kinetics of the Thermal Decomposition of Aluminum Sulphate using different calculation methods Journal of
Thermal Analysis 41:1,25.
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10. Aderemi, B.O, Oloro, E.F, Joseph, D, Oludipe, J. (2001) Kinetics of the Dealumination of Kankara Kaolin Clay, NJE vol.9 No1, pp 40-44.
11. Aderemi, B.O. and Oludipe, J.O. (2000). Dealumination of Kankara Kaolin ClayDevelopment of Governing Rate Equation. Nig. J. of Eng. 8(2) 22-30
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13. Perry R. H., Green, D. W. (1997) Perrys Chemical Engineering Handbook. McGraw-Hill Book Co. Pp 28 -36, 28-43.
14. Vogel A, I. (1999) Quantitative Inorganic Chemical Analysis. 4th edition pp.354.
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Well, for example, try to solve such problems as it was written - about
single crystals
So, let there are 2 single-byte registers A and B, which form a double-byte AB, a stack, a conditional jump in length + - 127 commands from the current point, unconditional - 255. 128 instructions in
RAM, 8K memory bank.
Commands for working with registers - bit shifts, addition.
Conditional transition can be if the flag goes over the border of the register?
What is going abroad? This is if you add 100 to 200. There will be an exit beyond the limit of the byte capacity. This is one that is 8 bit
Assume the first task. Implement subtraction. We only have addition.
Make AB, place the result in A.
Let us solve the problem for values up to 127. The high byte will be familiar with us
Accordingly, setting the high bit for A, we get -A
(-A) + B = AB
Problem solved
Can you make a solution for 255? And for 0xFFFFh?
Suppose we coped with the first task.
And how to change the values of A and B among themselves?
Can you do the multiplication? And the division? To facilitate the work - unsigned + a couple more registers have been added - DE
Do you know how to do the branching process with returning to one point using the stack?
But how to implement cos () and sin ()? :) | {"url":"https://sudonull.com/post/212332-Which-modern-programmers-are-able-to-solve-low-level-problems-or-tasks-for-programmers","timestamp":"2024-11-02T12:06:19Z","content_type":"text/html","content_length":"7930","record_id":"<urn:uuid:49513dfc-5e41-41b3-9a33-d8dbdd755af3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00186.warc.gz"} |
Solving Two Step Equations Worksheet Answers
Solving Two-Step Equations Worksheets is no longer a challenge for most students. This is the main reason why so many people, both adults and students, have trouble when it comes to equations that
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If you are worried about getting all the answers right, you can go ahead and use a solver. There are a few different types of solvers that you can use. One of the simplest ones is a spreadsheet that
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that you are unsure of how to solve the equation, you can simply copy some of the code from an online calculator onto a worksheet and then try to work through it.
However, if you want to know how to solve an equation that involves more than two steps, you will probably prefer to use a worksheet that has already been created for you. These worksheets are
available for most of the leading software programs that you are likely to use. Microsoft Excel is probably the best program that offers this type of solver. The reason why is because Excel is such
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Solving two-step equations by hand can be a very slow process. Sometimes, you will just spend hours trying to solve a seemingly simple equation. However, a worksheet solver can speed up the process
and even cut down on the time needed to solve complex problems. This is because these worksheets are specifically designed to help you solve problems. Rather than having to memorize formulas, you
type the equation into a worksheet that then performs the formula.
One of the nice things about using a worksheet solver is that you can store the solution to an Excel file and use it over again. Once you memorize how to solve an equation, perhaps once every few
months, you can store the solution to a worksheet and use it whenever you need to solve an equation. Worksheets can also provide you with information about the solutions to more complicated problems.
For example, if you enter the name of the partial derivative of a function, you can get the solution for the integral.
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Solving two-step equations can often be a daunting task. However, a worksheet solver can make the problem solving process a little bit easier. Instead of having to memorize equation solutions, type
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problems that were previously thought to be impossible.
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17,300 research outputs found
Since the event horizon of a black hole is a surface of infinite redshift, it might be thought that Hawking radiation would be highly sensitive to Lorentz violation at high energies. In fact, the
opposite is true for subluminal dispersion. For superluminal dispersion, however, the outgoing black hole modes emanate from the singularity in a state determined by unknown quantum gravity
processes.Comment: 5 pages, Talk presented at CPT01; the Second Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, 15-18 Aug. 200
Energy positivity is established for a class of solutions to Einstein-aether theory and the IR limit of Ho\v{r}ava gravity within a certain range of coupling parameters. The class consists of
solutions where the aether 4-vector is divergence free on a spacelike surface to which it is orthogonal (which implies that the surface is maximal). In particular, this result holds for spherically
symmetric solutions at a moment of time symmetry.Comment: 4 page
Local Lorentz invariance violation can be realized by introducing extra tensor fields in the action that couple to matter. If the Lorentz violation is rotationally invariant in some frame, then it is
characterized by an ``aether'', i.e. a unit timelike vector field. General covariance requires that the aether field be dynamical. In this paper we study the linearized theory of such an aether
coupled to gravity and find the speeds and polarizations of all the wave modes in terms of the four constants appearing in the most general action at second order in derivatives. We find that in
addition to the usual two transverse traceless metric modes, there are three coupled aether-metric modes.Comment: 5 pages; v2: Remarks added concerning gauge invariance of the waves and hyperbolicity
of the equations. Essentially the version published in PR
Bekenstein's Tensor-Vector-Scalar (TeVeS) theory has had considerable success as a relativistic theory of Modified Newtonian Dynamics (MoND). However, recent work suggests that the dynamics of the
theory are fundamentally flawed and numerous authors have subsequently begun to consider a generalization of TeVeS where the vector field is given by an Einstein-Aether action. Herein, I develop
strong-field solutions of the generalized TeVeS theory, in particular exploring neutron stars as well as neutral and charged black holes. I find that the solutions are identical to the neutron star
and black hole solutions of the original TeVeS theory, given a mapping between the parameters of the two theories, and hence provide constraints on these values of the coupling constants. I discuss
the consequences of these results in detail including the stability of such spacetimes as well as generalizations to more complicated geometries.Comment: Accepted for publication in Physical Review
If a black hole can accrete a body whose spin or charge would send the black hole parameters over the extremal limit, then a naked singularity would presumably form, in violation of the cosmic
censorship conjecture. We review some previous results on testing cosmic censorship in this way using the test body approximation, focusing mostly on the case of neutral black holes. Under certain
conditions a black hole can indeed be over-spun or over-charged in this approximation, hence radiative and self-force effects must be taken into account to further test cosmic censorship.Comment:
Contribution to the proceedings of the First Mediterranean Conference on Classical and Quantum Gravity (talk given by T. P. S.). Summarizes the results of Phys. Rev. Lett. 103, 141101 (2009),
arXiv:0907.4146 [gr-qc] and considers further example
We construct N = 2 chiral supergravity (SUGRA) which leads to Ashtekar's canonical formulation. The supersymmetry (SUSY) transformation parameters are not constrained at all and auxiliary fields are
not required in contrast with the method of the two-form gravity. We also show that our formulation is compatible with the reality condition, and that its real section is reduced to the usual N = 2
SUGRA up to an imaginary boundary term.Comment: 16 pages, late
The structure of boundaries between degenerate and nondegenerate solutions of Ashtekar's canonical reformulation of Einstein's equations is studied. Several examples are given of such "phase
boundaries" in which the metric is degenerate on one side of a null hypersurface and non-degenerate on the other side. These include portions of flat space, Schwarzschild, and plane wave solutions
joined to degenerate regions. In the last case, the wave collides with a planar phase boundary and continues on with the same curvature but degenerate triad, while the phase boundary continues in the
opposite direction. We conjecture that degenerate phase boundaries are always null.Comment: 16 pages, 2 figures; erratum included in separate file: errors in section 4, degenerate phase boundary is
null without imposing field equation | {"url":"https://core.ac.uk/search/?q=authors%3A(Jacobson%20T)","timestamp":"2024-11-09T10:24:06Z","content_type":"text/html","content_length":"119685","record_id":"<urn:uuid:368f5f1c-107b-4fa6-a4c5-fbb2dff76d8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00776.warc.gz"} |
Download Dynamical systems and processes by Weber M. PDF
By Weber M.
This e-book provides in a concise and available manner, in addition to in a standard surroundings, numerous instruments and strategies bobbing up from spectral idea, ergodic concept and stochastic
methods idea, which shape the foundation of and give a contribution interactively greatly to the present learn on almost-everywhere convergence difficulties. Researchers operating in dynamical
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written in a method obtainable to graduate scholars.
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Extra resources for Dynamical systems and processes
Sample text
Vm (θ ) − Vn (θ )| 2 n|θ | 2π 4π Q(θ, y)dy = 1/m (2) |θ | ≤ 1 m. Then, for the same reasons 1/n |θ| dy = (m − n)|θ | 2 1/m y 4 2 ≥ |Vm (θ ) − Vn (θ )| ≥ |Vm (θ ) − Vn (θ )|2 . π π Q(θ, y)dy = 1/m (3)
1 n 1/n > |θ | ≥ 1 m. 1/n This case is obvious since we have |Vm (θ ) − Vn (θ )| ≤ 2. Let f ∈ H , with spectral measure μf . Introduce a new measure, the spectral regularization of the measure μf
with respect to the kernel Q, defined by μˆ f (dy) = 4π π −π Q(θ, y)μf (dθ ) dy + 4μf (dy). 2) It is easy to verify that μˆ f ([0, 1]) ≤ 4(2π + 1)μf ([−π, π]) ≤ 4(2π + 1) f 2 .
Hence I (μf ) ≤ (q + 1)μf (−π, π] = (q + 1) f 2 . The second assertion of the proposition then follows from the previously established result. 3. 6) is optimal. One can indeed prove the following
statement, providing a minoration of the same order for the Littlewood–Paley square function associated to the moving averages. 6 Theorem. Let U be a unitary operator. Let φ(·) be a regular varying
function with Karamata index α ∈ (1, 2), and assume that the function φ(u)/u is increasing. Then there exists a constant c = c(φ), and a nondecreasing sequence of positive integers {np , p ≥ 1}, such
that for any element f ∈ H , one has ∞ Bnφp+1 (f ) − Bnφp (f ) 2 ≥c (θ )μf (dθ ).
4 |θ|. (iii) For any integers m ≥ n ≥ 1, |Vn (θ ) − Vm (θ )| ≤ π 4 |θ | (m − n). (iv) |Vn (θ) − Vm (θ )| ≤ 2 (m − n) /m. Proof. First observe that |Vx (θ )| ≤ 2 . x|eiθ −1| As for any −π ≤ θ ≤ π , |
eiθ − 1| = 2|θ | π 2| sin |θ| 2 | ≥ π , we deduce that |Vx (θ )| ≤ x|θ | , if −π ≤ θ ≤ π and x > 0. And |Vx (θ)| ≤ 1 if x is an integer. Hence (i). Now let −π ≤ θ ≤ π and put for any real x > 0, eixθ
− 1 ϕθ (x) = . x Then ϕθ (x) = iθ xeixθ −eixθ +1 , x2 and noting δ(u) := |iueiu − eiu + 1|2 , we have δ(u) = (1 − u sin u − cos u)2 + (u cos u − sin u)2 = 2[1 − u sin u − cos u] + u2 .
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Documentation/git-write-tree.txt - jrn/git - Git at Google
git-write-tree - Create a tree object from the current index
'git write-tree' [--missing-ok] [--prefix=<prefix>/]
Creates a tree object using the current index. The name of the new
tree object is printed to standard output.
The index must be in a fully merged state.
Conceptually, 'git write-tree' sync()s the current index contents
into a set of tree files.
In order to have that match what is actually in your directory right
now, you need to have done a 'git update-index' phase before you did the
'git write-tree'.
Normally 'git write-tree' ensures that the objects referenced by the
directory exist in the object database. This option disables this
Writes a tree object that represents a subdirectory
`<prefix>`. This can be used to write the tree object
for a subproject that is in the named subdirectory.
Part of the linkgit:git[1] suite | {"url":"https://googlers.googlesource.com/jrn/git/+/e326e520101dcf43a0499c3adc2df7eca30add2d/Documentation/git-write-tree.txt","timestamp":"2024-11-05T07:55:50Z","content_type":"text/html","content_length":"13111","record_id":"<urn:uuid:7800a1b7-2c17-44fa-8705-90a8b5a950ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00310.warc.gz"} |
Lie Algebras and Algebraic Groups
or 4 interest-free payments of $36.88 with
Aims to ship in 7 to 10 business days
The theory of groups and Lie algebras is interesting for many reasons. In the mathematical viewpoint, it employs at the same time algebra, analysis and geometry. On the other hand, it intervenes in
other areas of science, in particularindi?erentbranchesofphysicsandchemistry.Itisanactivedomain of current research. Oneofthedi?cultiesthatgraduatestudentsormathematiciansinterested in the theory
come across, is the fact that the theory has very much advanced, andconsequently,theyneedtoreadavastamountofbooksandarticlesbefore they could tackle interesting problems. One of the goals we wish to
achieve with this book is to assemble in a single volume the basis of the algebraic aspects of the theory of groups and Lie algebras. More precisely, we have presented the foundation of the study of
?nite-dimensional Lie algebras over an algebraically closed ?eld of characteristic zero. Here, the geometrical aspect is fundamental, and consequently, we need to use the notion of algebraic groups.
One of the main di?erences between this book and many other books on the subject is that we give complete proofs for the relationships between algebraic groups and Lie algebras, instead of admitting
them. We have also given the proofs of certain results on commutative al- bra and algebraic geometry that we needed so as to make this book as se- contained as possible. We believe that in this way,
the book can be useful for both graduate students and mathematicians working in this area. Let us give a brief description of the material treated in this book.
Industry Reviews
From the reviews:
"As Tauvel and Yu focus on algebraic groups, they approach Lie theory via algebraic geometry and even develop that subject from scratch ... . For the purpose at hand, Tauvel and Yu's work compares
favorably ... . Summing Up: Highly recommended. Upper-division undergraduates through professionals." (D.V. Feldman, Choice, 43:10, June 2006)
"The sheer volume of material covered herein should make this book an invaluable reference for people interested in, or teaching, Lie algebras or algebraic groups. It truly provides 'one stop
shopping' for someone needing a result or hard-to-find proof. ... I cannot even begin to imagine how much work must have gone into creating such a thorough and comprehensive reference, and I have no
doubt it will be an important and useful addition to the literature on this subject." (Mark Hunacek, The Mathematical Gazette, 90:19, 2006)
"The focus of this book is the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. ... the book is largely self-contained. ... the authors are
extremely knowledgeable in their subjects and the reader can profit from the wealth of material contained in this book. Therefore this book is an ideal reference source and research guide for
graduate students and mathematicians working in this area." (Benjamin Cahen, Zentralblatt MATH, Vol. 1068, 2005)
"The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. ... The aim of this book is to assemble in a single volume the algebraic aspects of the
theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on
commutative algebra and algebraic geometry are included." (L'Enseignement Mathematique, Vol. 51 (3-4), 2006)
"This introduction to Lie algebras andalgebraic groups aims to provide a full background to the subject. ... The book has an encyclopedic character, offering much else besides the actual subject."
(Mathematika, Vol. 52, 2005)
"The stated goal of the authors is to provide a 'foundation for the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero' in a self-contained work that
will be useful to 'both graduate students and mathematicians working in this area'. ... the book contains a wealth of detail and takes the reader from the basic classical concepts to the modern
borders of this still-active area. Complete proofs are given and the authors present their material clearly and concisely throughout." (Duncan Melville, MathDL, March, 2006)
"This book offers ... complete presentation of the theory of the topics in its title over an algebraically closed field of characteristic zero. Assuming only an undergraduate background in abstract
algebra, it covers in detail all the prerequisites that one needs for the theory of Lie algebras and algebraic groups together with the foundations of that theory. ... The book is well written and
easy to follow ... ." (William M. McGovern, SIAM Reviews, Vol. 48 (1), 2006)
"The theory of algebraic groups and Lie algebras is a deeply advanced and developed area of modern mathematics. ... The text is clearly written and the material is well organized and considered, so
the present book may be strongly recommended both to a beginner looking for a self-contained introduction to the theory of algebraic groups and Lie algebras, and to a specialist who wants to have a
systematic presentation of the theory." (Ivan V. Arzhantsev, Mathematical Reviews, Issue, 2006 c)
Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2
Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of
differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal
ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental
elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated
algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules
7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas,
groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan
decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities
on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7
Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties
12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and
projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters
14.4 Counterexamples 15. Morphisms and dimension 15.1 Criterion of affineness 15.2 Affine morphisms 15.3 Finite morphisms 15.4 Factorization and applications 15.5 Dimension of fibres of a morphism
15.6 An example 16. Tangent spaces 16.1 A first approach 16.2 Zariski tangent space 16.3 Differential of a morphism 16.4 Some lemmas 16.5 Smooth points 17. Normal varieties 17.1 Normal varieties 17.2
Normalization 17.3 Products of normal varieties 17.4 Properties of normal varieties 18. Root systems 18.1 Reflections 18.2 Root systems 18.3 Root systems and bilinear forms 18.4 Passage to the field
of real numbers 18.5 Relation between two roots 18.6 Base of a root system 18.7 Weyl chambers 18.8 Highest root 18.9 Closed subsets of roots 18.10 Weights 18.11 Graphs 18.12 Dynkin diagrams 18.13
Classification of root systems 19. Lie algebras 19.1 Generalities on Lie algebras 19.2 Representations 19.3 Nilpotent Lie algebras 19.4 Solvable Lie algebras 19.5 Radical and the largest nilpotent
ideal 19.6 Nilpotent radical 19.7 Regular linear forms 19.8 Cartan subalgebras 20. Semisimple and reductive Lie algebras 20.1 Semisimple Lie algebras 20.2 Examples 20.3 Semisimplicity of
representations 20.4 Semisimple and nilpotent elements 20.5 Reductive Lie algebras 20.6 Results on the structure of semisimple Lie algebras 20.7 Subalgebras of semisimple Lie algebras 20.8 Parabolic
subalgebras 21. Algebraic groups 21.1 Generalities 21.2 Subgroups and morphisms 21.3 Connectedness 21.4 Actions of an algebraic group 21.5 Modules 21.6 Group closure 22. Affine algebraic groups 22.1
Translations of functions 22.2 Jordan decomposition 22.3 Unipotent groups 22.4 Characters and weights 22.5 Tori and diagonalizable groups 22.6 Groups of dimension one 23. Lie algebra of an algebraic
group 23.1 An associative algebra 23.2 Lie algebras 23.3 Examples 23.4 Computing differentials 23.5 Adjoint representation 23.6 Jordan decomposition 24. Correspondence between groups and Lie algebras
24.1 Notations 24.2 An algebraic subgroup 24.3 Invariants 24.4 Functorial properties 24.5 Algebraic Lie subalgebras 24.6 A particular case 24.7 Examples 24.8 Algebraic adjoint group 25. Homogeneous
spaces and quotients 25.1 Homogeneous spaces 25.2 Some remarks 25.3 Geometric quotients 25.4 Quotient by a subgroup 25.5 The case of finite groups 26. Solvable groups 26.1 Conjugacy classes 26.2
Actions of diagonalizable groups 26.3 Fixed points 26.4 Properties of solvable groups 26.5 Structure of solvable groups 27. Reductive groups 27.1 Radical and unipotent radical 27.2 Semisimple and
reductive groups 27.3 Representations 27.4 Finiteness properties 27.5 Algebraic quotients 27.6 Characters 28. Borel subgroups, parabolic subgroups and Cartan subgroups 28.1 Borel subgroups 28.2
Theorems of density 28.3 Centralizers and tori 28.4 Properties of parabolic subgroups 28.5 Cartan subgroups 29. Cartan subalgebras, Borel subalgebras and parabolic subalgebras 29.1 Generalities 29.2
Cartan subalgebras 29.3 Application to semisimple Lie algebras 29.4 Borel subalgebras 29.5 Properties of parabolic subalgebras 29.6 More on reductive Lie algebras 29.7 Other applications 29.8 Maximal
subalgebras 30. Representations of semisimple Lie algebras 30.1 Enveloping algebra 30.2 Weights and primitive elements 30.3 Finite-dimensional modules 30.4 Verma modules 30.5 Results on existence and
uniqueness 30.6 A property of the Weyl group 31. Symmetric invariants 31.1 Invariants of finite groups 31.2 Invariant polynomial functions 31.3 A free module 32. S-triples 32.1 Theorem of
Jacobson-Morosov 32.2 Some lemmas 32.3 Conjugation of S-triples 32.4 Characteristic 32.5 Regular and principal elements 33. Polarizations 33.1 Definition of polarizations 33.2 Polarizations in the
semisimple case 33.3 A non-polarizable element 33.4 Polarizable elements 33.5 Theorem of Richardson 34. Results on orbits 34.1 Notations 34.2 Some lemmas 34.3 Generalities on orbits 34.4 Minimal
nilpotent orbit 34.5 Subregular nilpotent orbit 34.6 Dimension of nilpotent orbits 34.7 Prehomogeneous spaces of parabolic type 35. Centralizers 35.1 Distinguished elements 35.2 Distinguished
parabolic subalgebras 35.3 Double centralizers 35.4 Normalizers 35.5 A semisimple Lie subalgebra 35.6 Centralizers and regular elements 36. s -root systems 36.1 Definition 36.2 Restricted root
systems 36.3 Restriction of a root 37. Symmetric Lie algebras 37.1 Primary subalgebras 37.2 Definition of symmetric Lie algebras 37.3 Natural subalgebras 37.4 Cartan subspaces 37.5 The case of
reductive Lie algebras 37.6 Linear forms 38. Semisimple symmetric Lie algebras 38.1 Notations 38.2 Iwasawa decomposition 38.3 Coroots 38.4 Centralizers 38.5 S-triples 38.6 Orbits 38.7 Symmetric
invariants 38.8 Double centralizers 38.9 Normalizers 38.10 Distinguished elements 39. Sheets of Lie algebras 39.1 Jordan classes 39.2 Topology of Jordan classes 39.3 Sheets 39.4 Dixmier sheets 39.5
Jordan classes in the symmetric case 39.6 Sheets in the symmetric case 40. Index and linear forms 40.1 Stable linear forms 40.2 Index of a representation 40.3 Some useful inequalities 40.4 Index and
semi-direct products 40.5 Heisenberg algebras in semisimple Lie algebras 40.6 Index of Lie subalgebras of Borel subalgebras 40.7 Seaweed Lie algebras 40.8 An upper bound for the index 40.9 Cases
where the bound is exact 40.10 On the index of parabolic subalgebras References Index of notations Index of terminologies
ISBN: 9783642063336
ISBN-10: 3642063330
Series: Springer Monographs in Mathematics
Published: 21st October 2010
Format: Paperback
Language: English
Number of Pages: 674
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6 x 3.43
Weight (kg): 0.93
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How to Lookup Multiple Values in Excel: Step-By-Step
When working with large sets of data in Microsoft Excel, it can be challenging to find multiple values that match several conditions. Some built-in functions, e.g. VLOOKUP, were originally designed
to work with single values.
Several Excel functions can be combined to lookup multiple values. They include VLOOKUP, INDEX, MATCH, and IF functions. Current versions of Excel can use dynamic arrays while older versions use
array formulas.
This article will show you exactly how to use these functions in formulas that find multiple values in your data.
Let’s go!
How to Use VLOOKUP with Multiple Values
The VLOOKUP function is often used to find single values in a range of data. However, you can also look up multiple matches in Excel with this lookup formula.
By default, it will only return the first matching value it finds. However, you can modify the function to return multiple values by using an array formula.
What are Array Formulas?
An array formula is a formula that can perform calculations on arrays of data. It’s called an array formula because it can return an array of results, rather than a single value.
There are several easy steps when creating an array formula:
1. Select a range of cells to search.
2. Enter the formula in the formula bar.
3. Press Ctrl + Shift + Enter to complete it.
The syntax of an array formula is similar to that of a regular formula, but it includes curly braces {} around the formula. The curly braces indicate that the formula is an array formula and that it
will return an array of values.
The examples in this article will show you how to use array formulas correctly.
VLOOKUP Example
Our example has five items in column A of a worksheet:
• Apple
• Banana
• Bread
• Carrot
• Cherry
The task is to check if three specific fruits are in this list: apple, banana, and cherry.
Latest Versions of Excel
The VLOOKUP syntax depends on which version of Excel you are using.
The most recent versions (Excel 365 or Excel 2021) support dynamic arrays. This feature allows formulas to return multiple results that “spill” into adjacent cells.
This is the syntax (the equal sign starts the formula):
=VLOOKUP(lookup_value, table_array, col_index_num, [range_lookup])
• lookup_value: The value that you want to look up.
• table_array: The entire data table that you want to search.
• col_index_num: The table column number in the table_array that contains the data you want to return.
• range_lookup: Optional. Specifies whether you want an exact match or an approximate match.
Our specific example uses this formula:
When the formula is entered in cell B1, the results spill into cells C1 and D1. This picture shows the example in action:
Older Versions
If you’re using an older version of Excel that doesn’t support dynamic arrays (e.g., Excel 2019 or earlier), you’ll need to use a slightly different approach with an array formula.
Follow these steps:
1. Click on the cell where you want to display the results for the first item (e.g., column B).
2. Type the following formula without pressing Enter yet: =VLOOKUP({“apple”,”banana”,”cherry”}, A1:A5, 1, FALSE)
3. Press Ctrl + Shift + Enter to turn this into an array formula.
4. Copy cell B1 and paste it into the cell below (or use the fill handles).
When you use Ctrl + Shift + Enter, Excel adds curly braces around the formula. This indicates that it is an array formula.
Exact Match vs. Approximate Match
By default, the VLOOKUP function uses an approximate match. This means that it will return the closest match it can find, even if the cell values are not an exact match.
If you want to perform an exact match, you can set the range_lookup argument to FALSE.
Bear in mind that approximate matches work best with ordered numerical values. It is usually not appropriate when the cell value is text.
More About VLOOKUP
If you want to learn more about this versatile function, check out these articles:
Now that you’re set with the VLOOKUP function, let’s take a look at two other functions that can do what it does in a different way: INDEX and MATCH.
How to Use INDEX And MATCH to Lookup Multiple Values
You can combine the INDEX and MATCH functions together to find multiple values in multiple rows.
The INDEX function in Excel returns a value or reference to a cell within a specified range.
=INDEX(array, row_num, [column_num])
• array: The range of cells to be searched for the value.
• row_num: The row number within the array from which to return a value.
• column_num: (Optional) The column number within the array from which to return a value. If omitted, the function returns the entire row.
The MATCH function in Excel returns the position of a value within a specified range.
=MATCH(lookup_value, lookup_array, [match_type])
• lookup_value: The value to be searched for within the lookup_array.
• lookup_array: The range of cells to be searched for the lookup_value.
• match_type: (Optional) The type of match to be performed. If omitted, the function performs an exact match.
How to Use INDEX and MATCH Together in Excel 365
To use INDEX and MATCH together to look up multiple values in Excel, you need to use an array formula.
Working with the earlier sample data, this is the formula in Excel 365:
=INDEX(A1:A5, MATCH({“apple”,”banana”,”cherry”}, A1:A5, 0))
The above example breaks down as this:
• INDEX: this returns the value of a cell in a specified range based on a given row and column number. In this case, it will return the value from the range A1:A5.
• A1:A5: This is the defined table range where you’re searching for the value and from which the result will be returned.
• MATCH: this searches for a specified item in a range of cells and returns the relative position of that item in the range.
• {“apple”,”banana”,”cherry”}: this is the array constant containing the values you want to look up.
• A1:A5: this is the range where MATCH will search for the values from the array constant.
• 0: this is the match type for MATCH function. In this case, it’s 0, which means you’re looking for an exact match instead of a close match.
This picture shows the formula in action:
Working with Older Versions of Excel
If you’re using an older Excel file that doesn’t support dynamic arrays (e.g., Excel 2019 or earlier), you’ll need to use a different approach.
Because older versions don’t support the formulas “spilling” into adjacent cells, you will need to break out the usage into three separate formulas.
Follow these steps:
1. Click on the cell where you want the result for the first item (e.g., cell B1)
2. Type the below formula:
3. =INDEX(A1:A5, MATCH(“apple”, A1:A5, 0))
4. Press Enter to execute the formula.
5. Type this formula in cell B2: =INDEX(A1:A5, MATCH(“banana”, A1:A5, 0))
6. Type this formula in cell B3: =INDEX(A1:A5, MATCH(“cherry”, A1:A5, 0))
This picture shows cell reference B3:
INDEX and MATCH functions aren’t the only ones that can enable you to find multiple values. In the next section, we look at how you can use the IF function as an alternative.
How to Use the IF Function to Find Multiple Values
Another way to lookup multiple cell values based on certain criteria is to use the IF function with other functions.
The IF function allows you to test multiple conditions and return different results depending on the outcome of those tests.
For example, let’s say you have a table of sales data with columns for Product and Sales. You want to lookup and total the sales amount for two of the three products.
Current Versions of Excel
To find the sum of the Sales column where the product is either “Apple” or “Banana” using the IF function, you can use an array formula with IF, SUM, and OR functions.
Assuming your data starts in cell A1, use the following formula:
=SUM(IF((A2:A4=”Apple”)+(A2:A4=”Banana”), B2:B4, 0))
The section (A2:A4=”Apple”)+(A2:A4=”Banana”) creates an array that has a value of 1 if the cell in range A2:A4 contains “Apple” or “Banana”, and 0 otherwise.
The IF statement checks each element of the array argument. If the value is 1 (i.e., the product is either “Apple” or “Banana”), it takes the corresponding value in the Sales column (range B2:B4);
otherwise, it takes 0.
The SUM function adds up the values from the IF function, effectively summing the sales values for both “Apple” and “Banana”.
This picture shows the formula in action on the search range:
Older Versions of Excel
In Excel 2019 or earlier, you need to use an array formula. Follow these steps:
1. Type the formula but do not hit Enter.
2. Press Ctrl + Shift + Enter to make it an array formula.
Excel will add curly brackets {} around the formula, indicating it’s an array formula.
Next, we look at how you could use SUMPRODUCT to lookup several values based on your criteria. Let’s go!
How to Use SUMPRODUCT for Multiple Criteria
The SUMPRODUCT function also allows you to lookup multiple values based on multiple criteria.
Because it does not require the use of an array formula, the syntax is the same regardless of the version of Excel.
Using the same data as in the previous example, the formula looks like this:
=SUMPRODUCT((A2:A4=”Apple”)+(A2:A4=”Banana”), B2:B4)
The section (A2:A4=”Apple”)+(A2:A4=”Banana”) creates an array that has a value of 1 if the cell in range A2:A4 contains “Apple” or “Banana”, and 0 otherwise.
The SUMPRODUCT function multiplies the elements of the array with the corresponding elements in the Sales column (range B2:B4). It then adds up the resulting values, effectively summing the sales
values for both “Apple” and “Banana”.
The below formula shows it in action:
Excel functions are amazing when they work as expected, but sometimes you may run into errors. In the next section, we cover some of the common errors and how you can deal with them.
3 Common Errors with Lookup Functions
Lookup functions can sometimes return errors that can be frustrating and time-consuming to troubleshoot. The three most common errors you will encounter are:
1. #N/A Errors
2. #REF! Errors
3. Circular Errors
1. #N/A Errors
The #N/A error occurs when the lookup value cannot be found in the lookup array.
There are several reasons why this error may occur, including:
• the lookup value is misspelled or incorrect.
• the lookup array is not sorted in ascending order.
• the lookup value is not in the data set.
When the lookup value is not in the data set, this is useful information. However, inexperienced Excel users may think that #N/A means something has gone wrong with the formula. The next section
shows how to make this more user-friendly.
2. #REF! Errors
The #REF! error occurs when the lookup array or return array is deleted or moved.
This error can be fixed by updating the cell references in the lookup function.
3. Circular Errors
As you combine functions in complex formulas, Excel may tell you that you have a circular reference.
You’ll find these easier to investigate by using our guide to finding circular references in Excel.
How to Use IFERROR with Lookup Functions
The IFERROR function is a useful tool for handling errors in lookup functions. It allows you to specify a value or formula to return if the lookup function returns an error.
The syntax for the IFERROR function is as follows:
=IFERROR(value, value_if_error)
· value: value or formula you want to evaluate.
· Value_if_error: the value or formula to return if the first argument returns an error.
For example, let’s say you have a VLOOKUP function that is looking up multiple values in a table. In the picture below, one of the values does not exist in the searched data range.
As you can see, the #N/A error is displayed, which can be confusing to inexperienced Excel users.
Instead, you can use IFERROR to display a blank cell or a message that says “Not found” with this syntax:
=IFERROR(VLOOKUP(lookup_value, table_array, column_index, FALSE), “Not found”)
In this example, if the VLOOKUP function returns an error, the IFERROR function will return the message “Not found” instead.
This picture shows the formula in action. Column B has the missing value, while column C and column D have found matches.
We’ve covered a lot of ground so far, and you’re finally ready to learn more advanced techniques for lookups, which is the topic of the next section.
7 Advanced Lookup Techniques
Looking up multiple values in Excel can be a challenging task, especially when dealing with large datasets. You may encounter performance issues with slow processing.
There are seven advanced lookup techniques you can use to make the process easier and more efficient.
• Relative position lookup
• SMALL function
• ROW function
• FILTER Function
• Helper columns
• Dynamic Arrays
• Power Query
1. Relative Position Lookup
One of the simplest ways to lookup multiple values in Excel is by using relative position lookup. This involves specifying the row and column offsets from a given cell to locate the desired value.
For example, if you have a table of data and want to lookup a value that is two rows down and three columns to the right of a given cell, you can use the following formula:
=OFFSET(cell, 2, 3)
2. SMALL Function
Another useful technique for multiple value lookup is using the SMALL function. This function returns the nth smallest value in a range of cells.
By combining this function with other lookup functions like INDEX and MATCH, you can lookup multiple values based on specific criteria.
For example, the following formula looks up the second smallest value in a range of cells that meet a certain condition:
=INDEX(data, MATCH(SMALL(IF(criteria, range), 2), range, 0))
3. ROW Function
The ROW function can also be used for multiple value lookups in Excel. This function returns the row number of a given cell, which can be used to reference cells in a table of data.
For example, the following formula looks up a value in a table based on a unique identifier:
=INDEX(data, MATCH(unique_identifier, data[unique_identifier], 0), column_index_number)
4. FILTER Function
The FILTER function is not available in older versions of Excel.
In Excel 365, you can use it to filter a range of cells based on certain criteria, and return only the values that meet those criteria. This is the syntax and three arguments:
=FILTER(array, include, [if_empty])
• array: The specific data you want to filter.
• include: The include argument is the criteria or conditions you want to apply to the array.
• [if_empty] (optional): The value to return if no rows or columns meet the criteria specified in the include argument.
For example, the following formula works on the example data to find matches for two of the three items in the first column and sum their corresponding values in the second column.
=SUM(FILTER(B2:B4, (A2:A4=”Apple”)+(A2:A4=”Banana”)))
This picture shows how many rows are matched and the sum of the different values:
5. Helper Columns
You can use a helper column to join multiple fields together within a function like VLOOKUP.
Suppose you are working with first and last names in a separate sheet. You can concatenate them in a helper column that is referenced within the final formula.
6. Dynamic Arrays
As you’ve learned in earlier examples, Microsoft 365 users can take advantage of dynamic arrays for multiple value lookup in Excel.
Dynamic arrays allow you to return multiple values from a single formula, making it easier to lookup large amounts of data.
7. Power Query
Power Query is a powerful tool in Excel that can be used to return values based on multiple criteria.
For example, this video finds messy data in a spreadsheet and cleans it up.
Final Thoughts
That wraps up our deep dive into the art of looking up multiple values in Excel! Once you’ve mastered the VLOOKUP, INDEX, MATCH, and array formulas, you’ll find yourself breezing through complex data
sets like a hot knife through butter.
The key is to understand the syntax and the logic behind each formula. Keep in mind that these formulas can be complex, so it’s important to take your time and test your formulas thoroughly before
relying on them for important data analysis.
Excel is a powerful tool, but it’s only as good as the user. So keep honing those skills, keep experimenting, and you’ll soon be the master of multi-value lookups. Until next time, keep crunching
those numbers and making Excel do the hard work for you!
Develop a VBA application to automate repetitive tasks in Excel, increasing efficiency and reducing manual errors.
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GMAT Arithmetic Formulas | Arithmetic Cheat Sheet [PDF]
by Aug. 28, 2024 6216
To achieve a good score on the Quantitative section, it is essential that you memorize the necessary GMAT Math Formulas. Below you will be finding PDF for Arithmetic complete formulas required for
the GMAT Preparation. While preparing solving problems this PDF will be very useful. Befor that, one should be aware of the examination pattern clearly and have a proper GMAT preparation paln
Download Arithmetic Formulas For GMAT[PDF]
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GMAT Arithmetic Formulas:
Before we get started, we'll need to define a few terminology. A sequence, also known as a series, is a group of numbers that can be expressed in a certain order or can simply be a random set of
numbers. Arithmetic sequences, also known as arithmetic formulas or arithmetic series, are a set of integers that follow a specified pattern. The pattern is determined by multiplying each number in
the sequence by a specified quantity to get the next number in the sequence. The number added for each word in the sequence must be the same, and this number is known as the common difference.
These Arithmetic formulas are used in GMAT Quant Section. They can be used to determine patterns in architecture and also have uses in working with finances.
Various topics covered in this Arithmetic Formulas [PDF] are :
• Properties of Integers
• Fractions
• Decimals
• Real Numbers
• Ratio and Proportions
• Percentages
• Power and Roots of Numbers
• Descriptive Statistics Sets
• Counting Methods
• Discrete Probability
• Profit, Loss and Discounts
Check out the complete GMAT syllabus and Section-wise Preparation Tips
Arithmetic operations, such as add, subtract, multiply, and divide, are part of the basic Maths formulas. Algebraic identities also aid in the solution of equations. The following are some of the
It is necessary to know the first term of the sequence, the number of terms, and the common difference in order to calculate the formulas for an arithmetic sequence. Now,
Arithmetic Formulas
n^th Term Formula a[n] = a[1] + (n - 1)d
Sum of First n Terms S[n ]= n/2 (first term + last term)
Where, a[n] = n^th term that has to be found, a[1] = 1^st term in the sequence, n = Number of terms, d = Common difference, S[n] = Sum of n terms
A series with a constant difference between terms, such as 1, 5, 9, 13, 17 or 12, 7, 2, -3, -8, -13, -18. The first term is referred to as a1, the common difference is referred to as d, and the total
number of terms is n. While some sequences consist solely of random values, others follow a specific pattern to arrive at the sequence's terms. You will learn the Arithmetic Sequence Explicit formula
as well as a few solved examples to assist you comprehend this mathematical formula. The formula for Arithmetic Sequence Explicit is as follows:
a[n] = a[1] + (n - 1)d,
an is the nth
term in the sequence,
is the first term in the sequence, n is the term number, d is the common difference. We have covered all the arthmetics topics of formulas in the
. Check out the PDFs for
Ratio & Proportion Formulas
Profit and Loss Formulas
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Join GMATPOINT Telegram Channel
If you are starting your GMAT preparation from scratch, you should definitely check out the GMATPOINT
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Melinda Bellows
By Larry Bock Co-Founder of USA Science Science & Engineering Festival Like many who follow research developments in high technology, I am constantly amazed at the power of science, engineering,
technology and mathematics (STEM) to solve real-life problems -- especially problems across diverse venues and disciplines. Take for example the research of mathematician Lloyd Shapley and economist
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New Satellites Paint a Portrait of Plankton Spatial Variability
Posted by mmaheigan
· Saturday, April 2nd, 2016
The newest generation of satellites reveals plankton variability changes in character from uniform to chaotic at different spatial scales, reviving a classic question in oceanography. How does
plankton variability change at different spatial scales, and why?
New satellites, new insights
Satellite technologies can now collect images with resolution down to the scale of meters, presenting oceanographers data with unprecedented information about the fine-scale structure of plankton
communities in the surface ocean. In August 2015, there was significant media attention after two of the world’s most advanced satellites, Landsat 8 and Sentinel-2, published images of a
cyanobacteria (algal) bloom in the Baltic sea (Fig. 1). For scale, the images conveniently have boats in them (you really have to squint, or just zoom in – a little game of Where’s Waldo at sea).
While these images are beautiful in their own right, to an oceanographer they also illustrate the complexity of the biophysical interactions that drive plankton distributions. When we run computer
models to simulate e.g., how plankton communities might respond to a changing climate, we can’t replicate all of this variability, so we typically represent an X km × Y km square of ocean with a
single value (e.g., plankton concentration), which we consider as the average for that box; one peek at an image like this demonstrates that it’s difficult to justify this approach as doing full
justice to the system it’s simulating. Similarly, when we take samples out in the field, we often fill bottles with seawater and assume that sample represents a X km × Y km area around it. This image
suggests that taking a measurement off one side of the boat might give you a very different representation of that region than if you had taken it off the other side! These approaches are further
complicated by studies indicating that the variability we see in these images persists at microscopic scales.
This is not meant to needlessly criticize these approaches; oceanography is a challenging science, and we do the best we can. Often, these approaches can yield wonderful insights. These images just
draw attention to the fact that plankton spatial variability remains a fascinating and open problem in oceanography, which present-day technology puts us in good position to start addressing.
Characterizing variability
One way we can characterize such variability is by using a power spectral density (PSD), which allows us to quantify how much variability is contained at each scale in an image. Computing the PSD for
each of the above images is a straightforward exercise, thanks to modern computational capabilities. To draw an analogy, we can also compute the PSD for a painting by each of Rothko and Pollock
(Figs. 2a. and 2b., respectively); we might take the former to represent ’homogeneity’ and the latter to represent ’chaos’ (as Pollock’s paintings have been thought of for years). That is, imagine a
satellite looks down on a plankton bloom and sees a rather gargantuan painting of each type; how do these paintings compare with observed blooms, in terms of spatial variability?
The PSD has been computed for the red band of the RGB image of the Rothko painting, a black and white conversion of the Pollock painting, and for the green band of each of the satellite images.
Computing the PSD for other configurations did not change the result. The wavenumber k = 1 in this case corresponds to a wavelength λ ≈ 50 km. Wavenumbers have been rescaled to those of the
Sentinel-2 image, and PSDs have been normalized to their L2 norm.
Comparing power spectral densities
When we computed the PSDs for these four images (Figs. 1a, b and 2a, b), we found remarkable consistency (almost identical PSDs) between the two satellite images (Figs. 1a and b), which were taken
four days apart. This suggests that 1) the satellites are accurately and reproducibly capturing spatial bloom variability, and 2) bloom PSDs don’t change significantly from day to day. The PSDs from
the satellite images matched the Pollock spectrum at smaller spatial scales (i.e. high wavenumber) and the Rothko spectrum at larger spatial scales (i.e. low wavenumber) (Fig. 3). This raises the
question: why might this be happening? Also, at what scale does the ’Rothko-Pollock’ transition occur and why?
If the distribution of plankton was purely that of Brownian (random) motion, we’d expect a flatter PSD (i.e. a line with slope = -2). Another null hypothesis is that the distribution of plankton
might be set passively by advection of oceanic currents. In this case, we’d expect plankton distributions to have the same signature as temperature, which also has a PSD slope of -2. However, these
spectra (Fig. 3) have slopes that are steeper than -2 (closer to -2.5 or -3), so clearly there’s more afoot. The steeper slope of -3 at larger scales means that variability falls off faster as we
look at smaller scales, i.e. something about the plankton distribution is ’homogenizing’ at larger scales. Then, the PSDs get shallower at wavelengths of ~1 km, indicating that something kicks in at
sub-kilometer scales that introduces more variability. One way to think about this transition, which has been hypothesized since the 1970s (1), is that different processes can dominate at different
spatial scales. The specifics of the 70s manner of thinking aren’t quite compatible with these data, but the general concept is plausible. Plankton grow in response to light and nutrient conditions,
but also live in a turbulent environment. At large scales, growth occurs somewhat uniformly and is dominated by ambient light and nutrient conditions, whereas smaller-scale biophysical interactions
can introduce an additional source of variability in plankton growth. Biophysical variability can occur in many ways, including small-scale horizontal motions that can stir plankton patches into
filaments and small-scale vertical motions that can enhance growth locally by bringing up nutrients. In either case, these biophysical interactions are only observable at smaller scales.
Thus, at larger scales, the plankton will be distributed relatively homogeneously as uniform (light-/temperature-driven) growth wins out (. la Rothko), and at smaller scales, they will be distributed
heterogeneously as advective processes come into play (à la Pollock). The spatial scale at which this transition occurs is controversial and depends on many factors, though was originally
hypothesized to be ~1 km, which here appears plausible. See the vertical line in Fig. 3, which corresponds to a 1-km wavelength and appears to agree well with the scale of the observed transition
from Rothko-type to Pollock-type behavior.
Another thing to note is that these cyanobacterial mats (Fig. 1) are very thin and form just at the ocean surface –zoom in and you can see how the boat tracks cut through them. Thus, these patterns
may be representative of a different set of physical processes occurring only in the uppermost layer of the ocean.
While two satellite images of the same bloom may not be enough to verify the growth vs. turbulence hypothesis, ’Rothko-type’ versus ’Pollock-type’ behavior may not be quantitative enough descriptions
to satisfy any oceanographer, and the equally-complex third dimension isn’t included in these pictures, there is still a clear message here. The spatial resolution available from the newest
generation of satellites provides a novel opportunity to approach problems of scale in oceanography.
B. B. Cael (MIT Earth, Atmosphere and Planetary Sciences, Woods Hole Oceanographic Institution)
It is a pleasure to thank Bror Jonsson, Mick Follows, Bryan Kaiser, and Amala Mahadevan for useful discussion of this topic.
1. Denman, K.L., T. Platt, 1976. J. Marine Res. 34, 593-601. | {"url":"https://www.us-ocb.org/new-satellites-paint-a-portrait-of-plankton-spatial-variability/","timestamp":"2024-11-08T04:41:09Z","content_type":"text/html","content_length":"142107","record_id":"<urn:uuid:7c798b49-2d8d-4f12-bd22-a4f4d02b5ae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00656.warc.gz"} |
Chapter 6: Mathematics in a Conceptual PlayWorld
Leigh Disney
By reading and exploring the content of this chapter, you will learn:
• how to teach mathematics in play from within the imaginary play of children
• how to plan a Conceptual PlayWorld for mathematics
• how to pedagogical position yourself within the imaginary situation
Introduction – Jumping into the imaginary situation
Within this chapter, we will continue to follow our preschool teachers, Charlotte, the student teacher, and Yuwen, an experienced teacher. This chapter will focus on embedding mathematics concept
learning into children’s imaginative play using Fleer’s Conceptual PlayWorld model. The chapter will present a case study of Charlotte and Yuwen working collaboratively with their preschool children
to solve a mathematical measurement-related problem.
However, before returning to our protagonists, it’s important to consider how we think about teaching mathematics to young children, a topic that Charlotte has given much thought to. When Charlotte
reflects on her experience of learning mathematics, she thinks about the drill-and-practice style from her own primary school experience. Some of our earliest memories of mathematics reflect
countless hours spent completing worksheets or working on pre-designed worded problems. Charlotte worries that such practices will not fit within a play-based program as delivered within Yuwen’s
Practice reflection 6.1: What are your earliest memories of learning mathematics? Like Charlotte, do you recall endless worksheets or a range of mathematics manipulatives (i.e. counting blocks,
shapes, etc.)? Or, do you recall mathematics embedded in your play during everyday experiences (i.e. making biscuits with your parents, measuring quantities and making shapes)? Would you suggest that
these experiences were playful? Do you think mathematics concepts can be intentionally taught during play? Reflect on your memories and thoughts in your Fleer’s Conceptual PlayWorld thinking book.
Supporting mathematics learning in the teaching program
During planning time, Yuwen and Charlotte discuss the following week’s teaching program. Charlotte notices that the mathematical focus is on the concept of measurement. Yuwen has already embedded
several mathematical experiences into her program, mainly focusing on length measurement. This gives Charlotte an idea:
Charlotte: Yuwen, can we teach mathematical concepts using the Conceptual PlayWorld model?
Yuwen: Absolutely! What did you have in mind?
Charlotte: Well, I can see that in your teaching program, you focus on length measurement.
Yuwen: Do you remember the 5 characteristics of the Conceptual PlayWorld? Do you still have a copy of the Conceptual PlayWorld planning proforma?
Charlotte: I sure do! And I have a great story in mind!
Like other conceptual areas, when planning a Conceptual PlayWorld around mathematics, the chosen story must have a complex plot and a dramatic dimension. There are multiple mathematics-related
children’s books specifically related to mathematical concepts, such as The Doorbell Rang by Pat Hutchins, which uses the notion of sharing to explain an equal distribution. Another is One is a
Snail, Ten is a Crab by April Pulley Sayre and Jeff Sayre, a book focusing on number sense and skip counting. While great books for teaching children about specific mathematics concepts, stories like
this do not always contain complex plots or a range of characters that lend to the multi-layered problem-solving scenarios inherent to Fleer’s Conceptual PlayWorld. Therefore, choose your story
Research reading 6.1: See Chapter 1 for detailed information about the 5 characteristics of Fleer’s Conceptual PlayWorld model, and Appendix B for an example of a planning proforma designed for
infant and toddler children.
Practice reflection 6.2: What other books can you think of that may have inherent mathematical connections yet involve complex plots and a dramatic dimension? Record your ideas in your Fleer’s
Conceptual PlayWorld thinking book.
Charlotte proposes her choice of story:
Charlotte: The book I think would be great is Room on the Broom by Julia Donaldson. As we get closer to Halloween, I have heard the children talking about witches and fairies and know they are very
interested in this imaginative play. Plus, when I think about the book’s title, I think a link could be made to length measurement. You know, when we think about ‘room’, because, in the story, the
broom needs to be fixed after getting broken. This got me thinking: how long would a broom need to be to fit everyone?
Yuwen: Great! I like many of Julia Donaldson’s books and think your choice will work well. What I also appreciate is that you have considered the children’s interests. I’ve also noticed the children
getting excited about Halloween. This shows that you have thought about the first fundamental characteristic of the Conceptual PlayWorld.
Charlotte and Yuwen then spend time planning the Conceptual PlayWorld, considering each characteristic and how the concept of measurement can be learned within the meaningful conditions they intended
to create. When designing the space, the teachers decided that the children would need room to move and manipulatives to work with. Adjacent to the main teaching space, the preschool had a large area
where they decided to strategically place large items such as blocks (Figure 6.1), sticks, and milk crates so that when the children were deciding how to make a broom, they would have resources to
choose from. The teachers also created badges of frogs, dogs, cats, and witches so the children would be helped to identify with their character roles within the Room on the Broom story.
Figure 6.1.
A range of large blocks used for gross motor manipulation.
Note. Mathematical manipulatives can come in a range of sizes, shapes, and quantities. Image provided by the author.
Practice reflection 6.1: What are your earliest memories of learning mathematics? Like Charlotte, do you recall endless worksheets or a range of mathematics manipulatives (i.e. counting blocks,
shapes, etc.)? Or, do you recall mathematics embedded in your play during everyday experiences (i.e. making biscuits with your parents, measuring quantities, and making shapes)? Would you suggest
that these experiences were playful? Do you think mathematics concepts can be intentionally taught during play? Reflect on your memories and thoughts in your Fleer’s Conceptual PlayWorld thinking
Practice reflection 6.3: When you are next at an early learning centre, what manipulatives (large or small) do you have that could be used for mathematics learning? Make a list in your Fleer’s
Conceptual PlayWorld thinking book of what you have, which may transform within the Conceptual PlayWorld setting to be used in new and interesting ways.
Between the main teaching space and the large area was a door. Yuwen found a witch’s hat in the storeroom and stuck it to the door to act as the entry and exit to the Conceptual PlayWorld space (see
Figure 6.2).
Figure 6.2.
A transition device from the regular classroom, into the Conceptual PlayWorld space.
Note. Image provided by the author.
Situating mathematics into the storyline
The children each chose a character role before entering the Conceptual PlayWorld space for the first time. Charlotte decided to play the role of ‘Itchy Witchy’, and as the children entered the
Conceptual PlayWorld space, she would say a magic chant and wave a magic wand to support them as they entered through the door with the witch’s hat (Figure 6.3). Yuwen decided to take on the
character role of a dog, calling herself ‘Detective Dog’ because she was always carrying around a notepad and pen (Figure 6.3). Yuwen did this during the Conceptual PlayWorld so she could first, stay
in a character role and second, have easy access to documentation materials to note what the children said and did during the Conceptual PlayWorld. In this way, Yuwen was engaged in authentic
documentation practices.
Figure 6.3.
The teachers in their character roles.
Research reading 6.2: See Chapter 5 for insights into assessment practices capturing learning within the Conceptual PlayWorld.
After reading the story of Room on the Broom to the children, the teachers were keen to discover what aspect of the story the children were interested in and whether they could identify any problems
that needed to be fixed. Much to the teacher’s delight, the children immediately identified that the broom needed to be fixed, and thus, the conceptual problem was formed: How long do we need to make
the broom when we fix it?
To help dramatise the conceptual problem, the teachers crafted a letter, which they said was from ‘Ritchy Witchy’, the sister of Itchy Witchy (played by Charlotte). The letter (Figure 6.4) invited
Itchy Witchy and all her animal friends (played by the children) to come to Ritchy Witchy’s house for a party, but they needed to come together on one broom.
Figure 6.4.
Helping to dramatise the conceptual problem: a letter from the witch’s sister, Ritchy Witchy.
Letter transcript: Dear Itchy Witchy. I would like to invite you and your friends to our favourite holiday of the whole year. Please come on your broom, I hope it is long enough to fit you and all of
your friends. Also, please bring Detective Dog as there will be some Detective jobs for her to do. Love Ritchy Witchy. Image provided by the author.
The letter created the dramatised problem of creating a broom that could fit all the characters. We now pick up the Conceptual PlayWorld between the teachers and children as they collectively solve
the problem, and we will explain how the teachers pedagogically positioned themselves to support the children during the Conceptual PlayWorld experience (see Chapter 5 for more information on
pedagogical positioning).
Considering your pedagogical positioning to best support children’s mathematical discovery
As the children worked together, they used the large building blocks (see Figure 6.1) to create a new broom, where each block was a seat for a character. As this was a collective experience, the
children were never in the primordial-we position, as all the children were building within the storyline of the Conceptual PlayWorld and understood their roles. As the children worked, Charlotte
interacted with them, asking them simple questions and getting into the character role of a witch. During this time, both teachers were in the under position and the children in the above. In this
way, the children and teachers made an emotional connection with their characters, which led to the children’s motivation to eventually solve the conceptual problem of making the broom long enough to
fit all their friends.
While the children were certainly building a structure using the large blocks that was straight and resembling a new broom, many were more interested in their own character roles rather than solving
the mathematical problem. For example, the frogs were having a conversation about needing water on their delicate skin for the long journey; hence, they needed a tap attached to their seat. At that
moment, Charlotte feared that the mathematics concept of learning about length measurement was being lost, but then, a breakthrough occurred (Figure 6.5).
Figure 6.5.
Charlotte has a breakthrough in how to support children’s mathematics thinking – using multiple pedagogical positions.
Charlotte noticed how Yuwen was positioned, interacting with the children, seemingly noticing everything, and then asking a pointed question:
Yuwen: [Spotting 3 blocks joined together, asked the entire group] I wonder how many of our friends can fit on this section of the broom?
Charlotte: That’s such a good question Detective Dog, I’m not sure. Can anyone help figure this problem out?
Child A: [Walking over and pointing at each block] There are 1, 2, 3 blocks.
Charlotte: So, how many of our friends will fit on this section of the new broom?
Child B & C: [In unison] Three! Three friends!
Yuwen: So, how many blocks long does are new broom need to be? How many friends are here with us today?
In this scenario, the teachers had moved from being below the children to being equal with the children. They were now collectively problem-solving together, focusing on solving the mathematical
problem originally presented in the letter from Ritchy Witchy, to make one broom that could fit everyone to fly to the birthday party. The children had to determine how many characters needed to fit
onto the broom, which would let us know how long the broom needed to be. Yuwen (still in her character role as Detective Dog) asked the children to sit in a circle so that one of the children could
easily count all of the friends in the group. At this stage, the teachers began moving into the above position. They were showing children strategies that they could use to help solve the conceptual
Child A: [After moving around the circle and tapping each friend on the head, declared] 16, we have 16 friends.
Yuwen: 16; wow! That’s so many friends that need to fit onto the broom. So, if one block can only fit one friend, how many blocks do we need?
Multiple children: 16! 16 blocks!
Yuwen: Oh, I think I need to write this down in my Detective Dog book!
Charlotte: [Trying to capitalise on the teaching moment, in the above position, asked Child A] How do you know we need 16 blocks?
Child A: [Looking a little unsure of themselves] Everyone needs a seat…
Child B: [Stood up with a loud and enthusiastic voice] Because one block can only fit one friend!
Yuwen: According to my Detective Dog book, we have 16 friends. So, how many seats do we need on our broom, everybody?
Whole group: [Very loudly] 16!
Charlotte and Yuwen: That’s right!
The children then busily worked together to build a broom that was 16 blocks long. Once the children thought they were finished, the teachers asked them to count how long it was. The children then
collectively counted the length of the broom. The first attempt was 14 blocks long. The children then added 3 more blocks. Oh no, too long! The children finally made a broom of 16 blocks, and the
group took off on their new broom to Ritchy Witchy’s birthday party! This also acted as a transition into lunchtime, so the group exited the Conceptual PlayWorld setting.
In this way, the teachers had moved from the under (initially building an emotional connection to characters) to the equal (getting the children thinking mathematically) and finally to the above
position (demonstrating mathematical problem-solving techniques) within the Conceptual PlayWorld experience. It is also important to note that the children were never in an independent position. This
was because the teachers were inside the children’s play, fully aware of what the children were doing and part of the play scenario. Each pedagogical positioning allows the Conceptual PlayWorld to
continue and grow, motivating children to solve mathematics problems through imaginative scenarios and their character roles.
Practice reflection 6.4: While the focus was on measurement, what other mathematics concept could you see happening in the Room on the Broom Conceptual PlayWorld?
Charlotte was so pleased with how the Conceptual PlayWorld had run, and while there were moments of worry, both Yuwen and Charlotte agreed that the Conceptual PlayWorld had met their teaching agenda
and allowed for conceptual learning beyond their initial expectation.
In this chapter, Charlotte and Yuwen planned and implemented a Conceptual PlayWorld as part of their teaching program to specifically teach mathematics to children. By taking different pedagogical
positions, both teachers could take different roles within the collective play to allow children to be mathematical problem solvers and teach specific mathematical concepts.
Practice reflection 6.5: Bring together the elements of your Fleer’s Conceptual PlayWorlds thinking book. Hopefully, not only can you help Charlotte on her journey, but your thinking will provide you
with a springboard for embedding mathematics learning in Conceptual PlayWorlds. Carefully consider pedagogical positioning to engage children throughout the Conceptual PlayWorld to create an
emotional connection to their character roles and, ultimately, solve the mathematical conceptual problem. | {"url":"https://oercollective.caul.edu.au/why-play-works/chapter/chapter-6-mathematics-in-a-conceptual-playworld/","timestamp":"2024-11-04T02:51:06Z","content_type":"text/html","content_length":"91071","record_id":"<urn:uuid:fc906631-ff92-4868-81bf-a9694c0cef72>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00649.warc.gz"} |
Ioannis Kevrekidis : No Equations, No Variables, No Parameters, No Space, No Time -- Data, and the Crystal Ball Modeling of Complex/Multiscale Systems
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Ioannis Kevrekidis : No Equations, No Variables, No Parameters, No Space, No Time -- Data, and the Crystal Ball Modeling of Complex/Multiscale Systems (Dec 2, 2016 3:25 PM)
Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from
observations of the world (and some serious thinking!) first to selection of variables, then to equations for a model, and finally to the analysis of the model to make predictions. Good mathematical
models give good predictions (and inaccurate ones do not) --- but the computational tools for analyzing them are the same: algorithms that are typically operating on closed form equations.
While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations --- data, and appear to circumvent the serious
thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking into a crystal ball".
Yet the "serious thinking" is still there and uses the same --- and some new --- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is
now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this "new" path from data to predictions. It really is the same old path, but
it is traveled by new means.
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java backend to achieve addition, subtraction, multiplication, division and proportion calculation
In Java The math package provides the API class BigDecimal, which is used to accurately calculate the number of more than 16 significant bits.
BigDecimal creates objects, so you cannot directly perform mathematical operations on its objects using arithmetic operators such as +, -, *, /, etc.
First of all, you need to understand that BigDecimal has four construction methods:
//Creates an object with an integer value specified by the parameter
//Creates an object with the double value specified by the parameter
//Creates an object with an integer value specified by the parameter
//Creates an object with the value specified by the parameter as a string
//Here, two forms are compared. The first one directly writes the value of a number, and the second one is represented by a string
BigDecimal num1 = new BigDecimal(0.005);
BigDecimal num2 = new BigDecimal(1000000);
BigDecimal num3 = new BigDecimal(-1000000);
//Try to initialize in the form of string
BigDecimal num12 = new BigDecimal("0.005");
BigDecimal num22 = new BigDecimal("1000000");
BigDecimal num32 = new BigDecimal("-1000000");
1. Addition, subtraction, multiplication and division
BigDecimal result1 = num1.add(num2);
BigDecimal result12 = num12.add(num22);
//Subtraction subtract()
BigDecimal result2 = num1.subtract(num2);
BigDecimal result22 = num12.subtract(num22);
BigDecimal result3 = num1.multiply(num2);
BigDecimal result32 = num12.multiply(num22);
//Absolute value (ABS)
BigDecimal result4 = num3.abs();
BigDecimal result42 = num32.abs();
//Num1, num12 divisor, 20 exact decimal places, BigDecimal ROUND_ HALF_ Up rounding mode
BigDecimal result5 = num2.divide(num1,20,BigDecimal.ROUND_HALF_UP);
BigDecimal result52 = num22.divide(num12,20,BigDecimal.ROUND_HALF_UP);
Here, you can see the result by outputting all the results
There are differences here, which is why string is recommended for initialization
※ note:
1)System. out. The number in println () is double by default, and the decimal calculation of double is inaccurate.
2) When the BigDecimal class construction method is used to pass in the double type, the calculation result is also inaccurate!
Because not all floating-point numbers can be accurately represented as a double value, some floating-point numbers cannot be accurately represented as a double value, so it will be represented as
the value of the double type closest to it. You must use the constructor of the incoming String instead. This is explained in the constructor annotation of BigDecimal class.
Use the divide() parameter
When using the division function to divide, set various parameters, accurate decimal places and rounding mode, or an error will appear
We can see that the parameters configured for the divide function are as follows
//That is (BigDecimal divisor, int scale exact decimal places, int roundingMode rounding mode)
Public BigDecimal divide(BigDecimal divisor, int scale, int roundingMode)
But there are many rounding modes BigDecimal ROUND_ XXXX_ 30. What exactly does that mean
The eight rounding modes are explained below
Rounding away from zero.
Always increase the number before discarding the non-zero part (always add 1 to the number before the non-zero discarding part).
Note that this rounding mode never reduces the size of the calculated value.
Rounding mode close to zero.
Never increase the number before discarding a part (never increase the number before discarding a part by 1, i.e. truncate).
Note that this rounding mode never increases the size of the calculated value.
Rounding pattern close to positive infinity.
If BigDecimal is positive, the rounding behavior is the same as ROUND_UP is the same;
If negative, the rounding behavior is the same as round_ Same as down.
Note that this rounding mode never reduces the calculated value.
Rounding mode close to negative infinity.
If BigDecimal is positive, the rounding behavior is the same as round_ Same as down;
If the bold style is negative, the rounding behavior is the same as round_ Same as up.
Note that this rounding mode never increases the calculated value.
Round to the "closest" number. If the distance from two adjacent numbers is equal, it is the rounding mode of rounding up.
If the discard part > = 0.5, the rounding behavior is the same as ROUND_UP is the same; Otherwise, the rounding behavior is the same as round_ Same as down.
Note that this is the rounding pattern (rounding) that most of us learned in primary school.
Round to the "closest" number. If the distance from two adjacent numbers is equal, it is the rounding mode of rounding up.
If the discard part > 0.5, the rounding behavior is the same as ROUND_UP is the same; Otherwise, the rounding behavior is the same as round_ Same as down (rounded).
Round to the "closest" number, or even if the distance from two adjacent numbers is equal.
If the number to the left of the discarded part is odd, the rounding behavior is the same as ROUND_HALF_UP is the same;
If it is an even number, the rounding behavior is the same as round_ HALF_ Same as down.
Note that this rounding mode minimizes cumulative errors when repeating a series of calculations.
This rounding model, also known as "banker rounding method", is mainly used in the United States. Round to the nearest five.
If the previous digit is odd, round it, otherwise round it.
The following example is the result of this rounding method if 1 decimal place is reserved.
1.15>1.2 1.25>1.2
Assert that the requested operation has a precise result and therefore does not need to be rounded.
If this rounding mode is specified for an operation that obtains a precise result, an ArithmeticException is thrown.
For example: calculate the result of 1 ÷ 3 (the last ROUND_UNNECESSARY will report an error when the result is an infinite decimal)
2. Compare size
//The premise is that neither a nor b can be null
BigDecimal a = new BigDecimal("xx");
BigDecimal b = new BigDecimal("xx");
if(a.compareTo(b) == -1){
System.out.println("a less than b");
if(a.compareTo(b) == 0){
System.out.println("a be equal to b");
if(a.compareTo(b) == 1){
System.out.println("a greater than b");
if(a.compareTo(b) > -1){
System.out.println("a Greater than or equal to b");
if(a.compareTo(b) < 1){
System.out.println("a Less than or equal to b");
Good law: everything will be good in the end. If it's not good, it's not the end. | {"url":"https://programming.vip/docs/6213544d9b44c.html","timestamp":"2024-11-13T01:32:50Z","content_type":"text/html","content_length":"14014","record_id":"<urn:uuid:1ec64500-45af-4e42-9394-29e02585fc8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00305.warc.gz"} |
Research Coco- Cola And Assess Whether Or Not The Organization Has Outstanding Bonds Payable Or Has Invested In Bonds From Another Organization. Do You Support Their Choice To Use Bonds For Financing Or Investment Purposes? Why Or Why Not? - My Homework Help
Coca-Cola has no outstanding bonds and it does not own any bonds. Their primary financing source is instead equity capital, which they utilize to fund operations, growth initiatives and engage in
share repurchases for shareholders to return value. This choice is one I fully support because they retain ownership and control of their company while also having access to extra capital when
necessary. Coca-Cola is also able to avoid risks such as credit ratings or default if the market worsens by choosing equity over debt.
Moreover, using equity capital also helps keep Coca-Cola’s balance sheet healthy since there won’t be any added interest payments that need to be paid out each period. The company can still maximize
its profits, while maintaining a healthy financial position. This will build confidence with both investors and stakeholders. use the Capital Asset Pricing Model (CAPM) to evaluate a company’s stock?
The Capital Asset Pricing Model (CAPM) is used to evaluate a company’s stock by calculating its expected return given the risk-free rate and market risk premium. Comparing the expected returns of a
company’s stock with those from other companies in the same sector helps investors decide whether or not a particular stock is worthwhile. To calculate an expected return on a stock, you must first
determine its required return. This is equal to Risk Free Rate + beta x Market risk premium. The beta coefficient measures how much volatility an individual security has compared to an overall market
index such as the S&P 500. Investors can compare the returns of their investments to those from similar companies once this calculation has been made. | {"url":"https://myhomeworkhelp.org/sample/research-coco-cola-and-assess-whether-or-not-the-organization-has-outstanding-bonds-payable-or-has-invested-in-bonds-from-another-organization-do-you-support-their-choice-to-use-bonds-for-financing/","timestamp":"2024-11-02T18:50:54Z","content_type":"text/html","content_length":"55827","record_id":"<urn:uuid:5f91d33d-bf15-469c-aee8-de44dc6ab243>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00359.warc.gz"} |
2 digit multiplication games
2 Digit by 2 Digit Multiplication Bingo Game -
Multi-Digit Multiplication Games- Print and digital | Mathcurious
Interactive Math Lesson | Multiplying 2-Digit by 2-Digit Numbers
Math Game: Multiply 2-Digit by 1-Digit Numbers
FREE Printable Multiplication Spinner Game - Spin and Multiply!
Multi-digit multiplication online game (2nd, 3rd, and 4th grade math)
Double Digit Multiplication Practice: FREE Board Games
Octagon Multiplication (Multiply—2-Digit Factors) | Printable ...
Two Digit Multiplication Activities Your Students Will Love ...
Multi-Digit Multiplication Games- Print and digital | Mathcurious
Simple Multi Digit Multiplication Game {FREE}
2 digit by 2 digit multiplication games and worksheets
2 Digit by 1 Digit Multiplication Math Centers, Games, or Review ...
Double Digit Multiplication Practice: FREE Board Games
FREE! - 2 Digit By 2 Digit Multiplication Worksheets PDF Free
Multiplication (2-Digits Times 2-Digits)
Two Digit Multiplication Activities Your Students Will Love ...
Double Digit Multiplication Practice: FREE Board Games
Interactive Math Lesson | Multiplying 2-Digit by 2-Digit Numbers
Complete the Multiplication of Two 2-Digit Numbers Game - Math ...
Game for Practicing Multi-Digit Multiplication - Math Coachs Corner
2 Digit by 1 Digit and 2 Digit by 2 Digit Multiplication Bump Games
A Game of Long Multiplication - 2 digits by 2 digit (Fourth and ...
Double Digit Multiplication Practice: FREE Board Games
Two Digit Multiplication Activities Your Students Will Love ... | {"url":"https://worksheets.clipart-library.com/2-digit-multiplication-games.html","timestamp":"2024-11-13T06:14:43Z","content_type":"text/html","content_length":"21227","record_id":"<urn:uuid:037ea00e-3f85-4345-886d-6b75389196c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00440.warc.gz"} |
Nestedly Recursive Functions
Yet Another Ruliological Surprise
Integers. Addition. Subtraction. Maybe multiplication. Surely that’s not enough to be able to generate any serious complexity. In the early 1980s I had made the very surprising discovery that very
simple programs based on cellular automata could generate great complexity. But how widespread was this phenomenon?
At the beginning of the 1990s I had set about exploring this. Over and over I would consider some type of system and be sure it was too simple to “do anything interesting”. And over and over again I
would be wrong. And so it was that on the night of August 13, 1993, I thought I should check what could happen with integer functions defined using just addition and subtraction.
I knew, of course, about defining functions by recursion, like Fibonacci:
But could I find something like this that would have complex behavior? I did the analog of what I have done so many times, and just started (symbolically) enumerating possible definitions. And
immediately I saw cases with nested functions, like:
(For some reason I wanted to keep the same initial conditions as Fibonacci: f[1] = f[2] = 1.) What would functions like this do? My original notebook records the result in this case:
But a few minutes later I found something very different: a simple nestedly recursive function with what seemed like highly complex behavior:
I remembered seeing a somewhat similarly defined function discussed before. But the behavior I’d seen reported for that function, while intricate, was nested and ultimately highly regular. And, so
far as I could tell, much like with rule 30 and all the other systems I’d investigated, nobody had ever seen serious complexity in simple recursive functions.
It was a nice example. But it was one among many. And when I published A New Kind of Science in 2002, I devoted just four pages (and 7 notes) to “recursive sequences”—even though the gallery I made
of their behavior became a favorite page of mine:
A year after the book was published we held our first Wolfram Summer School, and as an opening event I decided to do a live computer experiment—in which I would try to make a real-time science
discovery. The subject I chose was nestedly recursive functions. It took a few hours. But then, yes, we made a discovery! We found that there was a nestedly recursive function simpler than the ones
I’d discussed in A New Kind of Science that already seemed to have very complex behavior:
Over the couple of decades that followed I returned many times to nestedly recursive functions—particularly in explorations I did with high school and other students, or in suggestions I made for
student projects. Then recently I used them several times as “intuition-building examples” in various investigations.
I’d always felt my work with nestedly recursive functions was unfinished. Beginning about five years ago—particularly energized by our Physics Project—I started looking at harvesting seeds I’d sown
in A New Kind of Science and before. I’ve been on quite a roll, with a few pages or even footnotes repeatedly flowering into rich book-length stories. And finally—particularly after my work last year
on “Expression Evaluation and Fundamental Physics”—I decided it was time to try to finish my exploration of nestedly recursive functions.
Our modern Wolfram Language tools—as well as ideas from our Physics Project—provided some new directions to explore. But I still thought I pretty much knew what we’d find. And perhaps after all these
years I should have known better. Because somehow in the computational universe—and in the world of ruliology—there are always surprises.
And here, yet again, there was indeed quite a surprise.
The Basic Idea
Consider the definition (later we’ll call this “P312”)
which we can also write as:
The first few values for f[n] generated from this definition are:
Continuing further we get:
But how are these values actually computed? To see that we can make an “evaluation graph” in which we show how each value of f[n] is computed from ones with smaller values of n, here starting from f
The gray nodes represent initial conditions: places where f[n] was sampled for n ≤ 0. The two different colors of edges correspond to the two different computations done in evaluating each f[n]:
Continuing to f[30] we get:
But what’s the structure of this graph? If we pull out the “red” graph on its own, we can see that it breaks into two path graphs, that consist of the sequences of the f[n] for odd and even n,
The “blue” graph, on the other hand, breaks into four components—each always a tree—leading respectively to the four different initial conditions:
And for example we can now plot f[n], showing which tree each f[n] ends up being associated with:
We’ll be using this same basic setup throughout, though for different functions. We’ll mostly consider recursive definitions with a single term (i.e. with a single “outermost f”, not two, as in
Fibonacci recurrences).
The specific families of recursive functions we’ll be focusing on are:
And with this designation, the function we just introduced is P312.
A Closer Look at P312 ( f[n_] := 3 + f[n – f[n – 2]] )
Let’s start off by looking in more detail at the function we just introduced. Here’s what it does up to n = 500:
It might seem as if it’s going to go on “seemingly randomly” forever. But if we take it further, we get a surprise: it seems to “resolve itself” to something potentially simpler:
What’s going on? Let’s plot this again, but now showing which “blue graph tree” each value is associated with:
And now what we see is that the f[–3] and f[–2] trees stop contributing to f[n] when n is (respectively) 537 and 296, and these trees are finite (and have sizes 53 and 15):
The overall structures of the “remaining” trees—here shown up to f[5000]—eventually start to exhibit some regularity:
We can home in on this regularity by arranging these trees in layers, starting from the root, then plotting the number of nodes in each successive layer:
Looking at these pictures suggests that there should be some kind of more-or-less direct “formula” for f[n], at least for large n. They also suggest that such a formula should have some kind of mod-6
structure. And, yes, there does turn out to be essentially a “formula”. Though the “formula” is quite complicated—and reminiscent of several other “strangely messy” formulas in other ruliological
cases—like Turing machine 600720 discussed in A New Kind of Science or combinator s[s[s]][s][s][s][s].
Later on, we’ll see the much simpler recursive function P111 (f[n_] := 1 + f[n – f[n – 1]]). The values for this function form a sequence in which successive blocks of length k have value k:
P312 has the same kind of structure, but much embellished. First, it has 6 separate riffled (“mod”) subsequences. Each subsequence then consists of a sequence of blocks. Given a value n, this
computes which subsequence this is on, which block for that subsequence it’s in, and where it is within that block:
So, for example, here are results for multiples of 1000:
For n = 1000 we’re not yet in the “simple” regime, we can’t describe the sequence in any simple way, and our “indices” calculation is meaningless. For n = 2000 it so happens that we are at block 0
for the mod-1 subsequence. And the way things are set up, we just start by giving exactly the form of block 0 for each mod. So for mod 1 the block is:
But now n = 2000 has offset 16 within this block, so the final value of f[2000] is simply the 16th value from this list, or 100. f[2001] is then simply the next element within this block, or 109. And
so on—until we reach the end of the block.
But what if we’re not dealing with block 0? For example, according to the table above, f[3000] is determined by mod-3 block 1. It turns out there’s a straightforward, if messy, way to compute any
block b (for mod m):
So now we have a way to compute the value, say of f[3000], effectively just by “evaluating a formula”:
And what’s notable is that this evaluation doesn’t involve any recursion. In other words, at the cost of “messiness” we’ve—somewhat surprisingly—been able to unravel all the recursion in P312 to
arrive at a “direct formula” for the value of f[n] for any n.
So what else can we see about the behavior of f[n] for P312? One notable feature is its overall growth rate. For large n, it turns out that (as can be seen by substituting this form into the
recursive definition and taking a limit):
One thing this means is that our evaluation graph eventually has a roughly conical form:
This can be compared to the very regular cone generated by P111 (which has asymptotic value
If one just looks at the form of the recursive definition for P312 it’s far from obvious “how far back” it will need to probe, or, in other words, what values of f[n] one will need to specify as
initial conditions. As it turns out, though, the only values needed are f[–3], f[–2], f[–1] and f[0].
How can one see this? In 3 + f[n – f[n – 2]] it’s only the outer f that can probe “far back” values. But how far it actually goes back depends on how much larger f[n – 2] gets compared to n. Plotting
f[n – 2] and n together we have:
And the point is that only for very few values of n does f[n – 2] exceed n—and it’s these values that probe back. Meanwhile, for larger n, there can never be additional “lookbacks”, because f[n] only
grows like
So does any P312 recursion always have the same lookback? So far, we’ve considered specifically the initial condition f[n] = 1 for all n ≤ 0. But what if we change the value of f[0]? Here are plots
of f[n] for different cases:
And it turns out that with f[0] = z, the lookback goes to –z for z ≥ 3, and to z – 4 for 1 ≤ z ≤ 2.
(If z ≤ 0 the function f[n] is basically not defined, because the recursion is trying to compute f[n] from f[n], f[n + 1], etc., so never “makes progress”.)
The case f[0] = 2 (i.e. z = 2) is the one that involves the least lookback—and a total of 3 initial values. Here is the evaluation graph in this case:
By comparison, here is the evaluation graph for the case f[0] = 5, involving 6 initial values:
If we plot the value of f[n] as a function of f[0] we get the following:
For n < 3 f[0], f[n] always has simple behavior, and is essentially periodic in n with period 3:
And it turns out that for any specified initial configuration of values, there is always only bounded lookback—with the bound apparently being determined by the largest of the initial values f[n
So what about the behavior of f[n] for large n? Just like in our original f[0] = 1 case, we can construct “blue graph trees” rooted at each of the initial conditions. In the case f[0] = 1 we found
that of the 4 trees only two continue to grow as n increases. As we vary f[0], the number of “surviving trees” varies quite erratically:
What if instead of just changing f[0], and keeping all other f[–k] = 1, we set f[n] = s for all n ≤ 0? The result is somewhat surprising:
For s ≥ 2, the behavior turns out to be simple—and similar to the behavior of P111.
So what can P312 be made to do if we change its initial conditions? With f[n] = 2 for n < 0, we see that for small f[0] the behavior remains “tame”, but as f[0] increases it starts showing its
typical complexity:
One question to ask is what set of values f[n] takes on. Given that the initial values have certain residues mod 3, all subsequent values must have the same residues. But apart from this constraint,
it seems that all values for f[n] are obtained—which is not surprising given that f[n] grows only like
The “P Family”: f[n_] := a + f[n – b f[n – c]]
P312 is just one example of the “P family” of sequences defined by:
Here is the behavior of some other Pabc sequences:
And here are their evaluation graphs:
P312 is the first “seriously complex” example.
P111 (as mentioned earlier) has a particularly simple form
which corresponds to the simple formula:
The evaluation graph in this case is just:
Only a single initial condition f[0] = 1 is used, and there is only a single “blue graph tree” with a simple form:
Another interesting case is P123:
Picking out only odd values of n we get:
This might look just like the behavior of P111. But it’s not. The lengths of the successive “plateaus” are now
with differences:
But this turns out to be exactly a nested sequence generated by joining together the successive steps in the evolution of the substitution system:
P123 immediately “gets into its final behavior”, even for small n. But—as we saw rather dramatically with P312—there can be “transient behavior” that doesn’t “resolve” until n is large. A smaller
case of this phenomenon occurs with P213. Above n = 68 it shows a simple “square root” pattern of behavior, basically like P111. But for smaller n it’s a bit more complicated:
And in this case the transients aren’t due to “blue graph trees” that stop growing. Instead, there are only two trees (associated with f[0] and f[–1]), but both of them soon end up growing in very
regular ways:
The “T Family”: f[n_] := a f[n – b f[n – c]]
What happens if our outermost operation is not addition, but multiplication?
Here are some examples of the behavior one gets. In each case we’re plotting on a log scale—and we’re not including T1xx cases, which are always trivial:
We see that some sequences have regular and readily predictable behavior, but others do not. And this is reflected in the evaluation graphs for these functions:
The first “complicated case” is T212:
The evaluation graph for f[50] in this case has the form:
And something that’s immediately notable is that in addition to “looking back” to the values of f[0] and f[–1], this also looks back to the value of f[–24]. Meanwhile, the evaluation graph for f[51]
looks back not only to f[0] and f[–1] but also to f[–3] and f[–27]:
How far back does it look in general? Here’s a plot showing which lookbacks are made as a function of n (with the roots of the “blue graph trees” highlighted):
There’s alternation between behaviors for even and odd n. But apart from that, additional lookbacks are just steadily added as n increases—and indeed the total number of lookbacks seems to follow a
simple pattern:
But—just for once—if one looks in more detail, it’s not so simple. The lengths of the successive “blocks” are:
So, yes, the lookbacks are quite “unpredictable”. But the main point here is that—unlike for the P family—the number of lookbacks isn’t limited. In a sense, to compute T212 for progressively larger n
, progressively more information about its initial conditions is needed.
When one deals with ordinary, unnested recurrence relations, one’s always dealing with a fixed lookback. And the number of initial conditions then just depends on the lookback. (So, for example, the
Fibonacci recurrence has lookback 2, so needs two initial conditions, while the standard factorial recurrence has lookback 1, so needs only one initial condition.)
But for the nested recurrence relation T212 we see that this is no longer true; there can be an unboundedly large lookback.
OK, but let’s look back at the actual T212 sequence. Here it is up to larger values of n:
Or, plotting each point as a dot:
Given the recursive definition of f[n], the values of f[n] must always be powers of 2. This shows where each successive power of 2 is first reached as a function of n:
Meanwhile, this shows the accumulated average of f[n] as a function of n:
This is well fit by 0.38 Log[n], implying that, at least with this averaging, f[n] asymptotically approximates n^0.26. And, yes, it is somewhat surprising that what seems like a very “exponential”
recursive definition should lead to an f[n] that increases only like a power. But, needless to say, this is the kind of surprise one has to expect in the computational universe.
It’s worth noticing that f[n] fluctuates very intensely as a function of n. The overall distribution of values is very close to exponentially distributed—for example with the distribution of
logarithmic values of f[n] for n between 9 million and 10 million being:
What else can we say about this sequence? Let’s say we reduce mod 2 the powers of 2 for each f[n]. Then we get a sequence which starts:
This is definitely not “uniformly random”. But if one look at blocks of sequential values, one can plot at what n each of the 2^b possible configurations of a length-b block first appears:
And eventually it seems as if all length-b blocks for any given b will appear.
By the way, whereas in the P family, there were always a limited number of “blue graph trees” (associated with the limited number of initial conditions), for T212 the number of such trees increases
with n, as more initial conditions are used. So, for example, here are the trees for f[50] and f[51]:
We’ve so far discussed T212 only with the initial condition f[n] = 1 for n ≤ 0. The fact that f[n] is always a power of 2 relies on every initial value also being a power of 2. But here’s what
happens, for example, if f(n) = 2^s for n ≤ 0:
In general, one can think of T212 as transforming an ultimately infinite sequence of initial conditions into an infinite sequence of function values, with different forms of initial conditions
potentially giving very different sequences of function values:
(Note that not all choices of initial conditions are possible; some lead to “f[n] = f[n]” or “f[n] = f[n + 1]” situations, where the evaluation of the function can’t “make progress”.)
The “Summer School” Sequence T311 (f[n_] := 3 f[n – f[n – 1]])
Having explored T212, let’s now look at T311—the original one-term nestedly recursive function discovered at the 2003 Wolfram Summer School:
Here’s its basic behavior:
And here is its evaluation graph—which immediately reveals a lot more lookback than T212:
Plotting lookbacks as a function of n we get:
Much as with T212, the total number of lookbacks varies with n in the fairly simple way (~ 0.44 n):
Continuing the T311 sequence further, it looks qualitatively very much like T212:
And indeed T311—despite its larger number of lookbacks—seems to basically behave like T212. In a story typical of the Principle of Computational Equivalence, T212 seems to have already “filled out
the computational possibilities”, so T311 “doesn’t have anything to add”.
The “S Family”: f[n_] := n – f[f[n – a] – b]
As another (somewhat historically motivated) example of nestedly recursive functions, consider what we’ll call the “S family”, defined by:
Let’s start with the very minimal case S10 (or “S1”):
Our standard initial condition f[n] = 1 for n ≤ 0 doesn’t work here, because it implies that f[1] = 1 – f[1]. But if we take f[n] = 1 for n ≤ 1 we get:
Meanwhile, with f[n] = 1 for n ≤ 3 we get:
The first obvious feature of both these results is their overall slope: 1/ϕ ≈ 0.618, where ϕ is the golden ratio. It’s not too hard to see why one gets this slope. Assume that for large n we can take
f[n] = σ n. Then substitute this form into both sides of the recursive definition for the S family to get σ n n – σ (σ (n – a) – b). For large n all that survives is the condition for the
coefficients of n
which has solution σ = 1/ϕ.
Plotting f[n] – n/ϕ for the case f[n] = 1 for n ≤ 1 we get:
The evaluation graph is this case has a fairly simple form
as we can see even more clearly with a different graph layout:
It’s notable that only the initial condition f[1] = 1 is used—leading to a single “blue graph tree” that turns out to have a very simple “Fibonacci tree” form (which, as we’ll discuss below, has been
known since the 1970s):
From this it follows that f[n] related to the “Fibonacci-like” substitution system
and in fact the sequence of values of f[n] can be computed just as:
And indeed it turns out that in this case f[n] is given exactly by:
What about when f[n] = 1 not just for n ≤ 1 but beyond? For n ≤ 2 the results are essentially the same as for n ≤ 1. But for n ≤ 3 there’s a surprise: the behavior is considerably more complicated—as
we can see if we plot f[n] – n/ϕ:
Looking at the evaluation graph in this case we see that the only initial conditions sampled are f[1] = 1 and f[3] = 1 (with f[2] only being reached if one specifically starts with f[2]):
And continuing the evaluation graph we see a mixture of irregularity and comparative regularity:
The plot of f[n] has a strange “hand-drawn” appearance, with overall regularity but detailed apparent randomness. The most obvious large-scale feature is “bursting” behavior (interspersed in an audio
rendering with an annoying hum). The bursts all seem to have approximately (though not exactly) the same structure—and get systematically larger. The lengths of successive “regions of calm” between
bursts (characterized by runs with Abs[f[n] – n/ϕ] < 3) seem to consistently increase by a factor ϕ:
What happens to S1 with other initial conditions? Here are a few examples:
So how does Sa depend on a? Sometimes there’s at least a certain amount of clear regularity; sometimes it’s more complicated:
As is very common, adding the parameter b in the definition doesn’t seem to lead to fundamentally new behavior—though for b > 0 the initial condition f[n] = 1, n ≤ 0 can be used:
In all cases, only a limited number of initial conditions are sampled (bounded by the value of a + b in the original definition). But as we can see, the behavior can either be quite simple, or can be
highly complex.
More Complicated Rules
Highly complex behavior arises even from very simple rules. It’s a phenomenon one sees all over the computational universe. And we’re seeing it here in nestedly recursive functions. But if we make
the rules (i.e. definitions) for our functions more complicated, will we see fundamentally different behavior, or just more of the same?
The Principle of Computational Equivalence (as well as many empirical observations of other systems) suggests that it’ll be “more of the same”: that once one’s passed a fairly low threshold the
computational sophistication—and complexity—of behavior will no longer change.
And indeed this is what one sees in nestedly recursive functions. But below the threshold different kinds of things can happen with different kinds of rules.
There are several directions in which we can make rules more complicated. One that we won’t discuss here is to use operations (conditional, bitwise, etc.) that go beyond arithmetic. Others tend to
involve adding more instances of f in our definitions.
An obvious way to do this is to take f[n_] to be given by a sum of terms, “Fibonacci style”. There are various specific forms one can consider. As a first example—that we can call ab—let’s look at:
The value of a doesn’t seem to matter much. But changing b we see:
12 has unbounded lookback (at least starting with f[n] = 1 for n ≤ 0), but for larger b, 1b has bounded lookback. In both 13 and 15 there is continuing large-scale structure (here visible in log
though this does not seem to be reflected in the corresponding evaluation graphs:
As another level of Fibonacci-style definition, we can consider ab:
But the typical behavior here does not seem much different from what we already saw with one-term definitions involving only two f’s:
(Note that aa is equivalent to a. Cases like 13 lead after a transient to pure exponential growth.)
A somewhat more unusual case is what we can call abc:
Subtracting overall linear trends we get:
For 111 using initial conditions f[1] = f[2] = 1 and plotting f[n] – n/2 we get
which has a nested structure that is closely related to the result of concatenating binary digit sequences of successive integers:
But despite the regularity in the sequence of values, the evaluation graph for this function is not particularly simple:
So how else might we come up with more complicated rules? One possibility is that instead of “adding f’s by adding terms” we can add f’s by additional nesting. So, for example, we can consider what
we can call S[3]1 (here shown with initial condition f[n] = 1 for n ≤ 3):
We can estimate the overall slope here by solving for x in x 1 – x^3 to get ≈ 0.682. Subtracting this off we get:
We can also consider deeper nestings. At depth d the slope is the solution to x 1 – x^d. Somewhat remarkably, in all cases the only initial conditions probed are f[1] = 1 and f[3] = 1:
As another example of “higher nesting” we can consider the class of functions (that we call a):
Subtracting a constant 1/ϕ slope we get:
The evaluation graph for 1 is complicated, but has some definite structure:
What happens if we nest even more deeply, say defining a functions:
With depth-d nesting, we can estimate the overall slope of f[n] by solving for x in
so that for the d = 3 case here the overall slope is the real root of or about 0.544. Subtracting out this overall slope we get:
And, yes, the sine-curve-like form of 5 is very odd. Continuing 10x longer, though, things are “squaring off”:
What happens if we continue nesting deeper? stays fairly tame:
However, already allows for more complicated behavior:
And for different values of a there are different regularities:
There are all sorts of other extensions and generalizations one might consider. Some involve alternate functional forms; others involve introducing additional functions, or allowing multiple
arguments to our function f.
An Aside: The Continuous Case
In talking about recursive functions f[n] we’ve been assuming—as one normally does—that n is always an integer. But can we generalize what we’re doing to functions f[x] where x is a continuous real
Consider for example a continuous analog of the Fibonacci recurrence:
This produces a staircase-like function whose steps correspond to the usual Fibonacci numbers:
Adjusting the initial condition produces a slightly different result:
We can think of these as being solutions to a kind of “Fibonacci delay equation”—where we’ve given initial conditions not at discrete points, but instead on an interval.
So what happens with nestedly recursive functions? We can define an analog of S1 as:
Plotting this along with the discrete result we get:
In more detail, we get
where now the plateaus occur at the (“Wythoff numbers”) .
Changing the initial condition to be x ≤ 3 we get:
Removing the overall slope by subtracting x/ϕ gives:
One feature of the continuous case is that one can continuously change initial conditions—though the behavior one gets typically breaks into “domains” with discontinuous boundaries, as in this case
where we’re plotting the value of f[x] as a function of x and the “cutoff” in the initial conditions f[x], x ≤ :
So what about other rules? A rule like P312 (f[n_] := 3 + f[n – f[n – 2]]) given “constant” initial conditions effectively just copies and translates the initial interval, and gives a simple order-0
interpolation of the discrete case. With initial condition f[x] = x some segments get “tipped”:
All the cases we’ve considered here don’t “look back” to negative values, in either the discrete or continuous case. But what about a rule like T212 (f[n_] := 2 f[n – 1 f[n – 2]]) that progressively
“looks back further”? With the initial condition f[x] = 1 for x ≤ 0, one gets the same result as in the discrete case:
But if one uses the initial condition f[x ] = Abs[x – 1] for x ≤ 0 (the Abs[x – 1] is needed to avoid ending up with f[x] depending on f[y] for y > x) one instead has
yielding the rather different result:
Continuing for larger x (on a log scale) we get:
Successively zooming in on one of the first “regions of noise” we see that it ultimately consists just of a large number of straight segments:
What’s going on here? If we count the number of initial conditions that are used for different values of x we see that this has discontinuous changes, leading to disjoint segments in f[x]:
Plotting over a larger range of x values the number of initial conditions used is:
And plotting the actual values of those initial conditions we get:
If we go to later, “more intense” regions of noise, we see more fragmentation—and presumably in the limit x f[x]:
For the S family, with its overall n/ϕ trend, even constant initial conditions—say for S1—already lead to tipping, here shown compared to the discrete case:
How Do You Actually Compute Recursive Functions?
Let’s say we have a recursive definition—like the standard Fibonacci one:
How do we actually use this to compute the value of, say, f[7]? Well, we can start from f[7], then use the definition to write this as f[6] + f[5], then write f[6] as f[5] + f[4], and so on. And we
can represent this using a evaluation graph, in the form:
But this computation is in a sense very wasteful; for example, it’s independently computing f[3] five separate times (and of course getting the same answer each time). But what if we just stored each
f[n] as soon as we compute, and then just retrieve that stored (“cached”) value whenever we need it again?
In the Wolfram Language, it’s a very simple change to our original definition:
And now our evaluation graph becomes much simpler:
And indeed it’s this kind of minimal evaluation graph that we’ve been using in everything we’ve discussed so far.
What’s the relationship between the “tree” evaluation graph, and this minimal one? The tree graph is basically an “unrolled” version of the minimal graph, in which all the possible paths that can be
taken from the root node to the initial condition nodes have been treed out.
In general, the number of edges that come out of a single node in a evaluation graph will be equal to the number of instances of the function f that appear on the right-hand side of the recursive
definition we’re using (i.e. 2 in the case of the standard Fibonacci definition). So this means that if the maximum length of path from the root to the initial conditions is s, the maximum number of
nodes that can appear in the “unrolled” graph is 2^s. And whenever there are a fixed set of initial conditions (i.e. if there’s always the same lookback), the maximum path length is essentially n
—implying in the end that the maximum possible number of nodes in the unrolled graph will be 2^n.
(In the actual case of the Fibonacci recurrence, the number of nodes in the unrolled graph is, or about 1.6^n.)
But if we actually evaluate f[7]—say in the Wolfram Language—what is the sequence of f[n]’s that we’ll end up computing? Or, in effect, how will the evaluation graph be traversed? Here are the
results for the unrolled and minimal evaluation graphs—i.e. without and with caching:
Particularly in the first case this isn’t the only conceivable result we could have gotten. It’s the way it is here because of the particular “leftmost innermost” evaluation order that the Wolfram
Language uses by default. In effect, we’re traversing the graph in a depth-first way. In principle we could use other traversal orders, leading to f[n]’s being evaluated in different orders. But
unless we allow other operations (like f[3] + f[3] f[3]) to be interspersed with f evaluations, we’ll still always end up with the same number of f evaluations for a given evaluation graph.
But which is the “correct” evaluation graph? The unrolled one? Or the minimal one? Well, it depends on the computational primitives we’re prepared to use. With a pure stack machine, the unrolled
graph is the only one possible. But if we allow (random-access) memory, then the minimal graph becomes possible.
OK, so what happens with nestedly recursive functions? Here, for example, are unrolled and minimal graphs for T212:
Here are the sequences of f[n]’s that are computed:
And here’s a comparison of the number of nodes (i.e. f evaluations) from unrolled and minimal evaluation graphs (roughly 1.2^n and 0.5 n, respectively):
Different recursive functions lead to different patterns of behavior. The differences are less obvious in evaluation graphs, but can be quite obvious in the actual sequence of f[n]’s that are
But although looking at evaluation sequences from unrolled evaluation graphs can be helpful as a way of classifying behavior, the exponentially more steps involved in the unrolled graph typically
makes this impractical in practice.
Primitive Recursive or Not?
Recursive functions have a fairly long history, that we’ll be discussing below. And for nearly a hundred years there’s been a distinction made between “primitive recursive functions” and “general
recursive functions”. Primitive recursive functions are basically ones where there’s a “known-in-advance” pattern of computation that has to be done; general recursive functions are ones that may in
effect make one have to “search arbitrarily far” to get what one needs.
In Wolfram Language terms, primitive recursive functions are roughly ones that can be constructed directly using functions like Nest and Fold (perhaps nested); general recursive functions can also
involve functions like NestWhile and FoldWhile.
So, for example, with the Fibonacci definition
the function f[n] is primitive recursive and can be written, say, as:
Lots of the functions one encounters in practice are similarly primitive recursive—including most “typical mathematical functions” (Plus, Power, GCD, Prime, …). And for example functions that give
the results of n steps in the evolution of a Turing machine, cellular automaton, etc. are also primitive recursive. But functions that for example test whether a Turing machine will ever halt (or
give the state that it achieves if and when it does halt) are not in general primitive recursive.
On the face of it, our nestedly recursive functions seem like they must be primitive recursive, since they don’t for example appear to be “searching for anything”. But things like the presence of
longer and longer lookbacks raise questions. And then there’s the potential confusion of the very first example (dating from the late 1920s) of a recursively defined function known not to be
primitive recursive: the Ackermann function.
The Ackermann function has three (or sometimes two) arguments—and, notably, its definition (here given in its classic form) includes nested recursion:
This is what the evaluation graphs look like for some small cases:
Looking at these graphs we can begin to see a pattern. And in fact there’s a simple interpretation: f[m, x, y] for successive m is doing progressively more nested iterations of integer successor
operations. f[0, x, y] computes x + y; f[1, x, y] does “repeated addition”, i.e. computes x × y; f[2, x, y] does “repeated multiplication”, i.e. computes x^y; f[3, x, y] does “tetration”, i.e.
computes the “power tower” Nest[x^#&, 1, y]; etc.
Or, alternatively, these can be given explicitly in successively more nested form:
And at least in this form f[m, x, y] involves m nestings. But a given primitive recursive function can involve only a fixed number of nestings. It might be conceivable that we could rewrite
f[m, x, y] in certain cases to involve only a fixed number of nestings. But if we look at f[m, m, m] then this turns out to inevitably grow too rapidly to be represented by a fixed number of
nestings—and thus cannot be primitive recursive.
But it turns out that the fact that this can happen depends critically on the Ackermann function having more than one argument—so that one can construct the “diagonal” f[m, m, m].
So what about our nestedly recursive functions? Well, at least in the form that we’ve used them, they can all be written in terms of Fold. The key idea is to accumulate a list of values so far
(conveniently represented as an association)—sampling whichever parts are needed—and then at the end take the last element. So for example the “Summer School function” T311
can be written:
An important feature here is that we’re getting Lookup to give 1 if the value it’s trying to look up hasn’t been filled in yet, implementing the fact that f[n] = 1 for n ≤ 0.
So, yes, our recursive definition might look back further and further. But it always just finds value 1—which is easy for us to represent without, for example, any extra nesting, etc.
The ultimate (historical) definition of primitive recursion, though, doesn’t involve subsets of the Wolfram Language (the definition was given almost exactly 100 years too early!). Instead, it
involves a specific set of simple primitives:
(An alternative, equivalent definition for recursion—explicitly involving Fold—is r[g_, h_] := Fold[{u, v} , Range[0, #1 – 1]] &.)
So can our nestedly recursive functions be written purely in terms of these primitives? The answer is yes, though it’s seriously complicated. A simple function like Plus can for example be written as
r[p[1], s], so that e.g. r[p[1], s][2,3]. Times can be written as r[z, c[Plus, p[1], p[3]]] or r[z, c[r[p[1], s], p[1], p[3]]], while Factorial can be written as r[c[s, z], c[Times, p[1], c[s, p
[2]]]]. But even Fibonacci, for example, seems to require a very much longer specification.
In writing “primitive-recursive-style” definitions in Wolfram Language we accumulated values in lists and associations. But in the ultimate definition of primitive recursion, there are no such
constructs; the only form of “data” is positive integers. But for our definitions of nestedly recursive functions we can use a “tupling function” that “packages up” any list of integer values into a
single integer (and an untupling function that unpacks it). And we can do this say based on a pairing (2-element-tupling) function like:
But what about the actual If[n ≤0, 1, ...] lookback test itself? Well, If can be written in primitive recursive form too: for example, r[c[s, z], c[f, c[s, p[2]]]][n] is equivalent to If[n ≤ 0, 1, f
So our nestedly recursive functions as we’re using them are indeed primitive recursive. Or, more strictly, finding values f[n] is primitive recursive. Asking questions like “For what n does f[n]
reach 1000?” might not be primitive recursive. (The obvious way of answering them involves a FoldWhile-style non-primitive-recursive search, but proving that there’s no primitive recursive way to
answer the question is likely very much harder.)
By the way, it’s worth commenting that while for primitive recursive functions it’s always possible to compute a value f[n] for any n, that’s not necessarily true for general recursive functions. For
example, if we ask “For what n does f[n] reach 1000?” there might simply be no answer to this; f[n] might never reach 1000. And when we look at the computations going on underneath, the key
distinction is that in evaluating primitive recursive functions, the computations always halt, while for general recursive functions, they may not.
So, OK. Our nestedly recursive functions can be represented in “official primitive recursive form”, but they’re very complicated in that form. So that raises the question: what functions can be
represented simply in this form? In A New Kind of Science I gave some examples, each minimal for the output it produces:
And then there’s the most interesting function I found:
It’s the simplest primitive recursive function whose output has no obvious regularity:
Because it’s primitive recursive, it’s possible to express it in terms of functions like Fold—though it’s two deep in those, making it in some ways more complicated (at least as far as the
Grzegorczyk hierarchy that counts “Fold levels” is concerned) than our nestedly recursive functions:
But there’s still an issue to address with nestedly recursive functions and primitive recursion. When we have functions (like T212) that “reach back” progressively further as n increases, there’s a
question of what they’ll find. We’ve simply assumed f[n] = 1 for n ≤0. But what if there was something more complicated there? Even if f[–m] was given by some primitive recursive function, say p[m],
it seems possible that in computing f[n] one could end up somehow “bouncing back and forth” between positive and negative arguments, and in effect searching for an m for which p[m] has some
particular value, and in doing that searching one could find oneself outside the domain of primitive recursive functions.
And this raises yet another question: are all definitions we can give of nestedly recursive functions consistent? Consider for example:
Now ask: what is f[1]? We apply the recursive definition. But it gives us f[1] = 1 – f[f[0]] or f[1] = 1 – f[1], or, in other words, an inconsistency. There are many such inconsistencies that seem to
“happen instantly” when we apply definitions. But it seems conceivable that there could be “insidious inconsistencies” that show up only after many applications of a recursive definition. And it’s
also conceivable that one could end up with “loops” like f[i] = f[i]. And things like this could be reasons that f[n] might not be a “total function”, defined for all n.
We’ve seen all sorts of complex behavior in nestedly recursive functions. And what the Principle of Computational Equivalence suggests is that whenever one sees complex behavior, one must in some
sense be dealing with computations that are “as sophisticated as any computation can be”. And in particular one must be dealing with computations that can somehow support computation universality.
So what would it mean for a nestedly recursive function to be universal? For a start, one would need some way to “program” the function. There seem to be a couple of possibilities. First, one could
imagine packing both “code” and “data” into the argument n of f[n]. So, for example, one might use some form of tupling function to take a description of a rule and an initial state for a Turing
machine, together with a specification of a step number, then package all these things into an integer n that one feeds into one’s universal nestedly recursive function f. Then the idea would be that
the value computed for f[n] could be decoded to give the state of the Turing machine at the specified step. (Such a computation by definition always halts—but much as one computes with Turing
machines by successively asking for the next steps in their evolution, one can imagine setting up a “harness” that just keeps asking for values of f[n] at an infinite progression of values n.)
Another possible approach to making a universal nestedly recursive function is to imagine feeding in a “program” through the initial conditions one gives for the function. There might well need to be
decoding involved, but in some sense what one might hope is that just by changing its initial conditions one could get a nestedly recursive function with a specific recursive definition to emulate a
nestedly recursive function with any other recursive definition (or, say, for a start, any linear recurrence).
Perhaps one could construct a complicated nestedly recursive function that would have this property. But what the Principle of Computational Equivalence suggests is that it should be possible to find
the property even in “naturally occurring cases”—like P312 or T212.
The situation is probably going to be quite analogous to what happened with the rule 110 cellular automaton or the s = 2, k = 3 596440 Turing machine. By looking at the actual typical behavior of the
system one got some intuition about what was likely to be going on. And then later, with great effort, it became possible to actually prove computation universality.
In the case of nestedly recursive functions, we’ve seen here examples of just how diverse the behavior generated by changing initial conditions can be. It’s not clear how to harness this diversity to
extract some form of universality. But it seems likely that the “raw material” is there. And that nestedly recursive functions will show themselves as able join so many other systems in fitting into
the framework defined by the Principle of Computational Equivalence.
Some History
Once one has the concept of functions and the concept of recursion, nestedly recursive functions aren’t in some sense a “complicated idea”. And between this fact and the fact that nestedly recursive
functions haven’t historically had a clear place in any major line of mathematical or other development it’s quite difficult to be sure one’s accurately tracing their history. But I’ll describe here
at least what I currently know.
The concept of something like recursion is very old. It’s closely related to mathematical induction, which was already being used for proofs by Euclid around 300 BC. And in a quite different vein,
around the same time (though not recorded in written form until many centuries later) Fibonacci numbers arose in Indian culture in connection with the enumeration of prosody (“How many different
orders are there in which to say the Sanskrit words in this veda?”).
Then in 1202 Leonardo Fibonacci, at the end of his calculational math book Liber Abaci (which was notable for popularizing Hindu-Arabic numerals in the West) stated—more or less as a recreational
example—his “rabbit problem” in recursive form, and explicitly listed the Fibonacci numbers up to 377. But despite this early appearance, explicit recursively defined sequences remained largely a
curiosity until as late as the latter part of the twentieth century.
The concept of an abstract function began to emerge with calculus in the late 1600s, and became more solidified in the 1700s—but basically always in the context of continuous arguments. A variety of
specific examples of recurrence relations—for binomial coefficients, Bernoulli numbers, etc.—were in fairly widespread use. But there didn’t seem to have yet been a sense that there was a general
mathematical structure to study.
In the course of the 1800s there had been an increasing emphasis on rigor and abstraction in mathematics, leading by the latter part of the century to a serious effort to axiomatize concepts
associated with numbers. Starting with concepts like the recursive definition of integers by repeated application of the successor operation, by the time of Peano’s axioms for arithmetic in 1891
there was a clear general notion (particularly related to the induction axiom) that (integer) functions could be defined recursively. And when David Hilbert’s program of axiomatizing mathematics got
underway at the beginning of the 1900s, it was generally assumed that all (integer) functions of interest could actually be defined specifically using primitive recursion.
The notation for recursively specifying functions gradually got cleaner, making it easier to explore more elaborate examples. And in 1927 Wilhelm Ackermann (a student of Hilbert’s) introduced (in
completely modern notation) a “reasonable mathematical function” that—as we discussed above—he showed was not primitive recursive. And right there, in his paper, without any particular comment, is a
nestedly recursive function definition:
In 1931 Kurt Gödel further streamlined the representation of recursion, and solidified the notion of general recursion. There soon developed a whole field of recursion theory—though most of it was
concerned with general issues, not with specific, concrete recursive functions. A notable exception was the work of Rózsa Péter (Politzer), beginning in the 1930s, and leading in 1957 to her book
Recursive Functions—which contains a chapter on “Nested Recursion” (here in English translation):
But despite the many specific (mostly primitive) recursive functions discussed in the rest of the book, this chapter doesn’t stray far from the particular function Ackermann defined (or at least
Péter’s variant of it).
What about the recreational mathematics literature? By the late 1800s there were all sorts of publications involving numbers, games, etc. that at least implicitly involved recursion (an example being
Édouard Lucas’s 1883 Tower of Hanoi puzzle). But—perhaps because problems tended to be stated in words rather than mathematical notation—it doesn’t seem as if nestedly recursive functions ever showed
In the theoretical mathematics literature, a handful of somewhat abstract papers about “nested recursion” did appear, an example being one in 1961 by William Tait, then at Stanford:
But, meanwhile, the general idea of recursion was slowly beginning to go from purely theoretical to more practical. John McCarthy—who had coined the term “artificial intelligence”—was designing LISP
as “the language for AI” and by 1960 was writing papers with titles like “Recursive Functions of Symbolic Expressions and Their Computation by Machine”.
In 1962 McCarthy came to Stanford to found the AI Lab there, bringing with him enthusiasm for both AI and recursive functions. And by 1968 these two topics had come together in an effort to use “AI
methods” to prove properties of programs, and in particular programs involving recursive functions. And in doing this, John McCarthy came up with an example he intended to be awkward—that’s exactly a
nestedly recursive function:
In our notation, it would be:
And it became known as “McCarthy’s 91-function” because, yes, for many n, f[n] = 91. These days it’s trivial to evaluate this function—and to find out that f[n] = 91 only up to n = 102:
But even the evaluation graph is somewhat large
and in pure recursive evaluation the recursion stack can get deep—which back then was a struggle for LISP systems to handle.
There were efforts at theoretical analysis, for example by Zohar Manna, who in 1974 published Mathematical Theory of Computation which—in a section entitled “Fixpoints of Functionals”—presents the
91-function and other nestedly recursively functions, particularly in the context of evaluation-order questions.
In the years that followed, a variety of nestedly recursive functions were considered in connection with proving theorems about programs, and with practical assessments of LISP systems, a notable
example being Ikuo Takeuchi’s 1978 triple recursive function:
But in all these cases the focus was on how these functions would be evaluated, not on what their behavior would be (and it was typically very simple).
But now we have to follow another thread in the story. Back in 1961, right on the Stanford campus, a then-16-year-old Douglas Hofstadter was being led towards nestedly recursive functions. As Doug
tells it, it all started with him seeing that squares are interspersed with gaps of 1 or 2 between triangular numbers, and then noticing patterns in those gaps (and later realizing that they showed
nesting). Meanwhile, at Stanford he had access to a computer running Algol, a language which (like LISP and unlike Fortran) supported recursion (though this wasn’t particularly advertised, since
recursion was still generally considered quite obscure).
And as Doug tells it, within a year or two he was using Algol to do things like recursively create trees representing English sentences. Meanwhile—in a kind of imitation of the Eleusis
“guess-a-card-rule” game—Doug was apparently challenging his fellow students to a “function game” based on guessing a math function from specified values. And, as he tells it, he found that functions
that were defined recursively were the ones people found it hardest to guess.
That was all in the early 1960s, but it wasn’t until the mid-1970s that Doug Hofstadter returned to such pursuits. After various adventures, Doug was back at Stanford—writing what became his book
Gödel, Escher, Bach. And in 1977 he sent a letter to Neil Sloane, creator of the 1973 A Handbook of Integer Sequences (and what’s now the Online Encyclopedia of Integer Sequences, or OEIS):
As suggested by the accumulation of “sequence ID” annotations on the letter, Doug’s “eta sequences” had actually been studied in number theory before—in fact, since at least the 1920s (they are now
usually called Beatty sequences). But the letter went on, now introducing some related sequences—that had nestedly recursive definitions:
As Doug pointed out, these particular sequences (which were derived from golden ratio versions of his “eta sequences”) have a very regular form—which we would now call nested. And it was the
properties of this form that Doug seemed most concerned about in his letter. But actually, as we saw above, just a small change in initial conditions in what I’m calling S1 would have led to much
wilder behavior. But that apparently wasn’t something Doug happened to notice. A bit later in the letter, though, there was another nestedly recursive sequence—that Doug described as a “horse of an
entirely nother color”: the “absolutely CRAZY” Q sequence:
Two years later, Doug’s Gödel, Escher, Bach book was published—and in it, tucked away at the bottom of page 137, a few pages after a discussion of recursive generation of text (with examples such as
“the strange bagels that the purple cow without horns gobbled”), there was the Q sequence:
Strangely, though, there was no picture of it, and Doug listed only 17 terms (which, until I was writing this, was all I assumed he had computed):
So now nestedly recursive sequences were out in the open—in what quickly became a very popular book. But I don’t think many people noticed them there (though, as I’ll discuss, I did). Gödel, Escher,
Bach is primarily a playful book focused on exposition—and not the kind of place you’d expect to find a new, mathematical-style result.
Still—quite independent of the book—Neil Sloane showed Doug’s 1977 letter to his Bell Labs colleague Ron Graham, who within a year made a small mention of the Q sequence in a staid academic math
publication (in a characteristic “state-it-as-a-problem” Erdös form):
There was a small and tight-knit circle of serious mathematicians—essentially all of whom, as it happens, I personally knew—who would chase these kinds of easy-to-state-but-hard-to-solve problems.
Another was Richard Guy, who soon included the sequence as part of problem E31 in his Unsolved Problems in Number Theory, and mentioned it again a few years later.
But for most of the 1980s little was heard about the sequence. As it later turns out, a senior British applied mathematician named Brian Conolly (who wasn’t part of the aforementioned tight-knit
circle) had—presumably as a kind of hobby project—made some progress, and in 1986 had written to Guy about it. Guy apparently misplaced the letter, but later told Conolly that John Conway and Sol
Golomb had worked on similar things.
Conway presumably got the idea from Hofstadter’s work (though he had a habit of obfuscating his sources). But in any case, on July 15, 1988, Conway gave a talk at Bell Labs entitled “Some Crazy
Sequences” (note the word “crazy”, just like in Hofstadter’s letter to Sloane) in which he discussed the regular-enough-to-be-mathematically-interesting sequence (which we’re calling G[3]111 here):
Despite its visual regularity, Conway couldn’t mathematically prove certain features of the wiggles in the sequence—and in his talk offered a $10,000 prize for anyone who could. By August a Bell Labs
mathematician named Colin Mallows had done it. Conway claimed (later to be contradicted by video evidence) that he’d only offered $1000—and somehow the whole affair landed as a story in the August 30
New York Times under the heading “Intellectual Duel: Brash Challenge, Swift Response”. But in any case, this particular nestedly recursive sequence became known as “Conway’s Challenge Sequence”.
So what about Sol Golomb? It turns out he’d started writing a paper—though never finished it:
He’d computed 280 terms of the Q sequence (he wasn’t much of a computer user) and noticed a few coincidences. But he also mentioned another kind of nestedly recursive sequence, no doubt inspired by
his work on feedback shift registers:
As he noted, the behavior depends greatly on the initial conditions, though is always eventually periodic—with his student Unjeng Cheng having found long-period examples.
OK, so by 1988 nestedly recursive functions had at least some notoriety. So what happened next? Not so much. There’s a modest academic literature that’s emerged over the last few decades, mostly
concentrated very specifically around “Conway’s Challenge Sequence”, Hofstadter’s Q function, or very similar “meta Fibonacci” generalizations of them. And so far as I know, even the first published
large-scale picture of the Q sequence only appeared in 1998 (though I had pictures of it many years earlier):
Why wasn’t more done with nestedly recursive functions? At some level it’s because they tend to have too much computational irreducibility—making it pretty difficult to say much about them in
traditional mathematical ways. But perhaps more important, studying them broadly is really a matter of ruliology: it requires the idea of exploring spaces of rules, and of expecting the kinds of
behavior and phenomena that are characteristic of systems in the computational universe. And that’s something that’s still not nearly as widely understood as it should be.
My Personal Story with Nestedly Recursive Functions
I think 1979 was the year when I first took recursion seriously. I’d heard about the Fibonacci sequence (though not under that name) as a young child a decade earlier. I’d implicitly (and sometimes
explicitly) encountered recursion (sometimes through error messages!) in computer algebra systems I’d used. In science, I’d studied fractals quite extensively (Benoit Mandelbrot’s book having
appeared in 1977), and I’d been exposed to things like iterated maps. And I’d quite extensively studied cascade processes, notably of quarks and gluons in QCD.
As I think about it now, I realize that for several years I’d written programs that made use of recursion (and I had quite a lot of exposure to LISP, and the culture around it). But it was in 1979—
having just started using C—that I first remember writing a program (for doing percolation theory) where I explicitly thought “this is using recursion”. But then, in late 1979, I began to design SMP
(“Symbolic Manipulation Program”), the forerunner of the modern Wolfram Language. And in doing this I quickly solidified my knowledge of mathematical logic and the (then-fledgling) field of
theoretical computer science.
My concept of repeated transformations for symbolic expressions—which is still the core of Wolfram Language today—is somehow fundamentally recursive. And by the time we had the first signs of life
for our SMP system, Fibonacci was one of our very first tests. We soon tried the Ackermann function too. And in 1980 I became very interested in the problem of evaluation order, particularly for
recursive functions—and the best treatment I found of it (though at the time not very useful to me) was in none other than the book by Zohar Manna that I mentioned above. (In a strange twist, I was
at that time also studying gauge choices in physics—and it was only last year that I realized that they’re fundamentally the same thing as evaluation orders.)
It was soon after it came out in 1979 that I first saw Douglas Hofstadter’s book. At the time I wasn’t too interested in its Lewis-Carroll-like aspects, or its exposition; I just wanted to know what
the “science meat” in it was. And somehow I found the page about the Q sequence, and filed it away as “something interesting”.
I’m not sure when I first implemented the Q sequence in SMP, but by the time we released Version 1.0 in July 1981, there it was: an external package (hence the “X” prefix) for evaluating
“Hofstadter’s recursive function”, elegantly using memoization—with the description I gave saying (presumably because that’s what I’d noticed) that its values “have several properties of randomness”:
Firing up a copy of SMP today—running on a virtual machine that still thinks it’s 1986—I can run this code, and easily compute the function:
I can even plot it—though without an emulator for a 1980s-period storage-tube display, only the ASCIIfied rendering works:
So what did I make of the function back in 1981? I was interested in how complexity and randomness could occur in nature. But at the time, I didn’t have enough of a framework to understand the
connection. And, as it was, I was just starting to explore cellular automata, which seemed a lot more “nature like”—and which soon led me to things like rule 30 and the phenomenon of computational
Still, I didn’t forget the Q sequence. And when I was building Mathematica I again used it as a test (the .tq file extension came from the brief period in 1987 when we were trying out “Technique” as
the name of the system):
When Mathematica 1.0 was released on June 23, 1988, the Q sequence appeared again, this time as an example in the soon-in-every-major-bookstore Mathematica book:
I don’t think I was aware of Conway’s lecture that occurred just 18 days later. And for a couple of years I was consumed with tending to a young product and a young company. But by 1991, I was
getting ready to launch into basic science again. Meanwhile, the number theorist (and today horologist) Ilan Vardi—yet again from Stanford—had been working at Wolfram Research and writing a book
entitled Computational Recreations in Mathematica, which included a long section on the analysis of Takeuchi’s nested recursive function (“TAK”). My email archive records an exchange I had with him:
He suggested a “more symmetrical” nested recursive function. I responded, including a picture that made it fairly clear that this particular function would have only nested behavior, and not seem
By the summer of 1991 I was in the thick of exploring different kinds of systems with simple rules, discovering the remarkable complexity they could produce, and filling out what became Chapter 3 of
A New Kind of Science: “The World of Simple Programs”. But then came Chapter 4: “Systems Based on Numbers”. I had known since the mid-1980s about the randomness in things like digit sequences
produced by successive arithmetic operations. But what about randomness in pure sequences of integers? I resolved to find out just what it would take to produce randomness there. And so it was that
on August 13, 1993, I came to be enumerating possible symbolic forms for recursive functions—and selecting ones that could generate at least 10 terms:
As soon as I plotted the “survivors” I could see that interesting things were going to happen:
Was this complexity somehow going to end? I checked out to 10 million terms. And soon I started collecting my “prize specimens” and making a gallery of them:
I had some one-term recurrences, and some two-term ones. Somewhat shortsightedly I was always using “Fibonacci-like” initial conditions f[1] = f[2] = 1—and I rejected any recurrence that tried to
“look back” to f[0], f[–1], etc. And at the time, with this constraint, I only found “really interesting” behavior in two-term recurrences.
In 1994 I returned briefly to recursive sequences, adding a note “solving” a few of them, and discussing the evaluation graphs of others:
When I finally finished A New Kind of Science in 2002, I included a list of historical “Close approaches” to its core discoveries, one of them being Douglas Hofstadter’s work on recursive sequences:
In retrospect, back in 1981 I should have been able to take the “Q sequence” and recognize in it the essential “rule 30 phenomenon”. But as it was, it took another decade—and many other explorations
in the computational universe—for me to build up the right conceptual framework to see this. In A New Kind of Science I studied many kinds of systems, probing them far enough, I hoped, to see their
most important features. But recursive functions were an example where I always felt there was more to do; I felt I’d only just scratched the surface.
In June 2003—a year after A New Kind of Science was published—we held our first summer school. And as a way to introduce methodology—and be sure that people knew I was fallible and approachable—I
decided on the first day of the summer school to do a “live experiment”, and try to stumble my way to discovering something new, live and in public.
A few minutes before the session started, I picked the subject: recursive functions. I began with some examples I knew. Then it was time to go exploring. At first lots of functions “didn’t work”
because they were looking back too far. But then someone piped up “Why not just say that f[n] is 1 whenever n isn’t a positive integer?” Good idea! And very easy to try.
Soon we had the “obvious” functions written (today Apply[Plus, ...] could be just Total[...], but otherwise there’s nothing “out of date” here):
In a typical story of Wolfram-Language-helps-one-think-clearly, the obvious function was also very general, and allowed a recurrence with any number of terms. So why not start with just one term? And
immediately, there it was—what we’re now calling T311:
And then a plot (yes, after Version 6 one didn’t need the Show or the trailing “;”):
Of course, as is the nature of computational constructions, this is something timeless—that looks the same today as it did 21 years ago (well, except that now our plots display with color by
I thought this was a pretty neat discovery. And I just couldn’t believe that years earlier I’d failed to see the obvious generalization of having “infinite” initial conditions.
The next week I did a followup session, this time talking about how one would write up a discovery like this. We started off with possible titles (including audience suggestions):
And, yes, the first title listed is exactly the one I’ve now used here. In the notebook I created back then, there were first some notes (some of which should still be explored!):
Three hours later (on the afternoon of July 11, 2003) there’s another notebook, with the beginnings of a writeup:
By the way, another thing we came up with at the summer school was the (non-nestedly) recursive function:
Plotting g[n + 1] – g[n] gives:
And, yes, bizarrely (and reminiscent of McCarthy’s 91-function) for n ≥ 396, g[n + 1] – g[n] is always 97, and g[n] = 38606 + 97 (n – 396).
But in any case, a week or so after my “writeups” session, the summer school was over. In January 2004 I briefly picked the project up again, and made some pictures that, yes, show interesting
structure that perhaps I should investigate now:
In the years that followed, I would occasionally bring nestedly recursive functions out again—particularly in interacting with high school and other students. And at our summer programs I suggested
projects with them for a number of students.
In 2008, they seemed like an “obvious interesting thing” to add to our Demonstrations Project:
But mostly, they languished. Until, that is, my burst of “finish this” intellectual energy that followed the launch of our Physics Project in 2020. So here now, finally, after a journey of 43 years,
I feel like I’ve been able to do some justice to nestedly recursive functions, and provided a bit more illumination to yet another corner of the computational universe.
(Needless to say, there are many, many additional questions and issues to explore. Different primitives, e.g. Mod, Floor, etc. Multiple functions that refer to each other. Multiway cases. Functions
based on rational numbers. And endless potential approaches to analysis, identifying pockets of regularity and computational reducibility.)
Thanks to Brad Klee for extensive help. Thanks also to those who’ve worked on nestedly recursive functions as students at our summer programs over the years, including Roberto Martinez (2003), Eric
Rowland (2003), Chris Song (2021) and Thomas Adler (2024). I’ve benefitted from interactions about nestedly recursive functions with Ilan Vardi (1991), Tal Kubo (1993), Robby Villegas (2003), Todd
Rowland (2003 etc.), Jim Propp (2004), Matthew Szudzik (2005 etc.), Joseph Stocke (2021 etc.), Christopher Gilbert (2024) and Max Niedermann (2024). Thanks to Doug Hofstadter for extensive answers to
questions about history for this piece. It’s perhaps worth noting that I’ve personally known many of the people mentioned in the history section here (with the dates I met them indicated): John
Conway (1984), Paul Erdös (1986), Sol Golomb (1981), Ron Graham (1983), Benoit Mandelbrot (1986), John McCarthy (1981) and Neil Sloane (1983). | {"url":"https://writings.stephenwolfram.com/2024/09/nestedly-recursive-functions/","timestamp":"2024-11-04T23:12:50Z","content_type":"text/html","content_length":"249141","record_id":"<urn:uuid:e97abfc8-6fc4-40ec-a5fc-1c538a358fb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00539.warc.gz"} |
Perimeter of Circle | Definition, Formula and Solved Examples
ArticlesMath ArticlesPerimeter of Circle
Perimeter of Circle
The distance along the boundary of a circle is the Circumference of the circle. In other words, it is the total length of the boundary. For example, if an athlete starts at a point on a circular
track, then, the distance he covers around the track to reach the same point again is the circumference of the track.
Definition and Formula of Circumference
The circumference formula is used to calculate the distance around the edge or boundary of a circle. It is expressed as:
Circumference = 2 x π x r
C = 2πr
Circumference (C) is the distance around the edge of the circle, measured in units such as meters (m) or centimeters (cm). π (pi) is a mathematical constant approximately equal to 3.14159 (often
rounded to 3.14). r is the radius of the circle, which is the distance from the center of the circle to any point on its edge.
The formula states that the circumference of a circle is equal to twice the product of π and the radius of the circle. The radius is multiplied by 2π because π represents the ratio of the
circumference to the diameter of a circle and multiplying it by the radius gives the circumference directly.
This formula is widely used in geometry, trigonometry, and various fields of science and engineering where circles are involved. It allows for the calculation of the circumference of a circle based
on its radius or vice versa.
Also Check
Solved Examples on Perimeter of Circle
Example 1: Finding the circumference when the radius is given
Given: Radius (r) = 5 cm
To find: Circumference (C)
Using the circumference formula: C = 2πr
C = 2 x 3.14 x 5
C ≈ 31.4 cm
Therefore, when the radius is 5 cm, the circumference of the circle is approximately 31.4 cm.
Example 2: Finding the radius when the circumference is given
Given: Circumference (C) = 12 m
To find: Radius (r)
Using the circumference formula: C = 2πr
12 = 2 x 3.14 x r
12 = 6.28r
r ≈ 1.91 m
Therefore, when the circumference is 12 m, the radius of the circle is approximately 1.91 m.
Example 3: Finding the circumference when the diameter is given
Given: Diameter (d) = 10 cm
To find: Circumference (C)
First, we need to find the radius using the formula:
radius = diameter / 2
r = 10 cm / 2 = 5 cm
Using the circumference formula: C = 2πr
C = 2 x 3.14 x 5
C ≈ 31.4 cm
Therefore, when the diameter is 10 cm, the circumference of the circle is approximately 31.4 cm.
Frequently Asked Question on Perimeter of Circle
What is the Circumference of a Circle?
The circumference of a circle is the distance around the outer boundary or edge of the circle. It is the total length of the curve that forms the circle. The circumference is directly related to the
size of the circle and can be found using the formula: Circumference = 2 x π x r
How to Calculate Diameter from Circumference?
o calculate the diameter of a circle when the circumference is known, you can use the following formula: Diameter = Circumference /π or d = C / π where: Diameter (d) is the distance across the
circle, passing through its center. Circumference (C) is the distance around the circle. π (pi) is a mathematical constant approximately equal to 3.14159 (often rounded to 3.14).
What are the steps involved in finding the circumference of a circle if its area is given?
To find the circumference of a circle when its area is given, you can follow these steps: Determine the formula for the area of a circle: The formula for the area of a circle is A = π r2, where A
represents the area and r is the radius of the circle. Rearrange the formula to solve for the radius: r = √(A / π) In this step, take the square root of (A / π) to find the radius. Calculate the
circumference using the radius: Circumference = 2 πr Substitute the value of the radius into the formula to find the circumference.
What is the value of pi?
The value of pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a simple fraction or a
finite decimal. Pi is approximately equal to 3.14159, but its decimal representation goes on infinitely without repeating.
Which formula is used to find the perimeter of a circle, when its area is given?
To find the perimeter (circumference) of a circle when its area is given, you need to use the following formula: Perimeter = 2 x √(π x Area)
What are the units of circumference?
Since a circle's circumference is the linear distance of the circle's edge, it describes a length. Therefore, the most common units of a circle's circumference are millimeter, centimeter, meter for
the metric system, and inch, feet, and yard for the imperial system.
How do you find the circumference of a 1m circle?
To calculate the circumference of a circle with a radius of 1 meter, simply follow these steps: Multiply the radius by 2 to get the diameter of 2 meters. Multiply the result by π, or 3.14 for an
estimation. And there you go; the circumference of a circle with a radius of 1 meter is 6.28 meters.
How to Calculate Diameter from Circumference?
The formula for circumference = diameter × π Or, diameter = circumference/π. So, the diameter of the circle in terms of circumference will be equal to the ratio of the circumference of the circle and
Find the circumference of a circle in terms of π, if the radius measures 3 cm.
We know that, Circumference = 2πr = 2π(3) = 6π Therefore, the circumference of a circle is 6π cm, if its radius is 3 cm. | {"url":"https://infinitylearn.com/surge/articles/perimeter-of-circle/","timestamp":"2024-11-09T13:22:08Z","content_type":"text/html","content_length":"172940","record_id":"<urn:uuid:63fc86e8-28d4-445b-8b42-a3b67506081f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00558.warc.gz"} |
Instructions for Precalculus Placement Test
The prerequisite for this course is completion of either:
- Shormann Algebra 2 or Saxon Algebra 2, 3rd Edition
- Or, any other Algebra 2 AND Geometry*
*If your student has not taken Geometry, instead of taking a stand-alone geometry course, Dr. Shormann recommends these students take Shormann Algebra 2 with Integrated Geometry. This will review and
develop fluency in their algebra skills while they learn the needed geometry concepts. Learn More: Algebra 1 & 2- No Geometry
To ensure readiness, we recommend all students who are new to Shormann Math to take a placement test.
1. How to Take the Test
• If the last math course taken was by Saxon or Shormann, see Saxon or Shormann Math
• A parent or teacher should supervise the test.
• Time Limit: 1 Hour
• No calculators and no help from a parent or teacher.
• The student should show all work on a separate piece of paper.
2. Print the Test: Placement Test for Precalculus
3. Use the Answer Key at the end of the Placement Test to grade it.
4. Placement Guide:
8+ Correct: Passed - ready for Shormann Precalculus with Trigonometry
7 or Less Correct: Failed - Shormann Algebra 2 with Integrated Geometry is recommended but take the placement test to ensure readiness. This course will build mastery and fluency in both geometry and
algebra, while providing excellent preparation for Precalculus, the PSAT, SAT, and ACT. It also includes a 3-6 week CLEP College Algebra prep course which earns 1 high school math credit and up to 3
college credits. If you pass the CLEP Practice exam, included in the course, you can list it on your transcript as Advanced Algebra. Take the Placement Test for Algebra 2
If you used Saxon Algebra 2, 3rd Edition Students
There are about 20 geometry concepts that were taught in Shormann Algebra 2 that were not taught in Saxon. To bridge this gap, we have created Geometry Prep Lessons to prepare you for Shormann
Precalculus. They are included in the Shormann Precalculus eLearning course and we recommend all Saxon Algebra 2 students complete them.
If you need help, CONTACT US HERE | {"url":"https://diveintomath.com/instructions-for-precalculus-placement-test/","timestamp":"2024-11-04T17:06:07Z","content_type":"text/html","content_length":"70308","record_id":"<urn:uuid:bec12729-583e-43da-ad23-65b792f94eed>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00533.warc.gz"} |
Prediction Time
Following up on my last post on Charles Murray's new book, real education, it's time to see what Murray predicts will be the results from the grand experiment he proposes:
On measures involving interpersonal and intrapersonal ability. I expect statistically significant but substantively modest gains. On measures of actual knowledge, the experimental group will
score dramatically higher than the members of the comparison group, perhaps 30-plus percentile points higher (technically more than a standard deviation). On measures of reading and math
achievement, the differences will be no more than 15 to 20 percentile points (about half a standard deviation). Three years after the experiment ends, all of the differences will have shrunk. The
differences in reading and math will be no more than 8 to 12 percentile points (no more than a third of a standard deviation) and may have disappeared altogether.
More formally, I predict that the magnitude of each academic effect will be a function of the g loading of the measure. Measures of retention of simple factual material have the lowest g loadings
and will show the largest gains. For highly g-loaded measures such as reading comprehension and math, what has been accomplished by the last half-century of preschool and elementary school will
be shown to be about as good as we can do, no matter how much money is spent.
This is a decent prediction. The I think that Murray overestimates the ease at which facts can be taught to and retained by low-IQ students and underestimates their ability with respect to math and
reading comprehension.
Facts are difficult to learn because facts must be mostly learned on a case by case basis which is not readily amenable to acceleration. Math and reading (decoding and comprehension) are easier to
teach because these skills, can be accelerated (even though teaching language and vocabulary remain problematic). But I knew that from the Follow through and the Baltimore Curriculum Project data.
The data shows that we can get at least about three-quarters to a standard deviation improvement on average by the end of elementary school, better if we discount the schools that are so incompetent
that they are unable to implement well-tested programs with fidelity.
Murray's point with respect to fade-out is well taken, but I'll leave that for another post.
4 comments:
>>For highly g-loaded measures such as reading comprehension and math, what has been accomplished by the last half-century of preschool and elementary school will be shown to be about as good as
we can do, no matter how much money is spent.<<
KD: I assume you disagree with the above quote.
Also, good instruction can change g--at least the way it is currently measured-- correct?
In my experience with Direct Instruction Corrective Reading Comprehension B this is almost certainly true.
KDeRosa said...
We have data up to about fifth/sixth grade that says otherwise, so I do disagree.
Past fifth grade the data is scant, so I have no basis for agreeing or disagreeing. I do think that most kids going through even a good instructional program like DI will continue to need
compensatory instruction for learning to continue at the same pace.
What we need, as Murray suggests, is a follow through program that takes us to 8th or 12th grade.
I am dubious that goo instruction can substantiaklly affect g after age 10 or so. I do, however, think that kids with low-g can learn more than they currently do with improved instruction.
okay, I give . . .
what's a layman's explanation of a g-load?
TangoMan said...
what's a layman's explanation of a g-load?
Here's a simple example that you can even field-test.
Digit Span versus Reverse Digit Span.
There is little correlation between g and remembering a string of numbers but there is a significant correlation between g and remembering those numbers in reverse order and processing them in
the same time-frame as the forward digit span.
All mental tasks are not equal in complexity. | {"url":"https://d-edreckoning.blogspot.com/2008/08/prediction-time.html","timestamp":"2024-11-03T21:45:41Z","content_type":"text/html","content_length":"74660","record_id":"<urn:uuid:f42865c4-e705-4acb-bb26-098280329395>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00858.warc.gz"} |
Differential Privacy is Vulnerable to Correlated Data -- Introducing Dependent Differential Privacy
[This post is joint work with Princeton graduate student Changchang Liu and IBM researcher Supriyo Chakraborty. See our paper for full details. — Prateek Mittal ]
The tussle between data utility and data privacy
Information sharing is important for realizing the vision of a data-driven customization of our environment. Data that were earlier locked up in private repositories are now being increasingly shared
for enabling new context-aware applications, better monitoring of population statistics, and facilitating academic research in diverse fields. However, sharing personal data gives rise to serious
privacy concerns as the data can contain sensitive information that a user might want to keep private. Thus, while on one hand, it is imperative to release utility-providing information, on the other
hand, the privacy of users whose data is being shared also needs to be protected.
What is differential privacy?
Among existing data privacy metrics, differential privacy (DP) has emerged has a rigorous mathematical framework for defining and preserving privacy, and has received considerable attention from the
privacy community. DP is used for protecting the privacy of individual users’ data while publishing aggregate query responses over statistical databases, and guarantees that the distribution of query
responses changes only slightly with the addition or deletion of a single tuple (entry or record) in the database. Thus, after observing the query response, an adversary is limited in its ability to
infer sensitive data about any tuple and in turn about the user contributing the tuple. For example, if the government aims to publish the average salary of individuals in New Jersey, DP can reveal
the approximate average salary of this region while protecting the privacy of any individual user’s salary. DP achieves this balance between utility and privacy by adding noise (perturbation) to the
true query result, for example, via the Laplace perturbation mechanism.
Differential privacy is vulnerable to data correlation!
To provide its guarantees, DP implicitly assumes that the data tuples in the database, each from a different user, are all independent. This is a weak assumption, especially because tuple dependence
occurs naturally in datasets due to social interactions between users. For example, consider a social network graph with nodes representing users, and edges representing friendship relationships. An
adversary can infer the existence of a social link between two users that are not explicitly connected in the graph using the existence of edges among other users in the graph ( since two friends are
likely to have many other friends in common). Similarly, private attributes in a user’s record can be inferred by exploiting the public attributes of other users sharing similar interests. An
adversary can easily infer a user’s susceptibility to a contagious disease using access to the noisy query result and also the fact that the user’s immediate family members are part of the database.
In an era of big-data, the tuples within a database present rich semantics and complex structures that inevitably lead to data correlation. Therefore, data correlation, i.e., the probabilistic
dependence relationship among tuples in the database, which is implicitly ignored in DP, can seriously deteriorate the privacy properties of DP.
Previous work has investigated this problem and proposed several interesting privacy metrics, such as zero-knowledge privacy, pufferfish privacy, membership privacy, inferential privacy, etc.
However, the privacy community is lacking effective perturbation mechanisms that can achieve these proposed privacy metrics.
In a paper that we presented at the Network and Distributed System Security Symposium (NDSS 2016), we first demonstrate a Bayesian attack on differentially private mechanisms using real-world
datasets (such as Gowalla location data). Our attack exploits the correlation between location information and social information of users in the Gowalla dataset. We show that it is possible to
infer a user’s sensitive location information from differentially private responses by exploiting her social relationships. When data is correlated, DP underestimates the amount of noise that
must be added to achieve a desired bound on the adversary’s ability to make sensitive inferences. Therefore, correlation (or dependency) among data tuples would break the expected privacy
How can we improve differential privacy to protect correlated data? Introducing dependent differential privacy.
Overall, our work generalizes the classic differential privacy framework to explicitly consider correlations in datasets, thus enhancing the applicability of the approach to a wide range of
datasets and applications. We formalize the notion of dependent differential privacy (DDP); DDP aims to explicitly incorporate probabilistic dependency constraints among tuples in the privacy
formulation. We also propose a perturbation mechanism to achieve the privacy guarantees in DDP; our approach extends the conventional Laplace perturbation mechanism by incorporating the
correlation between data tuples. To do so, we introduce the concept of dependent coefficient between two tuples, which aims to capture the correlation between data tuples from the perspective of
privacy. (The dependent coefficient between two tuples evaluates the ratio between the dependent indistinguishability of one tuple induced by the other tuple and the self indistinguishability of
the tuple itself.) Our perturbation mechanism can be applied for arbitrary data queries and we validated its effectiveness using real-world datasets.
What are future research directions for dependent differential privacy?
First, to form a deeper understanding of our dependent differential privacy framework, it would be interesting to explore the application of standard concepts in differential privacy to our
framework, such as local sensitivity and smooth sensitivity. Second, the effectiveness of our proposed perturbation mechanism depends on how well the dependence among data can be modeled and
computed. One limitation of our work is that the dependence coefficient is exactly known to both the adversary and the database owner. How to accurately compute the dependence coefficient and deal
with potential underestimation would be an interesting future work (note that the overestimation of dependence coefficient continues to provide rigorous DDP guarantees). Finally, in our dependent
differential privacy framework, we only considered pairwise correlations between tuples in a database. An important research direction is to generalize the pairwise correlations that we considered in
our paper to joint correlations in the dataset.
1. Frank McSherry says
I wrote up some follow-up comments, in longer form.
In particular, I have some questions about Section IV, including (i) which k-means algorithm you used, (ii) why the “information leaked” should be zero when epsilon is zero, and (iii) what the
weird shape of your figure 5b is due to.
□ Changchang Liu says
Hi Frank,
Thanks for your interests. We utilized the differentially private k-means algorithm in [14] (please see their related paragraphs prior to Section 5). For DP adversary, the information leakage
is zero under zero differential privacy. For DDP adversary, the information leakage is larger than zero even under zero differential privacy since the dependence relationship available to the
adversary would leak some information. Thanks.
□ Curious says
+1 — Thanks Frank, for the clear and lucid exposition. I look forward to the response by the authors.
2. Frank McSherry says
Hi! I have a few comments (I tried to post more, but they were eaten).
> To provide its guarantees, DP mechanisms assume that the data tuples (or records) in the database, each from a different user, are all independent.
This is not true. You are interested in a stronger guarantee, that a thing remains a secret, even when the subject of the secret no longer has the ability to keep it a secret (i.e. withholding
their data doesn’t maintain the secret). It is a fine thing to study, but it isn’t the guarantee differential privacy provides.
More details here, https://github.com/frankmcsherry/blog/blob/master/posts/2016-08-16.md, if that helps explain where I’m coming from.
3. Frank McSherry says
From your paper,
> To provide its guarantees, DP mechanisms assume that the data tuples (or records) in the database, each from a different user, are all independent.
This isn’t true. I think what you probably meant was “To provide *some other, stronger* guarantees, …” or something like that. But, best not to write things that aren’t true, because then other
people just copy them down as if they were true.
You seem interested in the guarantee that a thing remains a secret, even when the subject of the secret no longer has the agency to keep the secret (i.e. withholding their data doesn’t maintain
the secret). It is a fine thing to study, but it isn’t the guarantee differential privacy provides.
I personally think this may not even count as “privacy” any more; is learning that smoking leads to health risks in most people (so, likely you as well) a violation of your privacy, if we would
learn this with or without your data? This is probably a lot more debatable, but I hope we can agree that the value of being clear about what is and isn’t true is less debatable.
I wrote more about this in long form, if that is in any way helpful in explaining where I’m coming from:
Cheers, and thanks for the interesting work!
□ pmittal says
Hi Frank!
>> “You seem interested in the guarantee that a thing remains a secret, even when the subject of the secret no longer has the agency to keep the secret (i.e. withholding their data doesn’t
maintain the secret). It is a fine thing to study, but it isn’t the guarantee differential privacy provides.”
Indeed, this is the point of the blog post, and what we mean by “privacy for correlated data”. In scenarios where “data owner withholding their data” suffices to maintain the secret (i.e.
“independent tuples in the terminology used in our paper), stronger notions of privacy including dependent differential privacy, membership privacy, and inferential privacy seem to reduce to
differential privacy.
**However**, in scenarios where “data owner withholding their data” doesn’t suffice to maintain the secret (i.e. “privacy for correlated data”), what does differential privacy mean? Our work
on dependent differential privacy is a generalization in this context.
In my opinion, the latter scenario is the dominant scenario in many real-world datasets, including social data, communication data, genetic data etc. For example, removing my genetic
information from a genomic database may not suffice to protect the secret if a close relative is part of the database.
Currently, systems designers are applying differential privacy as a black-box to various datasets, without considering the difference between the two scenarios outlined above. We hope that
this post and our work inspires further research into stronger notions of privacy (and associated mechanisms!) for protecting user data.
Finally, +1 to Frank’s blogpost that nicely clarifies the difference between “your secrets” and “secrets about you” that seem to somewhat map to the two scenarios outlined above (?). The
challenge for privacy engineers is to clearly differentiate between the two when applying state-of-the-art privacy mechanisms, otherwise users may not get the privacy guarantees they might
expect. For example, there are number of papers that apply differential privacy to networked settings, without considering if the intended secrets could be leaked via correlations in the
(Hopefully this clarifies our perspective)
☆ Frank McSherry says
Right, so my point is that while
> In scenarios where “data owner withholding their data” suffices to maintain the secret (i.e. “independent tuples in the terminology used in our paper), stronger notions of privacy
including dependent differential privacy, membership privacy, and inferential privacy seem to reduce to differential privacy.
may be true, when the assumption of independence doesn’t exist (and it really shouldn’t, because assumptions like that are impossible to validate), differential privacy *still provides
its guarantees*. It doesn’t turn into a mathematical pumpkin or anything. Saying that its guarantees no longer hold is … =(
> **However**, in scenarios where “data owner withholding their data” doesn’t suffice to maintain the secret (i.e. “privacy for correlated data”), what does differential privacy mean?
It provides exactly the guarantee it says: no disclosures (or other bad things) happen as a result of participating in the computation. I have no idea if the disclosure will happen or
will not happen, and neither do you, because the data owner may have already disclosed enough information for others to discover the secret; unless you are going to micromanage the rest
of their lives, the others may eventually get around to learning the secret. However, if and when it happens, it will not be a result of mis-managing the individual data, and so the
person can participate in these sorts of computations without that specific concern.
> For example, removing my genetic information from a genomic database may not suffice to protect the secret if a close relative is part of the database.
That’s true, and it is a fun ethical question to determine if you have the right to suppress your relative’s genomic contribution in the interest of keeping your genomic data secret. In a
very real sense, your genomic information is not *your* secret. In either case, your *contribution* of genetic information is not what will lead to people learning anything specific about
your genome.
☆ Frank McSherry says
Just to try and make my point differently, here is an alternate abstract you could write that I think is less wrong and more on-message:
Differential privacy (DP) is a widely accepted mathematical framework for protecting data privacy. Simply stated, it guarantees that the distribution of query results changes only
slightly due to the modification of any one tuple in the database. However, this guarantee only conceals the data *from* each individual, it does not conceal the data *about* each
individual. Many datasets contain correlated records, where information from some records reflects on many others, and for which differential privacy does not guarantee that we will not
learn overly specific information about the participants.
In this paper, we make several contributions that not only demonstrate the feasibility of exploiting the above vulnerability but also provide steps towards mitigating it. …
4. Kamalika says
This is a very nice post! Thanks for pointing out this piece of work. We have some work currently on arxiv on this kind of privacy for correlated data that you might find relevant: http://
While we use a slightly different privacy framework (Pufferfish privacy), we provide a generalized notion of sensitivity for privacy in correlated data, and also a more specialized algorithm for
Bayesian networks. We were not aware of your work before, but we will make sure to cite you in an updated version of our paper.
□ pmittal says
Thanks Kamalika. Your technical report and the generalized notion of sensitivity look very interesting! | {"url":"https://freedom-to-tinker.com/2016/08/26/differential-privacy-is-vulnerable-to-correlated-data-introducing-dependent-differential-privacy/","timestamp":"2024-11-13T12:42:41Z","content_type":"application/xhtml+xml","content_length":"100147","record_id":"<urn:uuid:aea4ff6e-0bb0-444c-bfac-2acc1fbf42e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00228.warc.gz"} |
15 PSI to Liters per Minute Calculator - GEGCalculators
15 PSI to Liters per Minute Calculator
15 PSI cannot be directly converted to liters per minute (LPM) without knowing additional details such as pipe size and fluid characteristics. PSI measures pressure, while LPM measures flow rate. To
determine the LPM, you need to consider the specific system’s parameters and apply fluid dynamics principles.
15 psi to liters per minute Calculator
How do you convert PSI to litres per minute?
You cannot directly convert PSI (pounds per square inch) to liters per minute (LPM) because PSI is a measure of pressure, while LPM is a measure of flow rate. To determine LPM, you would need
additional information, such as the size of the pipe or the nozzle, and the pressure.
How do you convert PSI to flow rate?
To convert PSI to flow rate, you would need to know the characteristics of the system, including the size of the pipe or nozzle and the fluid being used. The flow rate can be calculated using the
principles of fluid dynamics and equations like the Bernoulli equation and the Darcy-Weisbach equation.
How many litres is 100 psi?
100 PSI is a measure of pressure and does not directly correspond to a specific volume like liters. The volume of fluid (in liters) that corresponds to 100 PSI depends on various factors, including
the size of the container and the type of fluid.
How many liters per minute is 150 psi?
Again, you cannot directly convert PSI to liters per minute without additional information about the system. PSI measures pressure, while liters per minute measures flow rate. The flow rate depends
on factors such as the size of the pipe and the fluid being used.
How do you calculate litres per minute?
To calculate liters per minute (LPM), you need to know the flow velocity (in meters per second) and the cross-sectional area of the pipe (in square meters). Multiply the velocity by the
cross-sectional area to get the flow rate in cubic meters per second, and then convert to LPM.
LPM = (Flow velocity in m/s) × (Cross-sectional area in m²) × 1000
How do I calculate PSI for a water pump?
To calculate PSI for a water pump, you need to know the flow rate (in gallons per minute, GPM) and the power of the pump (in horsepower, HP). You can use the formula:
PSI = (Flow rate in GPM × 2.31) / Horsepower
What is the flow rate for 60 PSI?
The flow rate for 60 PSI depends on various factors, including the size of the pipe and the type of fluid. You would need more information to determine the flow rate accurately.
How many PSI is equal to 1 GPM?
There is no direct conversion between PSI and GPM because they measure different aspects of a fluid system. PSI measures pressure, while GPM measures flow rate. The relationship between them depends
on the characteristics of the system, such as pipe size and fluid viscosity.
What is the PSI of water flow?
The PSI of water flow can vary widely depending on the specific application and the characteristics of the system. For example, household water pressure is typically around 40-60 PSI, but it can be
higher or lower depending on the location and local infrastructure.
How do you convert PSI to volume?
PSI cannot be directly converted to volume because PSI is a measure of pressure, not volume. To determine the volume, you would need additional information, such as the size of the container and the
type of fluid.
How many liters is 1900 PSI?
1900 PSI is a measure of pressure and does not directly correspond to a specific volume like liters. The volume of fluid (in liters) that corresponds to 1900 PSI depends on various factors, including
the size of the container and the type of fluid.
How much pressure should a liter have?
The pressure of a liter of fluid can vary widely depending on the specific application and the conditions. There is no fixed “pressure for a liter” because pressure is a measure of force per unit
area and can be influenced by various factors.
What is 4 liters per minute water pressure?
4 liters per minute is a measure of flow rate, not water pressure. Water pressure is typically measured in PSI (pounds per square inch) or Pascals (Pa).
How do you calculate flow rate?
Flow rate can be calculated using the formula:
Flow Rate (Q) = Area (A) × Velocity (V)
• Q is the flow rate (in liters per minute, GPM, or other units).
• A is the cross-sectional area of the pipe or channel (in square meters or square feet).
• V is the velocity of the fluid (in meters per second or feet per second).
How does pressure relate to flow rate?
Pressure and flow rate are related in fluid dynamics through various equations like the Bernoulli equation and the Darcy-Weisbach equation. Generally, an increase in pressure can result in an
increase in flow rate, but the relationship is influenced by factors such as pipe size, fluid viscosity, and system geometry.
What is 1 standard liter per minute?
One standard liter per minute (SLPM) is a unit of flow rate commonly used to measure the flow of gases. It is defined as one liter of gas flowing per minute under standard conditions (typically 0°C
and 1 atmosphere of pressure).
How do you calculate per liter?
To calculate a value “per liter,” you divide the quantity you’re interested in (e.g., price, concentration, flow rate) by the number of liters. For example, if you have 10 liters of a liquid and want
to find the cost per liter, you would divide the total cost by 10.
How do you calculate fluid per minute?
To calculate the flow rate of a fluid per minute, you need to measure the quantity of fluid that passes a specific point in a minute. Use appropriate units (e.g., liters per minute, gallons per
minute) based on your measurement.
How many PSI is a 1 HP water pump?
The PSI produced by a 1 HP (horsepower) water pump depends on the specific pump design, impeller size, and other factors. There is no fixed PSI value for a 1 HP pump, as it can vary widely. You would
need to consult the pump’s specifications or perform testing to determine the PSI output.
How much PSI is a 1hp pump?
The PSI produced by a 1 HP pump can vary widely depending on the pump’s design and application. It could range from a few PSI for a low-pressure application to several hundred PSI for high-pressure
systems. The specific PSI output would depend on the pump’s specifications.
How many PSI is 1 foot of water?
One foot of water column exerts approximately 0.4335 PSI (pounds per square inch) of pressure. This relationship is used to convert water column height into pressure units.
How many GPM is 80 PSI?
The flow rate (GPM) produced by 80 PSI of pressure depends on the specific system and its characteristics, such as pipe size and fluid viscosity. There is no direct conversion between PSI and GPM
without additional information.
Does PSI affect flow rate?
Yes, PSI (pressure) can affect flow rate in a fluid system. In general, an increase in pressure can result in an increase in flow rate, assuming all other factors remain constant. The relationship is
described by fluid dynamics equations.
How do you calculate flow rate in GPM?
To calculate flow rate in gallons per minute (GPM), you can use the formula:
Flow Rate (GPM) = (Area in square feet) × Velocity (feet per second)
How much PSI is 1500 GPM?
The pressure (PSI) generated by a flow rate of 1500 GPM depends on the specific system and its characteristics, such as pipe size, fluid viscosity, and pump performance. There is no fixed PSI value
for a given GPM without additional information.
How many GPM is 3000 PSI?
The flow rate (GPM) produced by 3000 PSI of pressure depends on the specific system and its characteristics, such as pipe size and fluid viscosity. There is no direct conversion between PSI and GPM
without additional information.
Will 1 PSI raise water?
Yes, an increase of 1 PSI in water pressure will cause the water level to rise. The relationship between pressure and height in a fluid column is typically described by the equation for hydrostatic
Pressure (PSI) = Density of Fluid × Gravitational Acceleration × Height (in feet)
So, an increase of 1 PSI would result in a small increase in the water’s height in a closed system.
How many GPM can a 1-inch pipe flow?
The flow rate of a 1-inch pipe depends on various factors, including the fluid being transported and the pressure. A rough estimate for water flowing through a 1-inch pipe at typical household
pressure (around 40-60 PSI) might be in the range of 20-50 GPM. However, the actual flow rate can vary based on many factors, and more precise calculations are needed for specific applications.
How do you convert pressure drop to GPM?
To convert pressure drop to GPM, you need to know the characteristics of the system, including the pipe size, length, and fluid properties. The relationship between pressure drop and flow rate is
typically determined through hydraulic calculations using the Darcy-Weisbach equation or other fluid dynamics principles.
How do you calculate flow in a pipe?
Flow in a pipe can be calculated using various fluid dynamics equations, such as the Darcy-Weisbach equation or the Hazen-Williams equation. These calculations take into account factors like pipe
diameter, length, fluid properties, and pressure.
How do you calculate water flow rate in a pipe?
To calculate water flow rate in a pipe, you can use the formula:
Flow Rate (GPM) = (Velocity in feet per second) × (Cross-sectional Area of the pipe in square feet)
What PSI is too high for water?
The ideal water pressure for a household or commercial building typically ranges between 40 PSI and 60 PSI. Pressure above 80 PSI is often considered too high and may require a pressure-reducing
valve (PRV) to prevent damage to plumbing fixtures and appliances.
What is PSI calculation?
PSI (Pounds per Square Inch) is a unit of pressure. The calculation for PSI depends on the context and the specific factors involved, but it is often determined using pressure measurement devices,
such as pressure gauges or transducers.
Can we convert pressure to volume?
Pressure and volume are related in fluid dynamics, but you cannot directly convert one to the other without additional information about the system’s characteristics, such as temperature, fluid
density, and container size.
How do you calculate PSI output?
To calculate the PSI output of a system, you need to measure the pressure using an appropriate pressure-measuring device, such as a pressure gauge. The PSI output is typically obtained directly from
the measurement.
How much is 100 PSI?
100 PSI (Pounds per Square Inch) is a unit of pressure and represents the force exerted by a fluid on a square inch of area. It is equivalent to approximately 6.895 kPa (kilopascals).
What is 1 PSI like?
1 PSI (Pound per Square Inch) is a relatively low pressure. It’s roughly equivalent to the pressure exerted by a column of water that’s about 2.31 feet (0.7 meters) in height. It’s used for various
applications, including measuring tire pressure, water pressure in household plumbing, and more.
Is 1500 PSI a lot?
1500 PSI (Pounds per Square Inch) is considered a high-pressure level in many applications. It is commonly used in industrial and commercial settings for tasks like high-pressure cleaning, hydraulic
systems, and some industrial machinery. However, whether it’s “a lot” or not depends on the specific context and application.
How many Litres per minute is good water pressure?
A good water pressure for residential use is typically in the range of 40 to 60 PSI, which can support a flow rate of around 5 to 10 gallons per minute (GPM) for most household fixtures and
appliances. However, the ideal flow rate can vary depending on individual needs and local plumbing standards.
Can pressure be measured in liters?
Pressure is typically measured in units such as Pounds per Square Inch (PSI), Pascals (Pa), or Bars (bar). It is not commonly measured in liters because liters are a unit of volume, whereas pressure
measures force per unit area.
Is 40 psi good water pressure?
40 PSI is generally considered acceptable water pressure for most residential applications. It provides adequate flow for household fixtures and appliances. However, specific needs may vary, and some
areas may have higher or lower water pressure due to local conditions.
Is 12 Litres per minute good water pressure?
12 Liters per minute (LPM) represents a relatively high flow rate for residential water use. It would typically provide good water pressure for most household fixtures and appliances, allowing for
efficient operation of showers, faucets, and more.
What is normal psi for residential water?
A normal PSI range for residential water supply is approximately 40 to 60 PSI. This range is suitable for most household plumbing systems and provides adequate water pressure for daily needs.
What is 50 psi water pressure?
50 PSI (Pounds per Square Inch) water pressure is considered a good pressure level for most residential applications. It provides sufficient flow and pressure for household fixtures and appliances to
function effectively.
What is the difference between water pressure and flow rate?
Water pressure is the force exerted by water on a unit area and is measured in units like PSI or Pascals. Flow rate, on the other hand, is the volume of water that passes through a pipe or fixture
per unit of time (e.g., GPM or LPM). Water pressure determines the speed at which water flows, while flow rate measures the quantity of water being delivered.
What is the flow rate of a 1-inch pipe?
The flow rate of a 1-inch pipe depends on various factors, including the type of fluid, the pressure, and the length and characteristics of the pipe. A 1-inch pipe can typically handle flow rates in
the range of 10 to 40 gallons per minute (GPM) for water.
What is the flow rate of 3/4-inch pipe?
The flow rate of a 3/4-inch pipe also depends on factors like fluid type and pressure. Generally, a 3/4-inch pipe can handle flow rates in the range of 6 to 30 gallons per minute (GPM) for water.
What is the equation for pressure to flow?
The relationship between pressure and flow in fluid dynamics is determined by various equations, such as the Darcy-Weisbach equation and the Bernoulli equation. These equations describe the complex
interplay between pressure, velocity, and other factors in fluid flow.
Do pumps create flow or pressure?
Pumps primarily create flow by increasing the pressure of a fluid, which then results in the flow of the fluid through a system. Pumps are used to overcome resistance and push fluids through pipes or
other conduits, which is why they are often associated with flow rate. However, they also indirectly affect pressure by increasing it to achieve the desired flow.
How do you calculate pipe size from flow rate and pressure?
Calculating the pipe size from flow rate and pressure involves using hydraulic calculations and fluid dynamics principles. It’s a complex process that depends on factors like the type of fluid,
desired flow rate, allowable pressure drop, and pipe material. Engineering software or tables specific to the type of pipe and fluid are typically used for such calculations.
What is liters per minute air flow?
Liters per minute (LPM) air flow is a measure of the rate at which air moves through a system or device. It is commonly used in applications like ventilation, HVAC (Heating, Ventilation, and Air
Conditioning), and medical equipment to quantify the volume of air being delivered or circulated.
What is normal liter flow?
Normal liter flow, often denoted as NLPM (Normal Liters Per Minute), refers to the flow rate of a gas (such as air or oxygen) at standard conditions, which are typically defined as 0 degrees Celsius
(32 degrees Fahrenheit) and 1 atmosphere of pressure (101.3 kPa). It is used in medical and industrial settings to specify the rate of gas delivery.
What is the unit of flow rate in litre?
The unit of flow rate in liters is typically denoted as LPM (Liters Per Minute) or LPS (Liters Per Second), depending on the context. It represents the volume of fluid or gas passing through a point
in a system per unit of time.
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Investigating Motion Using Video Processing Software | Blablawriting.com
Investigating Motion Using Video Processing Software
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• Pages: 5
• Word count: 1119
• Category: Software
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Part One Method:
We switched on the laptop and connected the web cam into the USB. Once this was done we made sure that the web cam was working correctly and as soon as this watched checked we began to set up the
practical part of the experiment. We then pressed record on the web cam using the software VISILOG and recorded the ball being thrown in the air next to a vertical ruler (one metre). Once this had
been done we stopped the recording and the replayed the video and once we were happy, using the software, we recorded the position of the ball frame by frame. Below are the results for the first part
of the experiment:
We can then draw a graph using the results and the first graph I have drawn (graph one) is of height of squash ball against time taken. Therefore, as we can see from the graph if a tangent is drawn
we can calculate the gradient. The gradient of both sides of the parabola graph is shown on graph one.
From the graph we can that as the squash ball is thrown the height of it increases as time increases and as it reaches its peak the ball speed is expected to decrease due to forces acting on
(gravity) and therefore the ball drops down again. The gradient of the graph also calculates the speed (which can be seen of the graph) of the ball because of the formula:
Distance = Speed * Time therefore Speed = Distance / Time
From these results a second graph of speed against time can also be made, which is graph two. Again from this graph we can draw tangents and then work out the gradient. Using the gradient we can also
work out the acceleration due to the formula:
Acceleration = change in speed / change in time Therefore acceleration = gradient.
Part Two Method:
We used the same equipment for the second stage of the experiment but instead of recording the ball thrown in the air we recorded it as a projectile. To measure the displacement we used metre rules
to measure horizontally as well as vertically. Below are the results from the experiment:
See next page!
From this table we can draw out a lot of information and from this I have produced a graph of horizontal displacement and vertical displacement (graph three). From the graph we can see that as
horizontal displacement increases so does vertical displacement until its peak of 0.34 m. once the squash ball reaches this height it begins to fall back down due forces acting on it. After the
vertical height reaches its peak it starts to decrease but the horizontal displacement continues to increase. From the graph we can also calculate the gradient, which I have done and this can be seen
on the graph.
Part Three Method:
We got a step ladder and to the top of it we attached a wooden beam to it using clamps. Then we got three springs and attached them together and once this was done we attached the springs to the
wooden beam using string. We then added a mass to the springs and recorded the oscillations it did using the camera. Below are the results for this:
See next page!
From the results (which is a rather long set of data) we can produce a graph and this graph is a sine wave. Within the wave, it can be seen that just about 4 oscillations have been made by the
spring. The time taken for 1 oscillation is called the period T. In this case the period T for 1 oscillation is about 1.31 seconds as shown on the graph. The number of oscillations per unit time is
the frequency, f = 1 / T.
Therefore, using the formula we can calculate the frequency of one oscillation, which is:
f = 1 / T
f = 1 / 1.31
f = 0.763358778
f = 0.76 Hz
Furthermore as the weights on the spring move about on a fixed point it means that the wave can be described as a single harmonic motion and the acceleration is proportional to its displacement (see
graph four for more details)
Therefore, in conclusion I have shown many things in this three part experiment. For each part, I have produced graphs and shown the results table that I have analysed as well. For the first part of
the experiment we can see that from the graphs the squash balls speed increases as it let go from the hand and then as it reaches its peak it begins to lose its speed and comes down, which is also
due to forces acting upon the graph.
Part two of the experiment is similar in respects and the difference is that it was a projectile instead. And the third part shows the spring oscillating from a ladder when a weight is attached to
it. For more on these look at the graphs I have produced and the analysis beneath each part of the experiment.
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Relative Humidity (RH) Calculator
Relative Humidity
Relative Humidity often abbreviated as RH, is the quantity which represents the amount of water vapor in the atmosphere or air. It is often represented in percentage. The ideal relative humidity
level is between 50% to 60%, however, the value becomes volatile based on the temperature. Human discomfort will arise when the relative humidity is getting very low. To find the amount of water
vapor (moisture) in the air, users can utilize the above formula for paperwork calculation or use this relative humidity calculator with the known input values of atmospheric temperature and dewpoint
of air. This calculator works well with input values in either degrees Fahrenheit or degrees Celsius scale. | {"url":"https://dev.ncalculators.com/meteorology/relative-humidity-calculator.htm","timestamp":"2024-11-12T18:39:14Z","content_type":"text/html","content_length":"32642","record_id":"<urn:uuid:4af2dfa2-e059-44ad-b716-0d5bfbc781cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00041.warc.gz"} |
What Is the Maturity Date on a Loan?
Photo Courtesy: Kiyoshi Hijiki/Getty Images
Whether you’re thinking of becoming a borrower or a lender, the maturity date on a loan is a key piece of information to know. Maturity dates come into play whether you’re attempting to pay off a
loan or cash in an investment like a government bond. We’ll give you a crash course in what a loan maturity date is and what you need to know about it.
What Is the Maturity Date on a Loan?
Photo Courtesy: thianchai sitthikongsak/Getty Images
To put it simply, a loan’s maturity date is the date when the loan must be paid in full. If you’re the borrower and have taken out a loan such as a mortgage, then your lender will most likely make
sure you stay well aware of the loan’s impending maturity date.
In the case of a mortgage, you’ll generally have two choices when the loan reaches maturity. You can either finish paying off the loan in full or attempt to refinance it with the lender. In the case
of secured loans, the maturity date is also when the lender will cease to have any authority over any assets the borrower may have provided as collateral.
If, on the other hand, you’re the lender then maturity dates tend to be a lot more fun. In this case, your loan’s maturity date means that the borrower has to repay you your principal, plus any
interest owed.
What Does Maturity Date Mean?
Photo Courtesy: Peter Gridley/Getty Images
When it comes to investing, a maturity date usually refers to the date when you’ll be able to reap the rewards of your investment. Generally, the two main types of investments you can make are either
equities or debt instruments. Equities refer to investing in something that you’ll own, such as stocks or real estate. Debt instruments refer to loans you give out in order to profit from interest.
Some common types of debt instruments that you can invest in include things like:
• Government loans such as treasury bonds, notes, and bills
• Savings accounts; while they may not seem like investments, they’re technically a loan to your bank. You earn interest in the form of an APY, even though it’s generally pretty low due to the fact
that you can take out the money any time.
• Certificates of Deposit (CDs)
• Corporate and municipal bonds
• Commercial Papers
How Do Loan Maturity Dates Work?
Photo Courtesy: RyanJLane/Getty Images
It depends on whether you are the borrower or the lender. If you’re the borrower, the maturity date is the final due date on the loan. The loan and any interest it’s incurred will ideally be paid off
in full unless you make arrangements to refinance. When the loan is paid off, the lender can no longer collect interest on it.
For this reason, you may be able to save yourself some money if you’re able to pay off a loan before the maturity date. Since the lender will no longer be able to collect interest from you, however,
you’ll want to check to make sure that they don’t impose early payment fees. If they do, you’ll want to compare them to the amount of money you’d save by dodging the remaining interest payments.
In the case of debt instruments, the maturity date is when you’ll get your investment plus any remaining interest back. Interest works differently depending on the type of debt instrument you’re
investing in. For example, treasury and municipal bonds pay interest twice a year for the duration of the loan. Savings bonds, on the other hand, payout both the principal and any interest acquired
over the life of the loan in one lump sum when they are cashed in.
How to Calculate Maturity Date
Photo Courtesy: warodom changyencham/Getty Images
Knowing the maturity date of a loan is also an important part of calculating the total amount the lender will ultimately receive when you factor in interest. This is called the maturity value and
it’s a helpful thing to know if you’re thinking of investing in a debt instrument. In order to go about these calculations, you’ll need to know several pieces of information:
• P= The original principal amount
• r= the interest rate per period on the loan
• n= the number of compounding intervals from the date the loan starts until it reaches its maturity date.
Once you have these numbers, you’ll be able to calculate V= the maturity value using the formula below.
Maturity Date Formula
Photo Courtesy: sod tatong/Getty Images
To calculate the maturity value, plug the numbers from above into the following formula:
V = P x (1 + r)^n
If you are using this formula to calculate the return you’ll get from investing in a debt instrument, it’s important to note that the maturity value will give you the return you’ll get overall.
Whether you’ll receive all of it on the maturity date will depend on the type of investment.
Some types of investments pay out interest twice every year, for example. In those instances, you’ll need to subtract the interest you’ll earn before the maturity date from the maturity value in
order to see how much you’ll actually receive in your final payment. In other words, when the maturity date arrives, you’ll usually only get one extra interest payment plus the initial principal on
the maturity date itself.
Loan Maturity Date Examples
Photo Courtesy: Milos Dimic/Getty Images
Let’s look at a quick example to give you an idea of how the formula works. Say that an investor named Bob invests $10,000 in a debt instrument that has a compounded interest rate of 8% per year. If
the loan’s maturity date is three years from the date of his investment, how much will he make from the loan?
In this example, Bob’s maturity value (V) would be calculated by using the following numbers:
P= $10,000
r= 8%
n= 3
So our formula would be:
V = 10,000 x (1 + 8%)^3
A bit of math reveals that upon Bob’s maturity date, his maturity value would be: $12,597.12. By subtracting his initial $10,000 principal, we can see that he’s earned $2,597.12
If you’re still a tad confused or if math just isn’t really your thing, rest assured that there are plenty of free maturity value calculators online that will handle the calculations for you. If your
investment is a government bond, then you can log into your account at treasurydirect.gov to track its value if it’s an electronic investment or get an update on your paper bond’s value on the tools
section of their website. | {"url":"https://www.askmoney.com/loans-mortgages/maturity-date-loan","timestamp":"2024-11-05T07:29:25Z","content_type":"text/html","content_length":"84215","record_id":"<urn:uuid:110e750a-0389-4b7a-a0e7-971ec03d384d>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00023.warc.gz"} |
Posting calculations
How much is it worth?
This is simple program for calculating historical money rates for Australia. It is intended to be a basic approach to calculating the relative value of money in Australia from 1850 to the present. It
is based on the Retail Price Index developed by the Australian Bureau of Statistics.
Obviously using the Retail Price Index is but one means of calculating relative values. More detailed discussion is available at these sites
Current value of money
Measuring worth
. The latter was the inspiration for this calculator. | {"url":"https://www.thomblake.com.au/secondary/hisdata/calculate.php","timestamp":"2024-11-08T14:28:32Z","content_type":"application/xhtml+xml","content_length":"8348","record_id":"<urn:uuid:0c6cbfdc-ac65-4bd4-9530-6930c97ea62c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00850.warc.gz"} |
Don't be (extra) afraid of math. It is just a language - GIAI
Don’t be (extra) afraid of math. It is just a language
Math in AI/Data Science is not really math, but a shortened version of English paragraph.
In science, researchers often ask 'plz speak in plain English', a presentation that math is just to explain science in more scientific way.
I liked math until high school, but it became an abomination during my college days. I had no choice but to make records of math courses on my transcript as it was one of the key factors for PhD
admission, but even after years of graduate study and research, I still don’t think I like math. I liked it when it was solving a riddle.
The questions in high school textbooks and exams are mostly about finding out who did what. But the very first math course in college forces you to prove a theorem, like 0+0=0. Wait, 0+0=0? Isn’t it
obvious? Why do you need a proof for this? I just didn’t eat any apple, so did my sister. So, nobody ate any apple. Why do you need lines of mathematical proof for this simple concept?
Then, while teaching AI/Data Science, I often claim that math equations in the textbook are just short version of long but plain English. I tell them “Don’t be afraid of math. It is just a language.”
Students are usually puzzled, and given a bunch of 0+0=0 like proof in the basic math textbooks for first year college courses, I get to grasp why my students showed no consent to the statement
(initially). So, let me illustrate my detailed back-up.
Math is just a language, but only in a certain context
Before I begin arguing math is a language, I would like to make a clear statement that math is not really a language as in academic defintion of language. The structure of math theorem and corollary,
for example, is not a replacement of paragraph with a leading statement and supporting examples. There might be some similarity, given that both are used to build logical thinking, but again, I am
not comparing math and language in 1-to-1 sense.
I still claim that math is a language, but in a certain context. My topic of study, along with many other closely related disciplines, usually create notes and papers with math jargons.
Mathematicians maybe baffled by me claiming that data science relies on math jargons, but almost all STEM majors have stacks of textbooks mostly covered with math equations. The difference between
math and non-math STEM majors is that the math equations in non-math textbooks have different meaning. For data science, if you find y=f(a,b,c), it means a, b, and c are the explanatory variables to
y by a non-linear regressional form of f. In math, I guess you just read it “y is a function of a, b, and c.”
My data science lecture notes usually are 10-15 pages for a 3-hour-long class. It might look too short for many of you, but in fact I need more time to cover the 15-pager notes. Why? For each page, I
condense many key concepts in a few math equations. Just like above statement “a, b, and c are the explanatory variables to y by a non-linear regressional form of f”, I read the equations in ‘plain
English’. In addition to that, I give lots of real life examples of the equation so that students can fully understand what it really means. Small variations of the equations also need hours to
Let me bring up one example. Adam, Bailey, and Charlie have worked together to do a group assignment, but it is unsure if they split the job equally. Say, you know exactly how the work was divided.
How can you shorten the long paragraph?
y=f(a,b,c) has all that is needed. Depending on how they divided the work, the function f is determined. If y is not a 0~100 scale grade but a 0/1 grade, then the function f has to reflect the
transformation. In machine learning (or any similar computational statistics), we require logistic/probit regressions.
In their assignment, I usually skip math equation and give a long story about Adam, Bailey, and Charlie. As an example, Charlie said he’s going to put together Adam’s and Bailey’s research at night,
because he’s got a date with his girlfriend in the afternoon. At 11pm, while Charlie was combining Adam’s and Bailey’s works, he found that Bailey almost did nothing. He had to do it by himself until
3am, and re-structured everything until 6am. We all know that Charlie did a lot more work than Bailey. Then, let’s build it in a formal fashion, like we scientists do. How much weight would you give
it to b and c, compared to a? How would you change the functional form, if Dana, Charlie’s girlfriend, helped his assignment at night? What if she takes the same class by another teacher and she has
already done the same assignment with her classmates?
If one knows all possibilities, y=f(a,b,c) is a simple and short replacement of above 4 paragraphes, or even more variations to come. This is why I call math is just a language. I am just a lazy guy
looking for the most efficient way of delivering my message, so I strictly prefer to type y=f(a,b,c) instead of 4 paragraphes.
Math is a univeral language, again only in a certain context
Teaching data science is fun, because it is like my high school math. Instead of constructing boring proof for seemingly an obvious theorem, I try to see hidden structures of data set and re-design
model according to the given problem. The diversion from real math is due to the fact that I use math as a tool, not as a mean. For mathematicians, my way of using math might be an insult, but I
often say to my students that we do not major math but data science.
Let’s think about medieval European countries when French, German, and Italian were first formed by the process of pidgin and creole. In case you are not familiar with two words, pidgin language is
to refer a language spoken by a children by parents without common tongue. Creole language is to refer a common language shared by those children. When parents do not share common tongue, children
often learn only part of the two languages and the family creates some sort of a new language for internal communication. This is called pidgin process. If it is shared by a town or a group of towns,
and become another language with its own grammar, then it is called creole process.
For data scientists, mathematics is not Latin, but French, German, or Italian, at best. The form is math (like Latin alphabet), but the way we use it is quite different from mathematicians. For major
European languages, for some parts, they are almost identical. For data science, computer science, natural science, and even economics, some math forms mean exactly the same. But the way scientists
use the math equations in their context is often different from others, just like French is a significant diversion from German (or vice versa).
Well-educated intellectuals in medieval Europe should be able to understand Latin, which must have helped him/her to travel across western Europe without much trouble in communication. At least basic
communication would have been possible. STEM students with heavy graduate course training should be able to understand math jargons, which help them to understand other majors’ research, at least
Latin was a universal language in medieval Europe, so as math to many science disciplines.
Math in AI/Data Science is just another language spoken only by data scientists
Having said all that, I hope you can now understand that my math is different from mathematician’s math. Their math is like Latin spoken by ancient Rome. My math is simply Latin alphabet to write
French, German, Italian, and/or English. I just borrowed the alphabet system for my own study.
When we have trouble understanding presentations with heavy math, we often ask the presentor, “Hey, can you please lay it out in plain English?”
The concepts in AI/Data Science can be, and should be able to be, written in plain English. But then 4 paragraphes may not be enough to replace y=f(a,b,c). If you need way more than 4 paragraphes,
then what’s the more efficient way to deliver your message? This is where you need to create your own language, like creole process. The same process occurs to many other STEM majors. For one, even
economics had decades of battle between sociology-based and math-based research methods. In 1980s, sociology line lost the battle, because it was not sharp enough to build the scientific logic. In
other words, math jargons were a superior means of communication to 4 paragraphes of plain English in scientific studies of economics. Now one can find sociology style economics only in a few British
universities. In other schools, those researchers can find teaching positions in history or sociology major. And, mainstream economists do not see them economists.
The field of AI/Data Science evolves in a similar fashion. For once, people thought software engineers are data scientists in that both jobs require computer programming. I guess now in these days
nobody would argue like that. Software engineers are just engineers with programming skills for websites, databases, and hardware monitoring systems. Data Scientists do create computer programs, but
it is not about websites or databases. It is about finding hidden patterns in data, building a mathematically robust model with explanatory variables, and predicting user behaviors by model-based
pattern analysis.
What’s still funny is that when I speak to another data scientists, I expect them to understand y=f(a,b,c), like “Hey, y is a function of a, b, and c”. I don’t want to lay it out with 4 paragraphes.
It’s not me alone that many data scientists are just as lazy as I am, and we want our counterparties to understand the shorter version. It may sound snobbish that we build a wall against non-math
speakers (depsite the fact that we also are not math majors), but I think this is an evident example that data scientists use math as a form of (creole) language. We just want the same language to be
spoken among us, just like Japanese speaking tourists looking for Japanese speaking guide. English speaking guides have little to no value to them.
Math in AI/Data Science can be, should be, and must be translated to ‘plain English’
A few years ago, I have created an MBA program for AI/Data Science that shares the same math-based courses with senior year BSc AI/Data Science, but does not require hard math/stat knoweldge. I only
ask them to borrow the concept from math heavy lecture notes and apply it to real life examples. It is because I wholeheartedly believe that the simple equation still can be translated to 4
paragraphes. Given that we still have to speak to each other in our own tongue, it should be and must be translated to plain language, if to be used in real life.
As an example, in the course, I teach cases of endogeneity, including measurement error, omitted variable bias, and simultaneity. For BSc students, I make them to derive mathematical forms of bias,
but for MBA students, I only ask them to follow the logic that what bias is expected for each endogenous case, and what are closely related life examples in business.
An MBA student tries to explain his company’s manufacture line’s random error that slows down automated process by measurement error. The error results in attenuation bias that under-estimates
mismeasured variable’s impact in scale. Had the product line manager knew the link between measurement error and attenuation bias, the loss of automation due to that error must have attracted a lot
more attention.
Like an above example, some MBA students in fact show way better performance than students in MSc in AI/Data Science, more heavily mathematical track. They think math track is superior, although many
of them cannot match math forms to actual AI/Data Science concepts. They fail not because they do not have pre-training in math, but because they just cannot read f(a,b,c) as work allocation model by
Adam, Bailey, and Charlie. They are simply too distracted to math forms.
During admission, there are a bunch of stubborn students with a die-hard claim that MSc or death, and absolutely no MBA. They see MBA a sort of blasphamy. But within a few weeks of study, they begin
to understand that hard math is not needed unless they want to write cutting edge scientific dissertations. Most students are looking for industry jobs, and the MBA with lots of data scientific
intuition was way more than enough.
The teaching medium, again, is ‘plain English’.
With the help of AI translator algorithms, I now can say that the teaching medium is ‘plain language’. | {"url":"https://giai.org/research/2024/04/dont-be-afraid-of-math-it-is-just-a-language/","timestamp":"2024-11-08T09:02:37Z","content_type":"text/html","content_length":"514062","record_id":"<urn:uuid:a8a122b7-4834-46c7-9913-91f154f223b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00877.warc.gz"} |
Output percentiles of multiple variables in a tabular format
A challenge for statistical programmers is getting data into the right form for analysis. For graphing or analyzing data, sometimes the "wide format" (each subject is represented by one row and many
variables) is required, but other times the "long format" (observations for each subject span multiple rows) is more useful. I've previously written about how to reshape data from wide to long format
It can be a challenge to get data in the right shape. For years I have wrestled with reformatting output data sets when computing descriptive statistics for multiple variables. One of the powers of
SAS software is that you can specify many variables in a procedure (such as MEANS or UNIVARIATE) and the procedure computes the requested statistics for each variable. However, to use that output in
a subsequent graph or analysis sometimes requires reshaping the output data set.
Since it is inconvenient to have to reshape data sets, this article describes three ways to produce descriptive statistics (in particular, percentiles) in a tabular form in which variables form
columns (or rows) and the statistics are arranged in the other dimension. This is usually the easiest shape to work with.
For standard percentiles, use PROC MEANS
As a canonical example, consider the the task of computing multiple percentiles for several variables when the underlying data are in a wide format. By default, both PROC MEANS and PROC UNIVARIATE
create the output data set in a less-than-optimal shape.
For commonly used percentiles (such as the 5th, 25th, 50th, 75th, and 95th percentiles), you can use PROC MEANS and the STACKODSOUTPUT option, which was introduced in SAS 9.3, to create an output
data set that contains percentiles for the multiple variables in a more convenient format. Compare the following two output data sets. The first output is usually harder to work with than the second:
/* default output data set. Yuck! */
proc means data=sashelp.cars noprint;
var mpg_city mpg_highway;
output out=MeansWidePctls P5= P25= P75= P95= / autoname;
proc print data=MeansWidePctls noobs; run;
/* Use the STACKODSOUTPUT option to get output in a more natural shape */
proc means data=sashelp.cars StackODSOutput P5 P25 P75 P95;
var mpg_city mpg_highway;
ods output summary=LongPctls;
proc print data=LongPctls noobs;run;
If I want to use the percentiles in a subsequent analysis, such as plotting the percentiles on a graph, I much prefer to work with the second set of output.
For nonstandard percentiles, use PROC STDIZE
For many years I have used PROC UNIVARIATE to compute custom percentiles. One application of custom percentiles is to compute a 95% bootstrap confidence interval, which requires computing the 2.5th
and 97.5th percentiles. However, when you are analyzing multiple variables PROC UNIVARIATE creates an output data set that is a single row, as shown in the following example:
/* default output data set. Yuck! */
proc univariate data=sashelp.cars noprint;
var mpg_city mpg_highway;
output out=UniWidePctls pctlpre=CityP_ HwyP_ pctlpts=2.5,15,65,97.5;
proc print data=UniWidePctls noobs; run;
There are many tricks, papers, and macros published that reshape or restructure the output from PROC UNIVARIATE. In my book, Simulating Data with SAS, I show one technique for reshaping the output.
(If you have a favorite method, link to it in the comment section.)
And so it was a surprise to find out that the easiest way to get customized percentiles in a tabular form is to not use PROC UNIVARIATE! It turns out that the STDIZE procedure also computes custom
percentiles, and the output is written in a tabular form. I knew that PROC STDIZE standardizes variables, but I didn't know that it computes custom percentiles.
The PROC STDIZE syntax is easy and very similar to the PROC UNIVARIATE syntax: use the PCTLPTS= option to specify the custom percentiles. Unlike PROC UNIVARIATE, you do not need PCTLPRE= option,
because the names of the variables are used for the output variables, and the percentiles are just additional rows that are added to the output data set. A typical example follows:
proc stdize data=sashelp.cars PctlMtd=ORD_STAT outstat=StdLongPctls
var mpg_city mpg_highway;
proc print data=StdLongPctls noobs;
where _type_ =: 'P';
I explicitly specified the PCTLMTD= option so that the algorithm uses the traditional "sort the data" algorithm for computing percentiles, rather than a newer one-pass algorithm. Although the
one-pass algorithm is very fast and well-suited for computing the median, it is not recommended for computing extreme percentiles such as the 2.5th and 97.5th.
Or use the SAS/IML language
Of course, no blog post from me is complete without showing how to compute the same quantity by using the SAS/IML language. The QNTL subroutine computes customized quantiles. (Recall that quantiles
and percentiles are essentially the same: The 0.05 quantile is the 5th percentile, the 0.25 quantile is the 25th percentile, and so forth.) By using the QNTL subroutine, the quantiles automatically
are packed into a matrix where each column corresponds to a variable and each row corresponds to a quantile, as follows:
proc iml;
varNames = {"mpg_city" "mpg_highway"};
use sashelp.cars;
read all var varNames into X;
close sashelp.cars;
Prob = {2.5, 15, 65, 97.5} / 100; /* prob in (0,1) */
call qntl(Pctls, X, Prob);
print Prob Pctls[c=varNames];
So there you have it. Three ways to compute percentiles (quantiles) so that the results are shaped in a tabular form. For standard percentiles, use PROC MEANS with the STACKODSOUTPUT option. For
arbitrary percentiles, use PROC STDIZE or PROC IML. If you use these techniques, the percentiles are arranged in a tabular form and you do not need to run any additional macro or DATA step to reshape
the output.
14 Comments
Many thanks - this PROC STDIZE tip is really simple/useful.
Is there a way to get 95% confidence intervals about the percentile estimates?
Yes, you can use the CIPCTLDF option on the PROC UNIVARIATE statement, which produces distribution-free confidence interals.
Suppose I want to calculate specific percentiles of multiple variables excluding 0 values or some other condition using where statement; how can I do that in single procedure; instead of
repeating same procedure over and over.
proc means data=sashelp.cars StackODSOutput P5 P25 P75 P95;
var mpg_city ;
where mpg_city >0 ;
ods output summary=LongPctls;
proc means data=sashelp.cars StackODSOutput P5 P25 P75 P95;
var mpg_hwy ;
where mpg_hwy >0 ;
ods output summary=LongPctls;
You can ask questions like this at the SAS Support Communities. Be sure to mention what SAS products you have available. The short answer is you can't do it with a WHERE clause because a
WHERE clause removes an entire observation whereas you are asking for a univariate filter for each variable. You can achieve your objective by a two-step process: replace all values <= 0 by a
missing value, then use the procedures in this blog post to compute the percentiles. However, in general it is not advisable to excluding part of your data, so the better approach is to
explain in the Community what you are trying to accomplish.
Very useful article!
How to find 95 percentile in grouping variables.
Like Branch wise 95 percentile.
is there a way to get Median in proc means with out using output statement in a dataset?
You can use the MEDIAN option on the PROC MEANS statement. If that doesn't answer your question, you can ask questions, post data, and have discussions on the SAS Support Communities.
Thanks for this. Is there a way to save the percentile output values as parameters that can be accessed by another PROC?
See https://blogs.sas.com/content/iml/2017/01/09/ods-output-any-statistic.html
What if I want to print the first 90 percentile of my data set?
1) Find the 90th percentile. Let's pretend the 90th percentile for your data is 57.3.
2) Use a WHERE clause in PROC PRINT to display only observations whose values are less than that value:
proc print data=Have;
where X <= 57.3; /* only values less than 90th pctl */
If you have additional questions, you can ask them on the SAS Support Community.
Thanks, @Rick for your help, '
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Caring For The Community
Posted on April 26, 2015 by annahaensch
As academic mathematicians, we spend a great deal of our days performing deeds of service to the mathematical community. Editing papers, organizing workshops, contributing to open-source software
initiatives. One could even argue that it is out of sheer benevolence to the mathematical community that we even write papers at all. In a blog I just discovered by mathematician and Clay Prize
winner, Michael Harris, the dynamics of this community get a very thorough and thoughtful analysis.
Mathematics without Apologies is, in the words of Michael Harris, “an unapologetic guided tour of the mathematical life.” Harris takes us on a tour of some of the very compelling — and highly fraught
— issues that arise when you start to think about the mathematical profession, and its community, from a sociological point of view. Michael Harris is a professor of mathematics at Université
Paris-Diderot and Columbia University.
In a multi-part post, Harris explains the much debated Elsevier boycott initiated by fellow mathematician and blogger, Tim Gowers. The boycott was proposed in 2012 in opposition to the extremely high
price tag the Elsevier puts on its research papers that are essentially donated to them. Harris addresses some of the alternative publishing models that people have proposed. One idea is that the
math community should be doing our own reviewing and publishing through Yelp-style real time user reviews. Harris is not so keen on this idea.
I don’t like the fact that unpaid Customer Reviews have undermined professions (as Tom Waits pointed out). I’m wary of replacing a practice that has evolved over several centuries to serve the
needs of the profession by a model of sociability that in less than a decade has led to the creation of massive fortunes and an enormous shift of power with practically no democratic oversight.
What I really appreciate about Harris’ blog is that it is very mindful of the community aspect of the mathematical profession. In one post he writes about Mathematics as a gift community, where we
are defined by the gifts that we bring to it. He reflects on the amount of time and effort that it takes to be a good community member, and that it should be in all of our best interest to improve
our community through deeds of service. I suppose this is true across all disciplines in academia, but it’s nice to think of things that way. It makes me happy to mow my mathematical lawn and take
out the trash to help my whole mathematical neighborhood look a bit more sparkly and clean.
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perplexus.info :: Probability : He stole my ball!
It seems natural to set up recursion relations, as is common in enumerative combinatoric problems of this kind. Then to work with generating function methods.
Maybe based on something like this:
• G1(N) = # of ways for the last player to get his ball, given that the second player takes his own colour.
• G2(N) = # of ways for the last player to get his ball, given that the second player can't take his own colour.
• B1(N) = # of ways for the last player to fail to get his ball, given that the second player takes his own colour.
• B2(N) = # of ways for the last player to fail to get his ball, if the second player doesn't have his own colour available to him.
where N = # of balls at the start.
Posted by FrankM on 2019-05-01 05:11:48 | {"url":"http://perplexus.info/show.php?pid=11703&cid=60982","timestamp":"2024-11-02T08:24:16Z","content_type":"text/html","content_length":"13111","record_id":"<urn:uuid:06b4b1e1-4b5e-4591-9733-203788c85887>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00308.warc.gz"} |
Hooke’s Law: Stress and Strain Revisited
131 Hooke’s Law: Stress and Strain Revisited
Learning Objectives
By the end of this section, you will be able to:
• Explain Newton’s third law of motion with respect to stress and deformation.
• Describe the restoration of force and displacement.
• Calculate the energy in Hook’s Law of deformation, and the stored energy in a string.
Figure 1. When displaced from its vertical equilibrium position, this plastic ruler oscillates back and forth because of the restoring force opposing displacement. When the ruler is on the left,
there is a force to the right, and vice versa.
Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate.
Consider, for example, plucking a plastic ruler to the left as shown in Figure 1. The deformation of the ruler creates a force in the opposite direction, known as a
restoring force
. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains
momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen
the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.
The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this
relationship between force and displacement was Hooke’s law: F = −kx.
Here, F is the restoring force, x is the displacement from equilibrium or deformation, and k is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring
force is in the direction opposite to the displacement.
Figure 2. (a) The plastic ruler has been released, and the restoring force is returning the ruler to its equilibrium position. (b) The net force is zero at the equilibrium position, but the ruler has
momentum and continues to move to the right. (c) The restoring force is in the opposite direction. It stops the ruler and moves it back toward equilibrium again. (d) Now the ruler has momentum to the
left. (e) In the absence of damping (caused by frictional forces), the ruler reaches its original position. From there, the motion will repeat itself.
The force constant k is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system. The units of k are newtons per
meter (N/m). For example, k is directly related to Young’s modulus when we stretch a string. Figure 3 shows a graph of the absolute value of the restoring force versus the displacement for a system
that can be described by Hooke’s law—a simple spring in this case. The slope of the graph equals the force constant k in newtons per meter. A common physics laboratory exercise is to measure
restoring forces created by springs, determine if they follow Hooke’s law, and calculate their force constants if they do.
Figure 3. (a) A graph of absolute value of the restoring force versus displacement is displayed. The fact that the graph is a straight line means that the system obeys Hooke’s law. The slope of the
graph is the force constant k. (b) The data in the graph were generated by measuring the displacement of a spring from equilibrium while supporting various weights. The restoring force equals the
weight supported, if the mass is stationary.
Example 1. How Stiff Are Car Springs?
Figure 4. The mass of a car increases due to the introduction of a passenger. This affects the displacement of the car on its suspension system. (credit: exfordy on Flickr)
What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0-kg person gets in?
Consider the car to be in its equilibrium position x=0 before the person gets in. The car then settles down 1.20 cm, which means it is displaced to a position x = −1.20 × 10^−2 m. At that point, the
springs supply a restoring force F equal to the person’s weight w = mg = (80.0 kg)(9.80 m/s^2) = 784 N. We take this force to be F in Hooke’s law. Knowing F and x, we can then solve the force
constant k.
Solve Hooke’s law, F = −kx, for k:
Substitute known values and solve k:
[latex]\begin{array}{lll}k&=&-\frac{784\text{ N}}{-1.20\times10^{-2}\text{ m}}\\\text{ }&=&6.53\times10^4\text{ N/m}\end{array}\\[/latex]
Note that F and x have opposite signs because they are in opposite directions—the restoring force is up, and the displacement is down. Also, note that the car would oscillate up and down when the
person got in if it were not for damping (due to frictional forces) provided by shock absorbers. Bouncing cars are a sure sign of bad shock absorbers.
Energy in Hooke’s Law of Deformation
In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is
deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a
spring is [latex]\text{PE}_{\text{el}}=\frac{1}{2}kx^2\\[/latex]. Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,
[latex]\text{PE}_{\text{el}}=\frac{1}{2}kx^2\\[/latex], where PE[el] is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement x from equilibrium and
a force constant k.
It is possible to find the work done in deforming a system in order to find the energy stored. This work is performed by an applied force F[app]. The applied force is exactly opposite to the
restoring force (action-reaction), and so F[app] = kx. Figure 5 shows a graph of the applied force versus deformation x for a system that can be described by Hooke’s law. Work done on the system is
force multiplied by distance, which equals the area under the curve or [latex]\frac{1}{2}kx^2\\[/latex] (Method A in Figure 5). Another way to determine the work is to note that the force increases
linearly from 0 to kx, so that the average force is [latex]\frac{1}{2}kx\\[/latex], the distance moved is x, and thus [latex]W=F_{\text{app}}d=\left(\frac{1}{2}kx\right)\left(x\right)=\frac{1}{2}kx^2
\\[/latex] (Method B in Figure 5).
Figure 5. A graph of applied force versus distance for the deformation of a system that can be described by Hooke’s law is displayed. The work done on the system equals the area under the graph or
the area of the triangle, which is half its base multiplied by its height, or [latex]W=\frac{1}{2}kx^2\\[/latex].
Example 2. Calculating Stored Energy: A Tranquilizer Gun Spring
We can use a toy gun’s spring mechanism to ask and answer two simple questions:
1. How much energy is stored in the spring of a tranquilizer gun that has a force constant of 50.0 N/m and is compressed 0.150 m?
2. If you neglect friction and the mass of the spring, at what speed will a 2.00-g projectile be ejected from the gun?
Figure 6. (a) In this image of the gun, the spring is uncompressed before being cocked. (b) The spring has been compressed a distance x, and the projectile is in place. (c) When released, the spring
converts elastic potential energy PE[el] into kinetic energy.
Strategy for Part 1
The energy stored in the spring can be found directly from elastic potential energy equation, because k and x are given.
Solution for Part 1
Entering the given values for k and x yields
[latex]\begin{array}{lll}\text{PE}_{\text{el}}&=&\frac{1}{2}kx^2=\frac{1}{2}\left(50.0\text{ N/m}\right)\left(0.150\text{ m}\right)^2=0.563\text{ N}\cdot\text{ m}\\\text{ }&=&0.563\text{ J}\end
Strategy for Part 2
Because there is no friction, the potential energy is converted entirely into kinetic energy. The expression for kinetic energy can be solved for the projectile’s speed.
Solution for Part 2
Identify known quantities:
KE[f] = PE[el] or [latex]\frac{1}{2}mv^2=\frac{1}{2}kx^2=\text{PE}_{\text{el}}=0.563\text{ J}\\[/latex]
Solve for v:
[latex]\displaystyle{v}=\left[\frac{2\text{PE}_{\text{el}}}{m}\right]^{1/2}=\left[\frac{2\left(0.563\text{ J}\right)}{0.002\text{ kg}}\right]^{1/2}=23.7\left(\text{J/kg}\right)^{1/2}\\[/latex]
Convert units: 23.7 m/s
Parts 1 and 2: This projectile speed is impressive for a tranquilizer gun (more than 80 km/h). The numbers in this problem seem reasonable. The force needed to compress the spring is small enough for
an adult to manage, and the energy imparted to the dart is small enough to limit the damage it might do. Yet, the speed of the dart is great enough for it to travel an acceptable distance.
Check your Understanding
Part 1
Envision holding the end of a ruler with one hand and deforming it with the other. When you let go, you can see the oscillations of the ruler. In what way could you modify this simple experiment to
increase the rigidity of the system?
You could hold the ruler at its midpoint so that the part of the ruler that oscillates is half as long as in the original experiment.
Part 2
If you apply a deforming force on an object and let it come to equilibrium, what happened to the work you did on the system?
It was stored in the object as potential energy.
Section Summary
• An oscillation is a back and forth motion of an object between two points of deformation.
• An oscillation may create a wave, which is a disturbance that propagates from where it was created.
• The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law: F = −kx,
where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
• Elastic potential energy PE[el] stored in the deformation of a system that can be described by Hooke’s law is given by [latex]{\text{PE}}_{\text{el}}=\frac{1}{2}kx^{2}\\[/latex].
Conceptual Questions
1. Describe a system in which elastic potential energy is stored.
Problems & Exercises
1. Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass). (a) What is the force constant of the spring in such a scale if it the
spring stretches 8.00 cm for a 10.0 kg load? (b) What is the mass of a fish that stretches the spring 5.50 cm? (c) How far apart are the half-kilogram marks on the scale?
2. It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg. (a)
What is the spring’s effective spring constant? (b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85 kg team?
3. One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg
plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?
4. (a) The springs of a pickup truck act like a single spring with a force constant of 1.30 × 10^5 N/m. By how much will the truck be depressed by its maximum load of 1000 kg? (b) If the pickup
truck has four identical springs, what is the force constant of each?
5. When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m. (a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?
6. A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it. (a) What is the force constant of the spring? (b) What is the
unloaded length of the spring?
deformation: displacement from equilibrium
elastic potential energy: potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring
force constant: a constant related to the rigidity of a system: the larger the force constant, the more rigid the system; the force constant is represented by k
restoring force: force acting in opposition to the force caused by a deformation
Selected Solutions to Problems & Exercises
1. (a) 1.23 × 10^3 N/m; (b) 6.88 kg; (c) 4.00 mm
3. (a) 889 N/m; (b) 133 N
5. (a) 6.53 × 10^3 N/m; (b) Yes | {"url":"https://library.achievingthedream.org/austinccphysics1/chapter/16-1-hookes-law-stress-and-strain-revisited/","timestamp":"2024-11-07T14:29:48Z","content_type":"text/html","content_length":"132693","record_id":"<urn:uuid:d529c3ff-182e-4a2f-aec6-8f082a8a41d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00741.warc.gz"} |
im trying to use the solve function for trigonometic equations.
To use the "solve" function on a TI-84 Plus calculator to solve a trigonometric equation and restrict the answer to the range of 0 to 2π, you can follow these steps:
1. Press the "MODE" button on the calculator.
2. Use the arrow keys to navigate to the "ANGLE" option in the second row of the mode menu.
3. Press the right arrow key to highlight the "DEGREE" option.
4. Press "ENTER" to select the "DEGREE" option if you want your answer in degrees. Alternatively, select the "RADIAN" option if you want your answer in radians.
5. Press the "Y=" button to enter the equation you want to solve.
6. Use the trigonometric functions on the calculator to enter the equation. For example, to enter the equation sin(x) = 1/2, you would type "sin(x) = 1/2".
7. Press the "2ND" button followed by the "CALC" button to access the "solve(" function.
8. Type the lower limit of the range you want to restrict your answer to, followed by a comma, then type the upper limit of the range followed by a comma, then type the variable name you used in
your equation. For example, if you want your answer to be in the range of 0 to 2π, and you used "x" as your variable, you would type "solve(sin(x) = 1/2,0,2π,x)".
9. Press "ENTER" to solve the equation and get the answer within the specified range.
The calculator will display the solution(s) within the specified range. If there is more than one solution, the calculator will display them all. | {"url":"https://www.calculatorti.com/calculator-help/5705/im-trying-use-the-solve-function-for-trigonometic-equations","timestamp":"2024-11-06T17:43:20Z","content_type":"text/html","content_length":"31619","record_id":"<urn:uuid:1f51d5af-3a65-4e23-9cfa-8484f0236360>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00781.warc.gz"} |
Profit Maximization: Marginal and Average Revenue
Here are some intro explanations and examples of microeconomic principles behind profit maximization, marginal, and average revenue. In our examples we will make some assumption in perfect
competition scenario.
Assumptions of perfect competition:
• Homogeneous product
• Perfect information
• many buyers and sellers
• No barriers to entry or exit
Perfect Competition Note
Given the above assumptions, no individual firm can influence the price of the product since their output represents an extremely small portion of the total output. Firms are said to be price-takers.
Revenues: Total Revenue (TR) = Price * Quantity
Marginal & Average Revenue
From this total revenue, we can derive:
1. Marginal Revenue: change in total revenue as a result of a 1-unit change in output. (Also the slope of the TR curve).
2. Average Revenue: Total revenue divided by total output. Can also be calculated by taking the slope of a ray from the origin and touching any point on the TR curve. But this ray is simply the TR
curve itself and we have calculated this slope as P*!
Costs of the Firm
What does the total cost curve look like? We need to address the issue of marginal returns over the possible output produced.
PROFITS : What will be thr profit-maximizing output?
Answer: it will lie (if you wanted to graph it you could), between Qo and Q1. Intuitively, if an additional unity adds more to revenues (MR) than it does to costs (MC), we should go ahead and produce
this unit.
Continue in this fashion until: MR = MC | {"url":"https://discusseconomics.com/blogs/dicusseconomics-blog/profit-maximization-marginal-and-average-revenue","timestamp":"2024-11-13T12:53:32Z","content_type":"text/html","content_length":"75132","record_id":"<urn:uuid:775aa2a2-ca99-4643-a7d1-847c924b6eee>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00745.warc.gz"} |
ACCA F9 September 2016 Exam Section C Question 31
Reader Interactions
1. Sir,
In part A) can we calculate the effective cost of refusing the discount and compare it with the cost of financing working capital? I calculated the effective cost of refusing the discount to be
6.2% which is greater than the cost of financing working capital and therefore we should accept the discount from the supplier. Please let me know if this approach can be used to solve this
question. Thanks
□ No, you should not use this approach because there is also the extra admin cost of 500 per year to be considered.
Have you watched my free lectures on the management of receivables and payables?
☆ Yes Sir I saw that and I have understood why this method cannot be used. Thank you Sir, your lecturers are very helpful.
☆ You are welcome, and thank you for the comment 馃檪
2. Hi John
On part A I assumed you would realise the new overdraft cost as 30/360 x 1,492,500 = 124,375 x 4% = 4975. Therefore the saving on overdraft would be old overdraft cost of 10,000 minus 4975 =
I know from the answer this is wrong but I am not sure why? After you take the early settlement discount the amount you are paying is now less.
Please explain where my thinking is wrong?
□ What you have done is fine and as I say in the free lecture the examiner always accepts either subtracting the discount or not subtracting it (even through the final answer is slightly
You must be logged in to post a comment. | {"url":"https://opentuition.com/acca/f9/acca-f9-september-2016-exam-section-c-question-31/","timestamp":"2024-11-08T01:28:37Z","content_type":"text/html","content_length":"80720","record_id":"<urn:uuid:f8b093ac-67b6-44cc-8538-57348a86b2cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00640.warc.gz"} |
Ring graphs and Goldberg's bound on chromatic index (Journal Article) | NSF PAGESAbstractAbstractAbstractAbstractAbstract
Let be a simple graph with maximum degree . A subgraph of is overfull if . Chetwynd and Hilton in 1986 conjectured that a graph with has chromatic index if and only if contains no overfull subgraph.
Let , be sufficiently large, and be graph on vertices with minimum degree at least . It was shown that the conjecture holds for if is even. In this paper, the same result is proved if is odd. As far
as we know, this is the first result on the Overfull Conjecture for graphs of odd order and with a minimum degree constraint.
more » « less | {"url":"https://par.nsf.gov/biblio/10110661-ring-graphs-goldberg-bound-chromatic-index","timestamp":"2024-11-02T01:19:53Z","content_type":"text/html","content_length":"244657","record_id":"<urn:uuid:2c44b323-15ae-4427-9fa0-f57d50fe07e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00008.warc.gz"} |
the time it takes to fly tfrom los angeles to new york varies
the time it takes to fly tfrom los angeles to new york varies inversely as the speed of the plane. if the trip takes 6 hours at 900 km/h how long would it take at 800 km/h
1 answer
(click or scroll down)
To solve this problem, we can use the inverse variation formula:
time = k/speed
where k is the constant of variation.
Given that the trip takes 6 hours at 900 km/h, we can plug in these values into the formula to find k:
6 = k/900
To solve for k, we can cross multiply:
k = 6 * 900
k = 5400
Now that we know k, we can use it to find the time it would take at 800 km/h:
time = 5400/800
time β 6.75 hours
Therefore, the trip would take approximately 6 hours and 45 minutes at 800 km/h. | {"url":"https://askanewquestion.com/questions/2428940","timestamp":"2024-11-02T11:24:08Z","content_type":"text/html","content_length":"15361","record_id":"<urn:uuid:340abe1c-de21-48d8-9ff0-6863f9db51a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00505.warc.gz"} |
Wall effects on Reiner-Rivlin liquid spheroid
An analysis is carried out to study the flow characteristics of creeping motion of an inner non-Newtonian Reiner-Rivlin liquid spheroid r = 1+ ∑[k=2]^∞α[k]G[k](cos θ), here α[k] is very small shape
factor and G[k] is Gegenbauer function of first kind of order k, at the instant it passes the centre of a rigid spherical container filled with a Newtonian fluid. The shape of the liquid spheroid is
assumed to depart a bit at its surface from the shape a sphere. The analytical expression for stream function solution for the flow in spherical container is obtained by using Stokes equation. While
for the flow inside the Reiner-Rivlin liquid spheroid, the expression for stream function is obtained by expressing it in a power series of S, characterizing the cross-viscosity of Reiner-Rivlin
fluid. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid and also the condition of impenetrability
and no-slip on the outer surface to the first order in the small parameter ε, characterizing the deformation of the liquid sphere. As an application, we consider an oblate liquid spheroid r = 1+2εG
[2](cos θ) and the drag and wall effects on the body are evaluated. Their variations with regard to separation parameter, viscosity ratio λ, cross-viscosity, i.e., S and deformation parameter are
studied and demonstrated graphically. Several well-noted cases of interest are derived from the present analysis. Attempts are made to compare between Newtonian and Reiner-Rivlin fluids which yield
that the cross-viscosity μ[c] is to decrease the wall effects K and to increase the drag D[N] when deformation is comparatively small. It is observed that drag not only varies with λ, but as η
increases, the rate of change in behavior of drag force increases also.
Reiner-Rivlin fluid; Gegenbauer function; stream functions; liquid spheroid; drag force; wall correction factor; spherical container | {"url":"https://acm.kme.zcu.cz/acm/article/view/268","timestamp":"2024-11-14T00:43:24Z","content_type":"text/html","content_length":"19057","record_id":"<urn:uuid:19d89dc6-8c0c-4807-8265-dd153500616a>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00783.warc.gz"} |
Practice exams | Achievable SIE
Achievable's SIE practice exams are a good estimate of your actual exam score. Our exams are generated from our exam question bank of over 2,600 question templates, with similar topic weightings and
phrasing that you can expect on the actual exam.
Achievable uses dynamic question templates to create multiple unique versions of a question. We intelligently randomize the question prompt, answer, distractors, and explanation. A single question
template can have hundreds of variations, ensuring that you thoroughly understand the concept and aren't just remembering an answer you've seen previously.
Ready for a practice exam? Check out Achievable's tips for success.
The actual exam will include an extra 10 experimental, unscored questions. | {"url":"https://app.achievable.me/study/finra-sie/exams","timestamp":"2024-11-06T17:49:24Z","content_type":"text/html","content_length":"577620","record_id":"<urn:uuid:8908798d-cc04-458a-9dbd-955af34d6e13>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00155.warc.gz"} |
Nonlinear parametric modelling and control for wave energy devices using numerical tank testing (NPM)
Project Description
Offshore floating wave energy converters (WECs) are a new and rapidly growing technology, driven by the need to reduce CO2 emissions and provide renewable energy sources. The design process of such a
device involves the use of computational tools and several stages of physical tank testing of scale models. Physical tank tests are generally expensive and can only provide insight into the physics
taking complex scaling effects into account. Full scale models cannot be used, as WECs are generally large in dimension and thereby too large for controlled tank tests. Different computational tools
exist, which are either linear or non-linear. Linear models, such as boundary element models, are used to describe the system of WEC-mooring-PTO-waves by means of a linear, predominantly
frequency-based, descriptions and are traditionally used in control design. The associated assumptions of inviscous fluid, irrotational flow, small waves and small body motion, however, are a major
limitation of this modelling approach, since WECs are designed to operate over wide wave amplitude ranges, experience large motions with waves breaking at the body and generating viscous drag and
vortex shedding.
With consideration of the full range of effects, the physics can be described using the Navier-Stokes equations. This branch of computational model falls into the category of computational fluid
dynamics (CFD). It is very computationally expensive and not suitable for use in the design process of WECs with regards to maximising the efficiency (through design iterations and development of
control algorithms) in real sea states, where long-time simulations are required.
This project aims to combine the strengths both types of modelling concepts. In particular, the mathematical description of the technical system including the main device, its auxiliaries such as the
mooring and power-take-off (PTO), and also the waves, that excite the device, is a necessity for the design of controllers but also when simulating the interaction of WECs in an array. Instead of
using linear coefficients in this model, we propose to generate non-linear parametric models using system identification techniques, based on WEC responses obtained from numerical tank tests
performed using CFD. These will take into account wave run-up in front of the structure, time varying degrees of submergence, which results in changes of buoyancy and restoring forces. It will also
model viscous effects, such as vortex shedding and drag, and turbulence.
The project is divided into three main sections. Within the first stage, an experimental setup will be developed, that can be used in a numerical wave tank in CFD. Such an experiment would need to
provide the data from which radiation effects due to the moving body, and diffraction effects due to waves exciting the WEC, can be identified. Possible solutions may include decay tests, where the
body is released from unstable positions and the response is measured. For the fixed body, i.e. to get diffraction results, waves of varying frequency may be generated and the surface elevations,
depending on the incoming wave, can be processed.
The next part of the project involves system identification. The CFD tests can be standardised and routines need to be developed that identify the necessary coefficients from the CFD results. The
Centre of Ocean Energy Research at NUI Maynooth has considerable experience, over more than 20 years, in the development and application of system identification methods for a wide range of
industrial applications. A crucial step in this stage involves the determination of appropriate model parameterisations which capture the essential nonlinear dynamics of families of WEC types. The
parametric description can then be used in the third stage of the project to design a true nonlinear control algorithm to optimise energy conversion. The benefit of such controllers is that they
respect the true nonlinear dynamics of WECs and are likely to give realistic energy maximisation in any irregular, non-linear sea.
Therefore, the project will provide an important step in extending the range of tractable computational techniques to the design and control of WECs, shortening the leap to expensive tank testing and
providing realistic mathematical models upon which to base energy-maximising control designs.
The final outcome of this project will be a modular software suite, which can be licensed to end users. Furthermore, a company will be set up, that can provide the proposed technology as a service.
The company can charge for the CFD model, system identification routine, development of the motion model and controller design separately. Also, as the CFD model is already available, a final
long-time full-scale simulation with the controller in place can be offered. This would be similar to the full scale device being deployed in controlled conditions.
Project Discussion No discussions posted at this time
You must be logged in to post a comment. | {"url":"https://coer.maynoothuniversity.ie/panorama/nonlinear-parametric-modelling-and-control-for-wave-energy-devices-using-numerical-tank-testing-npm/","timestamp":"2024-11-05T15:27:34Z","content_type":"text/html","content_length":"15356","record_id":"<urn:uuid:5c50b4cc-6875-462a-81b4-6f5bc2ba005c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00760.warc.gz"} |
Price of a Call Option Formula - Quant RL
Price of a Call Option Formula
Understanding the Basics of Call Options
In the world of options trading, call options are a popular investment vehicle that allows traders to capitalize on the potential upside of a security’s price movement. A call option gives the buyer
the right, but not the obligation, to purchase an underlying asset at a predetermined price, known as the strike price, before a specified expiration date. This flexibility makes call options an
attractive tool for investors seeking to manage risk, speculate on price movements, or generate income.
One of the primary benefits of call options is their ability to provide leverage, allowing traders to control a larger position with a smaller amount of capital. Additionally, call options can be
used to hedge against potential losses in a portfolio, providing a level of protection against market volatility. To fully understand the potential of call options, it’s essential to grasp the
underlying mechanics of their pricing, including the price of a call option formula, which will be explored in detail later in this guide.
How to Calculate the Price of a Call Option: A Step-by-Step Approach
The Black-Scholes model is a widely used mathematical formula for calculating the price of a call option. Developed by Fischer Black and Myron Scholes in 1973, this model provides a framework for
estimating the value of a call option based on several key variables. The formula for the Black-Scholes model is:
C = SN(d1) – Ke^(-rt)N(d2)
• C = the price of the call option
• S = the current stock price
• K = the strike price
• r = the risk-free interest rate
• t = the time to expiration
• N(d1) and N(d2) = the cumulative distribution functions of the standard normal distribution
• d1 and d2 = variables that can be calculated using the following formulas:
• d1 = ln(S/K) + (r + σ^2/2)t / σ√t
• d2 = d1 – σ√t
The Black-Scholes model takes into account the underlying stock price, strike price, risk-free interest rate, time to expiration, and volatility to estimate the price of a call option. By
understanding how to apply this formula, investors can gain a deeper appreciation for the factors that influence the price of a call option formula and make more informed investment decisions.
The Role of Volatility in Call Option Pricing
Volatility is a critical component in the calculation of the price of a call option formula. It represents the extent to which the underlying asset’s price fluctuates over a given period. In the
context of options trading, volatility is often referred to as the “fear factor” because it reflects the market’s uncertainty about the future direction of the underlying asset’s price.
A higher volatility means that the underlying asset’s price is more likely to fluctuate rapidly, making it more difficult to predict its future value. As a result, the price of a call option will
increase with higher volatility, as the option buyer is willing to pay a premium for the potential upside. Conversely, a lower volatility will result in a lower call option price, as the option buyer
is less likely to benefit from significant price movements.
The impact of volatility on call option prices can be seen in the Black-Scholes model, where it is represented by the symbol σ (sigma). An increase in σ will result in a higher call option price,
while a decrease in σ will result in a lower call option price. This is because the model takes into account the probability of the underlying asset’s price reaching the strike price at expiration,
which is affected by the level of volatility.
Understanding the role of volatility in call option pricing is essential for investors, as it can help them make more informed investment decisions. By recognizing the impact of volatility on the
price of a call option formula, investors can adjust their strategies to take advantage of market conditions and maximize their returns.
The Effect of Time Decay on Call Option Prices
Time decay is a critical concept in options trading that refers to the gradual decline in the value of an option over time. As the expiration date approaches, the option’s value decreases, making it
less valuable to the buyer. This phenomenon is also known as theta, which represents the rate of decline in the option’s value due to the passage of time.
The effect of time decay on call option prices is significant, as it can erode the value of the option even if the underlying asset’s price remains unchanged. This is because the option buyer is
paying a premium for the right to buy the underlying asset at a specified price, and as the expiration date approaches, the likelihood of the option expiring in the money decreases.
The price of a call option formula takes into account the time decay factor, which is reflected in the Black-Scholes model. The model assumes that the option’s value will decrease over time, and this
decrease is accelerated as the expiration date approaches. As a result, the price of a call option will decrease as the time to expiration decreases, all other factors being equal.
Understanding the effect of time decay on call option prices is essential for investors, as it can help them make more informed investment decisions. By recognizing the impact of time decay on the
price of a call option formula, investors can adjust their strategies to take advantage of market conditions and maximize their returns. For example, investors may choose to sell call options with a
shorter time to expiration to take advantage of the time decay factor, or they may opt for longer-term options to give themselves more time to benefit from potential price movements.
Interest Rates and Call Option Pricing: What’s the Connection?
Interest rates play a crucial role in the pricing of call options, as they affect the underlying asset’s value and the option’s premium. The relationship between interest rates and call option prices
is complex, but understanding it is essential for investors to make informed investment decisions.
In the Black-Scholes model, the risk-free interest rate is one of the key variables used to calculate the price of a call option formula. The model assumes that the option buyer can earn a risk-free
return by investing in a risk-free asset, such as a U.S. Treasury bond. As a result, the option buyer will demand a higher premium for the call option if interest rates are high, as they can earn a
higher return from a risk-free investment.
The impact of interest rates on call option prices can be seen in the following ways: an increase in interest rates will result in a higher call option price, while a decrease in interest rates will
result in a lower call option price. This is because higher interest rates increase the opportunity cost of holding the underlying asset, making the call option more valuable. Conversely, lower
interest rates decrease the opportunity cost, making the call option less valuable.
In addition to the risk-free interest rate, the yield curve also plays a role in call option pricing. The yield curve represents the relationship between interest rates and time to maturity, and it
can affect the price of a call option formula. For example, a steepening yield curve can increase the price of a call option, as it indicates higher interest rates in the future, which can increase
the underlying asset’s value.
Understanding the connection between interest rates and call option prices is essential for investors, as it can help them make more informed investment decisions. By recognizing the impact of
interest rates on the price of a call option formula, investors can adjust their strategies to take advantage of market conditions and maximize their returns.
Real-World Examples of Call Option Pricing in Action
Understanding the price of a call option formula is essential for investors, but seeing it in action can help solidify the concept. In this section, we’ll explore real-world examples of call option
pricing in action, including case studies and hypothetical examples.
Example 1: Tech Stock Call Option
Let’s say a tech company, XYZ Inc., is trading at $50 per share. An investor buys a call option to purchase 100 shares of XYZ Inc. at $55 per share, with an expiration date in three months. The price
of the call option formula is calculated using the Black-Scholes model, taking into account the underlying asset’s price, volatility, time to expiration, and interest rates. The calculated price of
the call option is $2.50 per share.
In this scenario, the investor is betting that the price of XYZ Inc. will rise above $55 per share within the next three months. If the price does rise, the investor can exercise the call option and
buy the shares at $55, selling them at the higher market price to profit. If the price doesn’t rise, the option will expire worthless, and the investor will lose the premium paid for the option.
Example 2: Index Call Option
Another example is an index call option on the S&P 500 index. An investor buys a call option to purchase the S&P 500 index at 3,500, with an expiration date in six months. The price of the call
option formula is calculated using the Black-Scholes model, taking into account the index’s current value, volatility, time to expiration, and interest rates. The calculated price of the call option
is $150 per contract.
In this scenario, the investor is betting that the S&P 500 index will rise above 3,500 within the next six months. If the index does rise, the investor can exercise the call option and buy the index
at 3,500, selling it at the higher market price to profit. If the index doesn’t rise, the option will expire worthless, and the investor will lose the premium paid for the option.
These examples demonstrate how the price of a call option formula is used in real-world trading scenarios. By understanding how to calculate the price of a call option, investors can make more
informed investment decisions and maximize their returns.
Common Mistakes to Avoid When Calculating Call Option Prices
Calculating the price of a call option formula can be complex, and even experienced investors can make mistakes. In this section, we’ll highlight common errors or misconceptions investors make when
calculating call option prices, including tips on how to avoid these mistakes.
Mistake 1: Ignoring Volatility
One common mistake investors make is ignoring the impact of volatility on call option prices. Volatility can significantly affect the price of a call option, and failing to account for it can lead to
inaccurate calculations. To avoid this mistake, investors should always consider the underlying asset’s volatility when calculating the price of a call option formula.
Mistake 2: Misusing the Black-Scholes Model
The Black-Scholes model is a widely used formula for calculating the price of a call option, but it’s not a one-size-fits-all solution. Investors should understand the assumptions and limitations of
the model and adjust it according to their specific needs. For example, the model assumes a constant volatility, which may not always be the case in real-world markets.
Mistake 3: Failing to Account for Time Decay
Time decay is a critical factor in call option pricing, as it affects the option’s value over time. Investors should always consider the time to expiration when calculating the price of a call option
formula, as it can significantly impact the option’s value.
Mistake 4: Overlooking Interest Rates
Interest rates can also impact the price of a call option, and investors should consider them when calculating the price of a call option formula. Failing to account for interest rates can lead to
inaccurate calculations and poor investment decisions.
Tips for Avoiding Mistakes
To avoid these common mistakes, investors should:
• Always consider the underlying asset’s volatility, time to expiration, and interest rates when calculating the price of a call option formula.
• Understand the assumptions and limitations of the Black-Scholes model and adjust it according to their specific needs.
• Use real-world examples and case studies to test their calculations and refine their strategies.
• Stay up-to-date with market trends and news to ensure their calculations are accurate and relevant.
By avoiding these common mistakes, investors can ensure accurate calculations and make informed investment decisions when using call options in their portfolios.
Advanced Call Option Pricing Strategies for Experienced Traders
For experienced traders, mastering the price of a call option formula is just the beginning. In this section, we’ll explore advanced strategies for using call options in combination with other
derivatives or investment vehicles to maximize returns and minimize risk.
Strategy 1: Call Option Spreads
A call option spread involves buying and selling call options with different strike prices or expiration dates. This strategy can help traders profit from volatility or time decay, while limiting
their exposure to potential losses. For example, a trader might buy a call option with a higher strike price and sell a call option with a lower strike price, profiting from the difference in
Strategy 2: Call Option Collars
A call option collar involves buying a call option and selling a put option with the same underlying asset and strike price. This strategy can help traders hedge against potential losses while still
benefiting from upside potential. For example, a trader might buy a call option on a stock and sell a put option on the same stock, limiting their potential losses while still profiting from a
potential price increase.
Strategy 3: Call Option Ladders
A call option ladder involves buying and selling call options with different strike prices, creating a “ladder” of potential profits. This strategy can help traders profit from volatility or time
decay, while limiting their exposure to potential losses. For example, a trader might buy a call option with a lower strike price and sell a call option with a higher strike price, profiting from the
difference in premiums.
Strategy 4: Call Option Ratio Backspreads
A call option ratio backspread involves buying and selling call options with different ratios, creating a “backspread” of potential profits. This strategy can help traders profit from volatility or
time decay, while limiting their exposure to potential losses. For example, a trader might buy two call options with a lower strike price and sell one call option with a higher strike price,
profiting from the difference in premiums.
By incorporating these advanced strategies into their trading repertoire, experienced traders can take their call option pricing skills to the next level, maximizing their returns and minimizing
their risk. | {"url":"https://quantrl.com/price-of-a-call-option-formula/","timestamp":"2024-11-04T21:55:23Z","content_type":"text/html","content_length":"75142","record_id":"<urn:uuid:e5be6a4b-864d-4756-b456-d94c382d2cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00445.warc.gz"} |
Display Boards
Design an arrangement of display boards in the school hall which fits the requirements of different people.
The challenge: arranging the display boards in the hall
A Year 5 class wants to display the results of their problem solving in the school hall.
They need 32 display boards - one each - and they are wondering how to arrange them.
Rules for setting up the boards:
• The boards can be joined in a straight line or at right angles.
• The boards will fall if there are more than four in a straight line.
• The final arrangement of the 32 display boards has to be a closed shape so that people can walk around the outside and view all the problem solving results displayed.
• The hall floor has square tiles. Each board, which is as long as the square tiles, needs to be placed on the edge of one of these squares so that it stands up.
See if you can work out how to arrange the boards to satisfy each of the following people.
This means you will need to find three different arrangements.
A. The kitchen staff would like to use the hall for school dinners. They would like the display to be as long and narrow as possible, and at one end of the hall.
B. Teachers would like to use the hall for PE. They would like the display to fit in a corner so that it leaves as much space as possible for PE.
Are the teachers right that the corner design takes up less space than the one at the end of the hall? Why or why not?
C. The Year 5 teacher thinks that the best viewing shape is the one that has as many long straight lines of four display boards in it as possible and they don't mind where it is in the room.
Further challenge
The headteacher would like the display to be very visible from whatever door people enter the hall. They would like the display to be as square as possible and in the middle of the hall.
A. Design a display for the headteacher that is as square as possible.
Explain how you have decided it is as square as possible.
B. Design a display for the headteacher that is as square as possible and has four lines of symmetry.
C. Design a display for the headteacher that is as square as possible, has four lines of symmetry and has an internal area of 40 square tiles.
How many arrangements can you find that fit these requirements?
This problem featured in a preliminary round of the Young Mathematicians' Award 2014.
Student Solutions
We haven't received many solutions to this task yet. The children below have tried to find the narrowest solution possible. If you think you've found a narrower one, please email us to let us know!
Some children at Beckwithshaw Primary School sent in their ideas. Ollie, Megan and Ryan said:
How we did it was we got 32 lolly sticks and placed them on 2 big sheets of A3 squared paper. After that, we looked for possible solutions keeping in mind the rules. We drew them in our books and
came to the conclusion of 1 possibility. Our answer showed all the rules and was a closed space. It was hard to keep it as narrow as possible but with trial and error we eventually solved it.
Jack and David sent in this solution for the narrowest display board:
Thank you all for sharing your ideas with us. Interestingly, Ollie, Megan and Ryan's last shape has a smaller area than Jack and David's shape, but Jack and David's shape is 'narrower' in the sense
that it is only four display boards wide at its widest point. I wonder which would be more helpful to have at the end of the hall?
Do you think it is possible to find an arrangement that is only three boards wide at its widest point? Or two?
Teachers' Resources
Why do this problem?
This problem puts the pupils in a spatial problem-solving situation. The amount of necessary knowledge is very little but perseverance in achieving the "best" arrangements will challenge many pupils.
Possible approach
Using 32 sticks/rods/matchsticks/straws have the children gathered around for a short discussion. It is good if you can use a squared background (flooring/large sheets of fairly big squared paper).
If you are presenting this to a small group of pupils then 2cm squared paper can be used together with cut straws. It would be good to be able to represent a much larger space than the arrangement
can cover so as to represent the whole of a school hall. The arrangement can be used to discuss the rules that are to be applied, particularly the "closed shape" and the sticking to the sides of
Following this start the pupils may be given access to many different pieces of equipment to help, particularly squared paper for recording.
Key questions
Tell me about what you're doing.
How did you decide that this is the best for ...?
Possible support
Some pupils may need help with moving the "display boards" around and keeping them in the position they've chosen. Some may need help with recording their ideas. | {"url":"https://nrich.maths.org/problems/display-boards","timestamp":"2024-11-11T16:47:29Z","content_type":"text/html","content_length":"47513","record_id":"<urn:uuid:efcce69c-0e26-4036-af3c-dee0e0b5b460>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00585.warc.gz"} |
Accounting Question - Custom Scholars
Accounting Question
11.2. For a particular SKU, the lead time is 6 weeks, the average demand is 90 units a week, and safetystock is 200 units. What is the average inventory if 10 weeks’ supply is ordered at one time?
What is the
order point?
11.4. Given the following data, calculate the average demand and the standard deviation.
11.6. The standard deviation of demand during the lead time is 100 units.
A. Calculate the safety stock required for the following service levels: 75%, 80%, 85%, 90%, 95%,
and 99.99%.
B. Calculate the change in safety stock required to increase the service levels from 75 to 80%, 80 to
85%, 85 to 90%, 90 to 95%, and 95 to 99.99%. What conclusion do you reach?
11.8. A company stocks an SKU with a weekly demand of 225 units and a lead time of 3 weeks.
Management will tolerate one stockout per year. If sigma for the lead time is 175 and the order quantity
is 800 units, what is the safety stock, the average inventory, and the order point?
11.19. A company that manufactures snow shovels has one plant and two distribution centers (DCs).
Given the following information for the two DCs, calculate the gross requirements, projected available,
and planned order releases for the two DCs and the gross requirements, projected available, and
planned order releases for the central warehouse.
11.21. A regional warehouse orders items once a week from a central warehouse. The truck arrives 3
days after the order is placed. The warehouse operates 5 days a week. For a particular brand and size of
chicken soup, the demand is fairly steady at 20 cases per day. Safety stock is set at 2 days’ supply.
A. What is the target level?
B. If the quantity on hand is 90 cases, how many should be ordered?
12.2. A company has 6500 cartons to store on pallets. Each pallet takes 30 cartons, and the cartons are
stored four high. How many pallet positions are needed?
12.4. A company has a warehouse with the dimensions shown in the following diagram. How many
pallets measuring 48″ × 40″ can be stored three high if there is to be a 2″ space between the pallets?
12.6. A company wants to store the following seven SKUs so there is 100% accessibility. Items are stored
on pallets that are stored four high.
A. How many pallet positions are needed?
B. What is the cube utilization?
C. If the company bought racking for storing the pallets, how many pallet positions are needed to
give 100% accessibility?
A. Which of the following items are within tolerance?
B. What is the percent accuracy by item?
12.10. A company does an ABC analysis of its inventory and calculates that out of 10,000 items, 19% can
be classified as A items, 30% as B items, and the remainder as C items. A decision is made that A items
are to be cycle counted twice a month, B items every 3 months, and C items once a year. Calculate the
total counts and the counts per day by classification. There are 250 working days per year. | {"url":"https://customscholars.com/accounting-question-190/","timestamp":"2024-11-06T17:31:42Z","content_type":"text/html","content_length":"53204","record_id":"<urn:uuid:6747b4e0-f212-4ccc-8aeb-a5b8076c15e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00376.warc.gz"} |
(i) Inductive reactance of the coil to neutralize capacitance
of 100% of the length of the line is 884.19 ohm
(ii) Inductive reactance of the coil to neutralize capacitance
of 100% of the length of 90% of the line is 982.44 ohm
(iii)Inductive reactance of the coil to neutralize capacitance
of 100% of the length of 80% of the line is 1105.24 ohm | {"url":"https://tbc-python.fossee.in/convert-notebook/Principles_of_Power_System/chapter26.ipynb","timestamp":"2024-11-05T15:10:50Z","content_type":"text/html","content_length":"213250","record_id":"<urn:uuid:7a327ebe-8479-4c9d-b8ca-619c5c958c11>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00750.warc.gz"} |
Tutorial of Scatter Plot in Base R Language - MLK - Machine Learning Knowledge
In this article, we will understand how to create Scatter Plot in Base R language. R is very popular for providing rich visualization options out of the box. Firs, we will understand the syntax of
plot() and then see how to use it for creating scatterplots.
Syntax of Scatter Plot plot() function in R
The basic syntax for scatterplot plot() function is shown below with commonly used parameters. The detailed syntax can be found here.
plot(x, y, main, xlab, ylab, axes)
• x – Denotes the data to be used for the x-axis of the plot
• y – Denotes the data to be used for the x-axis of the plot
• main – Assigns a title to the scatter plot
• xlab – Assigns label to the x-axis of the scatter plot
• ylab – Assigns label to the y-axis of the scatter plot
• axes – Denotes if both axes should be drawn on the plot.
Examples of Scatter Plot in R Language
The Dataset
We are going to use the famous Iris dataset for all our examples of scatter plots in R. This dataset has 5 features – Sepal Length, Sepal Width, Petal Length, Petal Width, and Species. There are 3
distinct Species, that go by the name Iris-Setosa, Iris-Versicolor, and Iris-Virginica.
Sepal.Length Sepal.Width Petal.Length Petal.Width Species
5.1 3.5 1.4 0.2 setosa
4.9 3.0 1.4 0.2 setosa
4.7 3.2 1.3 0.2 setosa
4.6 3.1 1.5 0.2 setosa
5.0 3.6 1.4 0.2 setosa
5.4 3.9 1.7 0.4 setosa
We assign the features Sepal Length, Petal length, and Species into x, y, and z variables respectively to use these features conveniently in the below examples.
In [1]:
x <- iris$Sepal.Length
y <- iris$Petal.Length
z <- iris$Species
Example 1: Basic Scatterplot in Base R
Let us start with a very basic scatter plot in R where we just pass the x and y parameters to the plot() function without any other options.
In [2]:
plot(x, y)
Example 2: Basic Scatterplot without Frame
We can disable the frame in the above example by using frame = False as shown below.
In [3]:
Example 3 : Scatterplot with Label and Titles
Now to include some more detail to our plot, we shall add the main title and corresponding labels. This can be done using the main, xlab, and ylab arguments.
In [4]:
plot(x, y, main = "Sepal length v/s Petal Length",
xlab = "Sepal length", ylab = "Petal Length", frame=FALSE)
Example 4: Changing Shapes & Size of Scatter Plot Markers in R
To plot our scatterplot with markers of different shapes, we can use the pch argument and pass our desired shape number. The size of the markers can be controlled with cex parameter
The below two examples show how two different types of markers.
In [5]:
plot(x, y, main = "Sepal length v/s Petal Length",
xlab = "Sepal length", ylab = "Petal Length",
pch = 17, cex = 1.5, frame=FALSE)
In [6]:
plot(x, y, main = "Sepal length v/s Petal Length",
xlab = "Sepal length", ylab = "Petal Length",
pch = 3, cex = 1.5 , frame=FALSE)
Example 5: Adding Legend to Scatter Plot
To add a legend to the scatter plot, we use a separate legend function. In the legend function, we can define the position, color, and size of the legends to be used for the scatter plot.
In this example, we are grouping the data in Group1 and Group2 based on the condition if the value of x is less than 6 or more. Using this, the legend is assigned.
In [7]:
group <- as.factor(ifelse(x < 6, "Group 1", "Group 2"))
plot(x, y,xlab = "Sepal length", ylab = "Petal Length", pch = 16, col =group, cex=1.5)
legend = c("Group 1", "Group 2"),
pch = 16)
Example 6: Scatter plot with regression line
We add a regression line to a scatter plot passing a lm object to the abline function. Alternatively, we can use locally-weighted polynomial regression. This non-parametric regression estimation can
be done with a lowess function.
In [8]:
plot(x, y, main = "Sepal length v/s Petal Length",
xlab = "Sepal length", ylab = "Petal Length",
pch = 19,col='brown')
abline(lm(y ~ x, data = mtcars), col = "blue", lwd=3)
In [9]:
plot(x, y,main = "Sepal length v/s Petal Length",
xlab = "Sepal length", ylab = "Petal Length",
pch = 19, col='blue',cex=1.2)
lines(lowess(x, y), col = "red", lwd=3)
Example 7: Scatter Plot with Groups
Using a grouping variable you can make a scatter plot by group passing the variable, as a factor, into the col argument. As a result, every group will be displayed with a distinct color.
In [10]:
group <- ifelse(x < 5.5, "small sepal length",
ifelse(x >7, "large speal length",
"medium sepal length"))
# Scatter plot
plot(x, y,
pch = 19,
col = factor(group),cex=1.5)
# Legend
legend = levels(factor(group)),
pch = 19,
col = factor(levels(factor(group))))
Example 8: Scatter plot matrix
When dealing with multiple variables it is useful to plot multiple scatter plots within a matrix, that will plot each variable against another to visualize the correlation between variables. You can
create a scatter plot in R with multiple variables, known as pairwise scatter plots or scatterplot matrix, with the pairs function.
In [11]:
pairs(~Sepal.Length + Sepal.Width + Petal.Length + Petal.Width,col = factor(iris$Species), data =iris)
In addition, in case your dataset contains a factor variable, you can specify the variable in the col argument as follows to plot the groups with a different color.
In [12]:
pairs(~Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, col = factor(iris$Species), pch = 19, data =iris)
Example 9: Calculate Correlation in Scatter plot
To display the relationship between two quantitative labels we use the correlation metric. Here, to find out the correlation between x and y variables we can use the corr() method in base R.
In [13]:
# Calculate correlation
Corr <- cor(x, y)
# Create the plot and add the calculated value
plot(x, y, pch =17,col='orange', cex=2)
text(paste("Correlation:", round(Corr, 2)), x =7, y =2, cex=1.8, col='red')
Example 10: Smooth scatterplot with smoothScatter function
The smoothScatter function is a base R function that creates a smooth color kernel density estimation of an R scatterplot.
In [14]:
smoothScatter(x, y, pch = 19,
transformation = function(x) x ^ 0.5, # Scale
colramp = colorRampPalette(c("#f7f7f7", "aquamarine"))) # Colors
Example 11: Heat map R scatter plot
With the smoothScatter function, you can also create a heat map. For that purpose, you will need to specify a color palette as follows:
In [15]:
smoothScatter(x, y, transformation = function(x) x ^0.9,
colramp = colorRampPalette(c("#000099", "#00FEFF", "#45FE4F",
"#FCFF00", "#FF9400", "#FF3100"))) | {"url":"https://machinelearningknowledge.ai/tutorial-of-scatter-plot-in-base-r-language/","timestamp":"2024-11-03T12:07:29Z","content_type":"text/html","content_length":"263212","record_id":"<urn:uuid:19bfc909-a62f-46f4-8de5-9a81afab227a>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00424.warc.gz"} |
Hacking the Law
First of all, I am neither a lawyer nor a trained ethicist. The following are a list of thought experiments related to “hacking” (i.e., testing the limits of) the law. Unless otherwise noted, I have
not done any research to confirm whether or not the questions posted herein are either novel or have already been answered. Although the following contains some material related to computers, I have
tried my best to write it in such a way as to be accessible to the widest audience.
Copyrighting a Number
Is it legal?
It is obviously legal to copyright an artistic work, like a digital photo. A digital photo, however, is really stored on a computer’s hard drive as a sequence of numbers, each representing the color
of a dot in the picture. This sequence of numbers could be summed such that it amounts to a single, unique number. Would it be legal for one to give that number—which uniquely represents the
copyrighted image—to a friend? The friend could then divide that number back into its sequence on the hard drive, thus reconstructing the original copyrighted picture. If copyrighting numbers is not
legal, then I do not see why what I just described would not be legal.
The issue is actually a bit more complicated than it seems.
It is entirely possible that the method used to convert the digital picture to a single number could be slightly modified (e.g., by adding 1 to the resulting number). If the recipient of the number
does not know that this was done then the resulting reconstructed picture will look like noise. If the recipient knows to subtract 1 from the number before reconstructing the picture, however, the
picture will be exactly the same as the copyrighted picture.
To add even more complication, it is entirely possible that, by adding 1 to the number, the improperly decoded picture might in fact become a completely different copyrighted picture.
1. Person X has a copyrighted picture, called picture A, that he/she legally owns.
2. X converts the picture to a number, $n$.
3. X sends the number $n+1$ to person Y.
Case 1:
• Y converts the number $n-1$ back to a picture, resulting in picture A.
Case 2:
• Y converts the number $n$ to a picture, resulting in a completely different picture B.
• Picture B turns out to be copyrighted by person Z.
• Neither person X nor person Y have ever even seen picture B before.
At what point is copyright lost?
Related to copyrighting a number is the following.
When the picture is represented as a sequence of numbers (representing the colors of the individual dots in the picture), it is possible to increment each of the colors of the individual dots. For
example, let’s say the dot in the upper left corner of picture A is currently black. We could iteratively increment the color of that dot so that it eventually becomes white (going through a sequence
of lightening grays in the process). We could even increment all of the dots in the picture at the same time.
Now, let’s say picture A is a photo of the Mona Lisa of which we do not own the copyright. Picture B is a photo of the Empire State Building that you took and of which therefore own the copyright.
Both of the pictures have the same dimensions; therefore each dot in picture A has a corresponding dot in picture B.
Now, we iteratively increment the dots in A such that they all move toward the color of their corresponding dot in picture B. Let’s call the result of this picture C. At the beginning, C will look
exactly like picture A. At the end, C will look exactly like picture B. In the middle of the process, C will look like a linear combination of A and B.
Question 1
At what point during the “morph” from A to B will the “copyright” of picture C transition from that of picture A to picture B?
Question 2
Is there any point during the process that picture C might not be protected by either picture A or picture B’s copyrights? | {"url":"https://www.sultanik.com/blog/Hacking_the_law","timestamp":"2024-11-06T07:41:22Z","content_type":"text/html","content_length":"25860","record_id":"<urn:uuid:3ac4db34-556c-4e1c-9ae7-1e8b2e7cf31b>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00369.warc.gz"} |
College of Science and Mathematics
Department of Mathematics
Bachelor of Science in Mathematics - Integrated Credential Option
The requirement for entrance to the major and minor programs is completion of two years of algebra as well as courses in geometry and trigonometry, or a sequence of courses containing their
equivalents, such as MATH 3 (College Algebra) and 5 (Trigonometry). It is strongly recommended that such study be completed before entrance to the university.
Incoming Integrated Credential Option students are eligible for the Mathematics Teaching Scholars Honors Program
Major Requirements (53 units) and Additional Requirements (8 units)
Core curriculum (35 units)
MATH 75 (or 75A and B), 76, 77, 81 (15 units)
MATH 101 (4 units)
MATH 111 (4 units)
MATH 151, 152 (8 units)
MATH 171 (4 units)
Option specific electives (18 units)
Math 143 (4 units)
MATH 145 (3 units)
MATH 149S (4 units)
MATH 161 (3 units)
MATH 123 (3 units)
MATH 193D (1 unit)
Additional requirements (8 units)
CSCI 40 (4 units)
PHYS 4A (3 units)
PHYS 4AL (see note 1) (1 unit)
General Education requirements (48 units) (see notes 1, 3, 4 and 5)
Other requirements (9 units)
American Government and Institutions (PLSI 2), Multicultural and International (MI), and Upper-division writing (see note 2)
Credential requirements (34 units)
CI 151 (3 units)
CI 152 (3 units)
CI 161 (3 units)
LEE 157 (3 units)
EHD 154A (1 unit)
EHD 155A (4 units)
EHD 155B (10 units)
SPED 158 (3 units)
LEE 156 (3 units)
EHD 154B (1 unit)
Total (134 units)
The culminating experience for the Integrated Credential Option is MATH 149S.
Advising Notes
1. PHYS 4AL is required for the Integrated Credential Option. PHYS 4A and 4AL will satisfy the General Education (GE) Breadth requirement.
2. Students in the Integrated Credential Option are required to take ANTH 105W as a writing course that also satisfies the Multicultural/International Requirement.
3. Three units of MATH 75 (or MATH 75A) also will satisfy the G.E. Foundation B4 requirement and Math 123 will satisfy G.E. Area IB requirement. CI 152 is double counted as a Credential requirement
and G.E. ID requirement for the Integrated Credential option.
4. Thirteen units of courses are double counted (B1/3, B4, IB, and ID), and 3 units of G.E. requirements are waived (A3) for the Integrated Credential option.
5. See Mathematics Road Map at:
Road Map
It is strongly recommended, to all math majors, to have an advising session at least once a semester. If you do not have an assigned advisor, please talk to the staff at the Mathematics department
office (PB 381).
Grade Requirements
All courses required as prerequisites for a mathematics course must be completed with a grade of C or better before registration will be permitted. All courses taken to fulfill major or minor
requirements must be completed with a grade of C or better. | {"url":"https://csm.fresnostate.edu/math/degrees/bs/integrated.html","timestamp":"2024-11-10T11:25:13Z","content_type":"text/html","content_length":"32528","record_id":"<urn:uuid:65295581-0eb9-47fe-878b-7de284cca3d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00657.warc.gz"} |
Tech Tip: How to Use Expressions for Numeric Fields in Onshape
You can use expressions in any numeric feature in Onshape, including features, dimensions, variables, and even configuration tables. Expressions evaluate using FeatureScript and the result must
contain no units or units to the first power (such as length, but not area or volume).
Most mathematical functions are allowed in expressions. These include operators such as exponent (^), multiplication (*), division (/), addition (+), and subtraction (-). They also include more
complex functions such as ceil, floor, round, exp, sqrt, abs, max, min, and log. A full list of functions are listed in the Standard Library Documentation under math.
To add more flexibility, conditional statements are also allowed. One way to do this is an array with a lookup value as shown below:
The position requests the value at position 3. The result is 8 because the position count starts at 0.
In a practical example for a pipe flange, the position is denoted by the class of flange. A 300# flange sets the #Class variable to 1.
The #Class variable is then used to determine the dimensions based on the Nominal Pipe Size and an array.
Another form of conditional statement uses a ternary operator (similar to an IF statement). This uses a condition, Value if True, and Value if False, as shown below:
In this example, if the number of USB ports chosen for a configuration is greater than 3, it will evaluate to 3. Otherwise, it will evaluate to 2. After plugging the formula into the instance count
for computer fans, you can see that the number of fans updates appropriately.
Expressions are a powerful tool in Onshape that can cut down on configured features and add more automation to your design variants.
Interested in learning more Onshape Tech Tips? You can review the most recent technical blogs here. | {"url":"https://www.onshape.com/en/resource-center/tech-tips/tech-tip-how-to-use-expressions-for-numeric-fields-in-onshape","timestamp":"2024-11-09T16:45:30Z","content_type":"text/html","content_length":"19539","record_id":"<urn:uuid:ecd3d7b0-7d94-424a-8987-8c22c6d31982>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00234.warc.gz"} |
how hard is partial differential equations
We first look for the general solution of the PDE before applying the initial conditions. since we are assuming that u(t, x) is a solution to the transport equation for all (t, x). In general,
partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which
all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables. Active 2 years, 11 months ago. Differential Equations 2 : Partial
Differential Equations amd Equations of Mathematical Physics (Theory and solved Problems), University Book, Sarajevo, 2001, pp. 2 An equation involving the partial derivatives of a function of more
than one variable is called PED. So the partial differential equation becomes a system of independent equations for the coefficients of : These equations are no more difficult to solve than for the
case of ordinary differential equations. In the above six examples eqn 6.1.6 is non-homogeneous where as the first five equations … Using linear dispersionless water theory, the height u (x, t) of a
free surface wave above the undisturbed water level in a one-dimensional canal of varying depth h (x) is the solution of the following partial differential equation. How hard is this class? pdex1pde
defines the differential equation (See [2].) Get to Understand How to Separate Variables in Differential Equations As indicated in the introduction, Separation of Variables in Differential Equations
can only be applicable when all the y terms, including dy, can be moved to one side of the equation. A Partial Differential Equation commonly denoted as PDE is a differential equation containing
partial derivatives of the dependent variable (one or more) with more than one independent variable. The simple PDE is given by; ∂u/∂x (x,y) = 0 The above relation implies that the function u(x,y) is
independent of x which is the reduced form of partial differential equation formulastate… It was not too difficult, but it was kind of dull. For multiple essential Differential Equations, it is
impossible to get a formula for a solution, for some functions, they do not have a formula for an anti-derivative. . This course is known today as Partial Differential Equations. It was not too
difficult, but it was kind of dull.
Even though Calculus III was more difficult, it was a much better class--in that class you learn about functions from R^m --> R^n and what the derivative means for such a function. So, we plan to
make this course in two parts – 20 hours each. How to Solve Linear Differential Equation? Combining the characteristic and compatibility equations, dxds = y + u,
(2.11), dyds = y, (2.12), duds = x − y
(2.13). The unknown in the diffusion equation is a function u(x, t) of space and time.The physical significance of u depends on what type
of process that is described by the diffusion equation. It is used to represent many types of phenomenons like sound, heat, diffusion, electrostatics, electrodynamics, … . Analytic Geometry deals
mostly in Cartesian equations and Parametric Equations. User account menu • Partial differential equations? The definition of Partial Differential Equations (PDE) is a differential equation that has
many unknown functions along with their partial derivatives. A partial differential equation requires, d) an equal number of dependent and independent variables. Differential equations (DEs) come in
many varieties. We plan to offer the first part starting in January 2021 and … Maple is the world leader in finding exact solutions to ordinary and partial differential equations. Pro Lite, CBSE
Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. In the equation, X is the independent variable. Press question mark to learn the rest of the keyboard
shortcuts. For example, dy/dx = 9x. For eg. . Differential Equations • A differential equation is an equation for an unknown function of one or several variables that relates the values of the
function itself and of its derivatives of various orders. And we said that this is a reaction-diffusion equation and what I promised you is that these appear in, in other contexts. Compared to
Calculus 1 and 2. Partial Differential Equations. endstream endobj 1993 0 obj <>stream Most often the systems encountered, fails to admit explicit solutions but fortunately qualitative methods were
discovered which does provide ample information about the … In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations
called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent
variables. Now, consider dds (x + uy) = 1y dds(x + u) − x + uy2 dyds , = x + uy − x + uy = 0. The derivation of partial differential equations from physical laws usually brings about simplifying
assumptions that are difficult to justify completely. It is used to represent many types of phenomenons like sound, heat, diffusion, electrostatics, electrodynamics, … thats why first courses focus
on the only easy cases, exact equations, especially first order, and linear constant coefficient case. We stressed that the success of our numerical methods depends on the combination chosen for the
time integration scheme and the spatial discretization scheme for the right-hand side. There are many "tricks" to solving Differential Equations (ifthey can be solved!). Hence the derivatives are
partial derivatives with respect to the various variables. This is a linear partial differential equation of first order for µ: Mµy −Nµx = µ(Nx −My). to explain a circle there is a general equation: (x
– h)2 + (y – k)2 = r2. L u = ∑ ν = 1 n A ν ∂ u ∂ x ν + B = 0 , {\displaystyle Lu=\sum _ {\nu =1}^ {n}A_ {\nu } {\frac {\partial u} {\partial x_ {\nu }}}+B=0,} where the coefficient matrices Aν and
the vector B may depend upon x and u. (diffusion equation) These are second-order differential equations, categorized according to the highest order derivative. 258. Would it be a bad idea to take
this without having taken ordinary differential equations? In algebra, mostly two types of equations are studied from the family of equations. In the previous notebook, we have shown how to transform
a partial differential equation into a system of coupled ordinary differential equations using semi-discretization. Such a method is very convenient if the Euler equation is of elliptic type.
Scientists and engineers use them in the analysis of advanced problems. Furthermore, the classification of Partial Differential Equations of Second Order can be done into parabolic, hyperbolic, and
elliptic equations. Do you know what an equation is? This defines a family of solutions of the PDE; so, we can choose φ(x, y, u) = x + uy, Example 2. Log In Sign Up. It is also stated as Linear
Partial Differential Equation when the function is dependent on variables and derivatives are partial. See Differential equation, partial, complex-variable methods. Press J to jump to the feed.
differential equations in general are extremely difficult to solve. You can classify DEs as ordinary and partial Des. How hard is this class? This Site Might Help You. We solve it when we discover
the function y(or set of functions y). In addition, we give solutions to examples for the heat equation, the wave equation and Laplace’s equation. A method of lines discretization of a PDE is the
transformation of that PDE into an ordinary differential equation. Press question mark to learn the rest of the keyboard shortcuts. An ode is an equation for a function of pdepe solves partial
differential equations in one space variable and time. In this eBook, award-winning educator Dr Chris Tisdell demystifies these advanced equations. Viewed 1k times 0 $\begingroup$ My question is why
it is difficult to find analytical solutions for these equations. Method of Lines Discretizations of Partial Differential Equations The one-dimensional heat equation. But first: why? If the partial
differential equation being considered is the Euler equation for a problem of variational calculus in more dimensions, a variational method is often employed. What is the intuitive reason that
partial differential equations are hard to solve? First, differentiating ƒ with respect to x … Likewise, a differential equation is called a partial differential equation, abbreviated by pde, if it
has partial derivatives in it. Introduction to Differential Equations with Bob Pego. This book examines the general linear partial differential equation of arbitrary order m. Even this involves more
methods than are known. The most common one is polynomial equations and this also has a special case in it called linear equations. Separation of Variables, widely known as the Fourier Method, refers
to any method used to solve ordinary and partial differential equations. So in geometry, the purpose of equations is not to get solutions but to study the properties of the shapes. Get to Understand
How to Separate Variables in Differential Equations What To Do With Them? The movement of fluids is described by The Navier–Stokes equations, For general mechanics, The Hamiltonian equations are
used. There are two types of differential equations: Ordinary Differential Equations or ODE are equations which have a function of an independent variable and their derivatives. Differential
Equations 2 : Partial Differential Equations amd Equations of Mathematical Physics (Theory and solved Problems), University Book, Sarajevo, 2001, pp. This chapter presents a quasi-homogeneous partial
differential equation, without considering parameters.It is shown how to find all its quasi-homogeneous (self-similar) solutions by the support of the equation with the help of Linear Algebra
computations. Vedantu 1. A differential equation having the above form is known as the first-order linear differential equation where P and Q are either constants or … These are used for processing
model that includes the rates of change of the variable and are used in subjects like physics, chemistry, economics, and biology. A Differential Equation can have an infinite number of solutions as a
function also has an infinite number of antiderivatives. And different varieties of DEs can be solved using different methods. Some courses are made more difficult than at other schools because the
lecturers are being anal about it. Sorry!, This page is not available for now to bookmark. The ‘=’ sign was invented by Robert Recorde in the year 1557.He thought to show for things that are equal,
the best way is by drawing 2 parallel straight lines of equal lengths. Equations are considered to have infinite solutions. Ordinary and Partial Differential Equations. Partial differential equations
arise in many branches of science and they vary in many ways. The precise idea to study partial differential equations is to interpret physical phenomenon occurring in nature. There are Different
Types of Partial Differential Equations: Now, consider dds (x + uy) = 1y dds(x + u) − x + uy, The general solution of an inhomogeneous ODE has the general form: u(t) = u. In this book, which is
basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators. Vedantu academic counsellor will be calling you
shortly for your Online Counselling session. In elementary algebra, you usually find a single number as a solution to an equation, like x = 12. Partial Differential Equation helps in describing
various things such as the following: In subjects like physics for various forms of motions, or oscillations. to explain a circle there is a general equation: (x – h). A PDE for a function u(x1,……xn)
is an equation of the form The PDE is said to be linear if f is a linear function of u and its derivatives. This is a linear differential equation and it isn’t too difficult to solve (hopefully). You
can classify DEs as ordinary and partial Des. They are a very natural way to describe many things in the universe. What are the Applications of Partial Differential Equation? Thus, they learn an
entire family of PDEs, in contrast to classical methods which solve one instance of the equation. Ask Question Asked 2 years, 11 months ago. 5. Most of the time they are merely plausibility
arguments. A linear ODE of order n has precisely n linearly independent solutions. The reason for both is the same. Using differential equations Radioactive decay is calculated. All best, Mirjana I'm
taking both Calc 3 and differential equations next semester and I'm curious where the difficulties in them are or any general advice about taking these subjects? An equation is a statement in which
the values of the mathematical expressions are equal. Therefore, each equation has to be treated independently. To apply the separation of variables in solving differential equations, you must move
each variable to the equation's other side. The general solution of an inhomogeneous ODE has the general form: u(t) = uh(t) + up(t). A topic like Differential Equations is full of surprises and
fun but at the same time is considered quite difficult. Differential equations involve the differential of a quantity: how rapidly that quantity changes with respect to change in another. Since we
can find a formula of Differential Equations, it allows us to do many things with the solutions like devise graphs of solutions and calculate the exact value of a solution at any point. Nonlinear
differential equations are difficult to solve, therefore, close study is required to obtain a correct solution. While I'm no expert on partial differential equations the only advice I can offer is
the following: * Be curious but to an extent. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. For
virtually all functions ƒ ( x, y) commonly encountered in practice, ƒ vx; that is, the order in which the derivatives are taken in the mixed partials is immaterial. To apply the separation of
variables in solving differential equations, you must move each variable to the equation's other side. The following is the Partial Differential Equations formula: We will do this by taking a Partial
Differential Equations example. Press J to jump to the feed. Homogeneous PDE: If all the terms of a PDE contains the dependent variable or its partial derivatives then such a PDE is called
non-homogeneous partial differential equation or homogeneous otherwise. In case of partial differential equations, most of the equations have no general solution. Here are some examples: Solving a
differential equation means finding the value of the dependent […] Don’t let the name fool you, this was actually a graduate-level course I took during Fall 2018, my last semester of undergraduate
study at Carnegie Mellon University.This was a one-semester course that spent most of the semester on partial differential equations (alongside about three weeks’ worth of ordinary differential
equation theory). . Section 1-1 : Definitions Differential Equation. If a hypersurface S is given in the implicit form. Learn differential equations for free—differential equations, separable
equations, exact equations, integrating factors, and homogeneous equations, and more. the constant coefficient case is the easiest becaUSE THERE THEY BEhave almost exactly like algebraic equations.
Partial differential equations form tools for modelling, predicting and understanding our world. If you're seeing this message, it means we're having trouble loading external resources on our
website. Included are partial derivations for the Heat Equation and Wave Equation. As a general rule solving PDEs can be very hard and we often have to resort to numerical methods. Download for
offline reading, highlight, bookmark or take notes while you read PETSc for Partial Differential Equations: Numerical Solutions in C and Python. This is not a difficult process, in fact, it occurs
simply when we leave one dimension of … Differential equations have a derivative in them. Algebra also uses Diophantine Equations where solutions and coefficients are integers. Differential equations
are the equations which have one or more functions and their derivatives. Calculus 2 and 3 were easier for me than differential equations. All best, Mirjana The number $ k $ and the number $ l $ of
coefficients $ a _ {ii} ^ {*} ( \xi ) $ in equation (2) which are, respectively, positive and negative at the point $ \xi _ {0} $ depend only on the coefficients $ a _ {ij} ( x) $ of equation (1). It
is used to represent many types of phenomenons like sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation, and quantum mechanics. If you need a refresher on
solving linear first order differential equations go back and take a look at that section . Polynomial equations are generally in the form P(x)=0 and linear equations are expressed ax+b=0 form where
a and b represents the parameter. A variable is used to represent the unknown function which depends on x. The RLC circuit equation (and pendulum equation) is an ordinary differential equation, or
ode, and the diffusion equation is a partial differential equation, or pde. by Karen Hao archive page There are many other ways to express ODE. Alexander D. Bruno, in North-Holland Mathematical
Library, 2000. This is intended to be a first course on the subject Partial Differential Equations, which generally requires 40 lecture hours (One semester course). Maple 2020 extends that lead even
further with new algorithms and techniques for solving more ODEs and PDEs, including general solutions, and solutions with initial conditions and/or boundary conditions. • Ordinary Differential
Equation: Function has 1 independent variable. The derivatives re… As a consequence, differential equations (1) can be classified as follows. While I'm no expert on partial differential equations the
only advice I can offer is the following: * Be curious but to an extent. This example problem uses the functions pdex1pde, pdex1ic, and pdex1bc. The differential equations class I took was just about
memorizing a bunch of methods. PETSc for Partial Differential Equations: Numerical Solutions in C and Python - Ebook written by Ed Bueler. The definition of Partial Differential Equations (PDE) is a
differential equation that has many unknown functions along with their partial derivatives. Would it be a bad idea to take this without having taken ordinary differential equations? User account menu
• Partial differential equations? Example 1: If ƒ ( x, y) = 3 x 2 y + 5 x − 2 y 2 + 1, find ƒ x, ƒ y, ƒ xx, ƒ yy, ƒ xy 1, and ƒ yx. Log In Sign Up. RE: how hard are Multivariable calculus (calculus
III) and differential equations? A differential equationis an equation which contains one or more terms which involve the derivatives of one variable (i.e., dependent variable) with respect to the
other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable For example, dy/dx = 5x A differential equation that contains derivatives
which are either partial derivatives or ordinary derivatives. In this chapter we introduce Separation of Variables one of the basic solution techniques for solving partial differential equations. A
partial differential equation has two or more unconstrained variables. Partial differential equations can describe everything from planetary motion to plate tectonics, but they’re notoriously hard to
solve. I find it hard to think of anything that’s more relevant for understanding how the world works than differential equations. Partial Differential Equations I: Basics and Separable Solutions We
now turn our attention to differential equations in which the “unknown function to be deter-mined” — which we will usually denote by u — depends on two or more variables. And different varieties of
DEs can be solved using different methods. . Differential equations are the key to making predictions and to finding out what is predictable, from the motion of galaxies to the weather, to human
behavior. If a differential equation has only one independent variable then it is called an ordinary differential equation. • Partial Differential Equation: At least 2 independent variables. Today
we’ll be discussing Partial Differential Equations. Pro Lite, Vedantu There are many ways to choose these n solutions, but we are certain that there cannot be more than n of them. Here are some
examples: Solving a differential equation means finding the value of the dependent […] Now isSolutions Manual for Linear Partial Differential Equations . Partial Differential Equations: Analytical
Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners’ course for graduate students. Sometimes we can get a
formula for solutions of Differential Equations. The Navier-Stokes equations are nonlinear partial differential equations and solving them in most cases is very difficult because the nonlinearity
introduces turbulence whose stable solution requires such a fine mesh resolution that numerical solutions that attempt to numerically solve the equations directly require an impractical amount of
computational power. The differential equations class I took was just about memorizing a bunch of methods. Two C1-functions u(x,y) and v(x,y) are said to be functionally dependent if det µ ux uy vx
vy ¶ = 0, which is a linear partial differential equation of first order for u if v is a given … That's point number two down here. -|���/�3@��\���|{�хKj���Ta�ެ�ޯ:A����Tl��v�9T����M���۱� m�m�e�r�T�� ձ$m
For this reason, some branches of science have accepted partial differential equations as … 40 . On its own, a Differential Equation is a wonderful way to express something, but is hard to use.. For
eg. In addition to this distinction they can be further distinguished by their order. The first definition that we should cover should be that of differential equation.A differential equation is any
equation which contains derivatives, either ordinary derivatives or partial derivatives. Even more basic questions such as the existence and uniqueness of solutions for nonlinear partial differential
equations are hard problems and the resolution of existence and uniqueness for the Navier-Stokes equations in three spacial dimensions in particular is … For instance, an ordinary differential
equation in x(t) might involve x, t, dx/dt, d 2 x/dt 2 and perhaps other derivatives. Homogeneous PDE: If all the terms of a PDE contains the dependent variable or its partial derivatives then such a
PDE is called non-homogeneous partial differential equation or homogeneous otherwise. Even though we don’t have a formula for a solution, we can still Get an approx graph of solutions or Calculate
approximate values of solutions at various points. Pro Lite, Vedantu (i) Equations of First Order/ Linear Partial Differential Equations, (ii) Linear Equations of Second Order Partial Differential
Equations. We also just briefly noted how partial differential equations could be solved numerically by converting into discrete form in both space and time. No one method can be used to solve all of
them, and only a small percentage have been solved. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor. Even though Calculus III was
more difficult, it was a much better class--in that class you learn about functions from R^m --> R^n and … This is the book I used for a course called Applied Boundary Value Problems 1. The
complicated interplay between the mathematics and its applications led to many new discoveries in both. Well, equations are used in 3 fields of mathematics and they are: Equations are used in
geometry to describe geometric shapes. Differential Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. Publisher
Summary. H���Mo�@����9�X�H�IA���h�ޚ�!�Ơ��b�M���;3Ͼ�Ǜ�`�M��(��(��k�D�>�*�6�PԎgN �`rG1N�����Y8�yu�S[clK��Hv�6{i���7�Y�*�c��r�� J+7��*�Q�ň��I �v��$R� J��������:dD��щ֢+f;4Рu@�wE{ٲ�Ϳ�]�|0p��#h�Q�L�@�&�
`fe����u,�. Ordinary and partial differential equations: Euler, Runge Kutta, Bulirsch-Stoer, stiff equation solvers, leap-frog and symplectic integrators, Partial differential equations: boundary
value and initial value problems. The definition of Partial Differential Equations (PDE) is a differential equation that has many unknown functions along with their partial derivatives. A central
theme is a thorough treatment of distribution theory. The examples pdex1, pdex2, pdex3, pdex4, and pdex5 form a mini tutorial on using pdepe. For example, u is the concentration of a substance if the
diffusion equation models transport of this substance by diffusion.Diffusion processes are of particular relevance at the microscopic level in … 258. Having a good textbook helps too (the calculus
early transcendentals book was a much easier read than Zill and Wright's differential equations textbook in my experience). (y + u) ∂u ∂x + y ∂u∂y = x − y in y > 0, −∞ < x < ∞. Analysis - Analysis -
Partial differential equations: From the 18th century onward, huge strides were made in the application of mathematical ideas to problems arising in the physical sciences: heat, sound, light, fluid
dynamics, elasticity, electricity, and magnetism. Read this book using Google Play Books app on your PC, android, iOS devices. Differential equations (DEs) come in many varieties. So, to fully
understand the concept let’s break it down to smaller pieces and discuss them in detail. In addition to this distinction they can be further distinguished by their order. The partial differential
equation takes the form. We will show most of the details but leave the description of the solution process out. Free ebook http://tinyurl.com/EngMathYT Easy way of remembering how to solve ANY
differential equation of first order in calculus courses. In the above six examples eqn 6.1.6 is non-homogeneous where as the first five equations … The degree of a partial differential equation is
the degree of the highest order derivative which occurs in it after the equation has been rationalized, i.e made free from radicals and fractions so for as derivatives are concerned. YES!
Especially first order differential equations: Numerical solutions in C and Python - eBook written by Ed Bueler as.! Following is the easiest because there they BEhave almost exactly like algebraic
equations in case of partial differential arise! Integrating factors, and only a small percentage have been solved function also has an number... Can get a formula for solutions of differential
equations of DEs can be further by... Be done into parabolic, hyperbolic, and homogeneous equations, integrating factors, and elliptic equations case the. The family of equations and pdex5 form a
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equation and Wave equation and Wave equation and i... Geometry deals mostly in Cartesian equations and this also has a special in... The precise idea to study partial differential equations ( PDE )
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maintaining the expected level of it.... The PDE before applying the initial conditions equations for free—differential equations, integrating factors, and.. The examples pdex1, pdex2, pdex3, pdex4,
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Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor
Properties expressible in small fragments of the theory of the hyperfinite II${}_{1}$ factor
Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 37-47.
We show that any II${}_{1}$ factor that has the same 4-quantifier theory as the hyperfinite II${}_{1}$ factor $ℛ$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and
has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $ℛ$. In the same spirit,
we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $ℛ$ is an
infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $ℛ$ has the same 3-quantifier theory
as an infinitely generic embeddable factor.
Published online:
Classification: O3C66, 46L10
Keywords: Continuous model theory, von Neumann algebras, II$_1$ factors, factorial commutant embedding problem
Author's affiliations:
Isaac Goldbring ^1; Bradd Hart ^2
Copyrights: The authors retain unrestricted copyrights and publishing rights
author = {Isaac Goldbring and Bradd Hart},
title = {Properties expressible in small fragments of the theory of the hyperfinite {II}$_1$ factor},
journal = {Confluentes Mathematici},
pages = {37--47},
publisher = {Institut Camille Jordan},
volume = {12},
number = {2},
year = {2020},
doi = {10.5802/cml.67},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.67/}
TY - JOUR
AU - Isaac Goldbring
AU - Bradd Hart
TI - Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor
JO - Confluentes Mathematici
PY - 2020
SP - 37
EP - 47
VL - 12
IS - 2
PB - Institut Camille Jordan
UR - https://cml.centre-mersenne.org/articles/10.5802/cml.67/
DO - 10.5802/cml.67
LA - en
ID - CML_2020__12_2_37_0
ER -
%0 Journal Article
%A Isaac Goldbring
%A Bradd Hart
%T Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor
%J Confluentes Mathematici
%D 2020
%P 37-47
%V 12
%N 2
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.67/
%R 10.5802/cml.67
%G en
%F CML_2020__12_2_37_0
Isaac Goldbring; Bradd Hart. Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor. Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 37-47. doi : 10.5802/cml.67. https://cml.centre-mersenne.org/articles/10.5802/cml.67/
[1] S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli. Factorial commutants and the generalized Jung property for II${}_{1}$ factors. Preprint. arXiv 2004.02293.
[2] I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov. Model theory for metric structures, in Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note
Ser. 350, 315–427. Cambridge Univ. Press, Cambridge, 2008. | DOI | Zbl
[3] N. Brown. Topological dynamical systems associated to II${}_{1}$ factors. Adv. Math. 227 (2011), 1665–1699. With an appendix by N. Ozawa. | DOI | Zbl
[4] A. Connes and V. Jones. Property T for von Neumann algebras. Bull. London Math. Soc. 17 (1985), 57–62. | DOI | MR | Zbl
[5] I. Farah, I. Goldbring, B. Hart, and D. Sherman. Existentially closed II${}_{1}$ factors. Fund. Math. 233 (2016), 173–196. | DOI | MR | Zbl
[6] I. Farah, B. Hart, and D. Sherman. Model theory of operator algebras III: Elementary equivalence and II${}_{1}$ factors. Bull. London Math. Soc. 46 (2014), 1–20. | DOI | Zbl
[7] I. Goldbring. On Popa’s factorial commutant embedding problem. Proc. Amer. Math. Soc., to appear. | DOI | MR | Zbl
[8] I. Goldbring. Spectral gap and definability, Beyond First Order Model Theory Vol. 2, to appear.
[9] I. Goldbring and B. Hart. The universal theory of the hyperfinite II${}_{1}$ factor is not computable. Preprint. arXiv 2004.02299.
[10] I. Goldbring and B. Hart. On the theories of McDuff’s II${}_{1}$ factors. Int. Math. Res. Notices 27 (2017), 5609–5628. | DOI | Zbl | {"url":"https://cml.centre-mersenne.org/articles/10.5802/cml.67/","timestamp":"2024-11-12T12:56:45Z","content_type":"text/html","content_length":"44730","record_id":"<urn:uuid:e81a252e-803f-49bb-8847-92b9a8a66700>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00851.warc.gz"} |
Tensor (intrinsic definition)
(Redirected from Tensor rank)
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In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties
can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general
relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.
Definition via tensor products of vector spaces[edit]
Given a finite set { V[1], ..., V[n] } of vector spaces over a common field F, one may form their tensor product V[1] ⊗ ... ⊗ V[n], an element of which is termed a tensor.
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
${\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}$
where V^∗ is the dual space of V.
If there are m copies of V and n copies of V^∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m + n. The tensors of order
zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V^∗ (for this reason the last two spaces
are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
${\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}$
The type (1, 1) tensors
${\displaystyle V\otimes V^{*}}$
are isomorphic in a natural way to the space of linear transformations from V to V. A bilinear form on a real vector space V, V × V → R, corresponds in a natural way to a type (0, 2) tensor in
${\displaystyle V^{*}\otimes V^{*}.}$
An example of such a bilinear form may be defined, termed the associated metric tensor (or sometimes misleadingly the metric or inner product), and is usually denoted g.
Tensor rank[edit]
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor (Hackbusch 2012, pp. 4)) is a tensor that can be written as a product of tensors of the form
${\displaystyle T=a\otimes b\otimes \cdots \otimes d}$
where a, b, ..., d are nonzero and in V or V^∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the
minimum number of simple tensors that sum to T (Bourbaki 1989, II, §7, no. 8).
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the
highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is d^n−1 when each product is of n vectors from a vector space of finite-dimensional vector space of
dimension d.
The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the
minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
${\displaystyle A=vw^{\mathrm {T} }.}$
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
${\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}$
In indices, a tensor of rank 1 is a tensor of the form
${\displaystyle T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .}$
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3
or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). Computational tasks such as the efficient
multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
${\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}\,}$
for given inputs x[i] and y[j]. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known (Knuth 1998, pp. 506–508).
Universal property[edit]
The space ${\displaystyle T_{n}^{m}(V)}$ can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that
many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of
priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach
carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
${\displaystyle f:V_{1}\times V_{2}\times \cdots \times V_{N}\to \mathbf {R} }$
is multilinear if it is linear in each argument. The space of all multlinear mappings from the product V[1] × V[2] × ... × V[N] into W is denoted L^N(V[1],V[2],...,V[N]; W). When N = 1, a multilinear
mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V; W).
The universal characterization of the tensor product implies that, for each multilinear function
${\displaystyle f\in L^{m+n}(\underbrace {V,V,\dots ,V} _{m},\underbrace {V^{*},V^{*},\dots ,V^{*}} _{n};W)}$
there exists a unique linear function
${\displaystyle T_{f}\in L(V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*};W)}$
such that
${\displaystyle f(v_{1},\dots ,v_{m},\alpha _{1},\dots ,\alpha _{n})=T_{f}(v_{1}\otimes \cdots \otimes v_{m}\otimes \alpha _{1}\otimes \cdots \otimes \alpha _{n})}$
for all v[i] ∈ V and α[i] ∈ V^∗.
Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism
${\displaystyle T_{n}^{m}(V)\cong L(V^{*}\otimes \dots \otimes V^{*}\otimes V\otimes \dots \otimes V;\mathbb {R} )\cong L^{m+n}(V^{*},\dots ,V^{*},V,\dots ,V;\mathbb {R} ).}$
In the formula above, the roles of V and V^∗ are reversed. In particular, one has
${\displaystyle T_{0}^{1}(V)\cong L(V^{*};\mathbb {R} )\cong V}$
${\displaystyle T_{1}^{0}(V)\cong L(V;\mathbb {R} )=V^{*}}$
${\displaystyle T_{1}^{1}(V)\cong L(V;V).}$
Tensor fields[edit]
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the
concept of a tensor that varies from point to point on the manifold. | {"url":"https://static.hlt.bme.hu/semantics/external/pages/tenzorszorzatok/en.wikipedia.org/wiki/Tensor_rank.html","timestamp":"2024-11-07T00:02:16Z","content_type":"text/html","content_length":"94172","record_id":"<urn:uuid:5621605a-26d9-4a71-9411-be1af9888176>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00240.warc.gz"} |
Some illustrative examples
Having succeeded in finding one single matrix for each automaton which summarizes important properties of the distribution of its ancestors, it is worthwhile examining some typical cases; because
the or), then one which has no Garden of Eden but which is not reversible ( exclusive or), and finally one which is reversible ( right shift). In due course we shall encounter nontrivial
reversible automata.
Harold V. McIntosh | {"url":"http://delta.cs.cinvestav.mx/~mcintosh/newweb/ra/node25.html","timestamp":"2024-11-12T18:40:01Z","content_type":"text/html","content_length":"3330","record_id":"<urn:uuid:2ca76ae3-745d-4da3-8fe8-9f75148f2865>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00269.warc.gz"} |
Density Expansions of Transport Coefficients for Gases
From Scholarpedia
Jay Robert Dorfman (2011), Scholarpedia, 6(9):9628. doi:10.4249/scholarpedia.9628 revision #136879 [link to/cite this article]
The determination of density expansions for transport coefficients for dilute and moderately dense gases has been an active field of research in the kinetic theory of gases for at least a century,
beginning, perhaps, with the fundamental work of David Enskog in 1911 (S. Chapman and T. G. Cowling, 1971). A density expansion of a transport or thermodynamic property of a dilute and moderately
dense gas composed of particles with central, short-ranged forces, and obeying classical mechanics, is usually thought to be a power series expansion of such quantities in powers of the reduced
density of the gas, ~n = na^^d, where n=N/V is the number density of N particles in a contained of volume V, a is a characteristic diameter of a molecule of the gas, and d is the number of spatial
dimensions of the system. The thermodynamic properties of dilute and moderately dense gases can indeed be expressed to a high degree of approximation by means of such series expansions, called
‘‘virial expansions’’, in powers of the gas density where the lowest order term is the ideal gas limit. The coefficients of the powers are determined by the potential energies of interactions among
small groups of particles considered in isolation from the other particles in the gas (J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, 1964; L. Reichl, 1998). The ideal gas term is obtained if one
ignores all interactions between the particles, the first correction to it involves two particle interactions, the next term, three particle interactions, and so on. The same situation does not
obtain if one attempts to apply similar methods to obtain useful expansions for transport properties, such as the coefficients of shear and bulk viscosity, thermal conductivity, diffusion and so on,
in series of powers of the gas density, with lowest order term the values obtained by use of the Boltzmann transport equation. Instead, due to the presence of long-range and long-time dynamical
correlations between the particles produced by correlated sequences of collisions, the coefficients in any power series expansion diverge after a few terms (J. R. Dorfman and H. van Beijeren, 1977).
The order in density of the first divergent term in the series depends upon the spatial dimensions of the system. The physical reason for these divergences has its origin in the attempt to express
transport properties as power series expansions in the density of the gas. Each term in this expansion is determined by the dynamics of small groups of particles, considered in isolation from the
rest of the particles in the gas. An important dynamical property of particles in the gas, the mean free path, is treated in an inappropriate way in this expansion. In a gas, each particle travels
only a mean free path between collisions, on the average. However, dynamical processes in which particles are treated as if they can travel freely for arbitrarily large distances determine the terms
in the power series expansion. This situation suggests that the terms in the density expansion should be combined in such a way as to correctly incorporate the mean free path damping of the
likelihood that a particle would travel a large distance between successive collisions. This can be accomplished by summing the most divergent terms in each order of density in the expansion (K.
Kawasaki and I. Oppenheim, 1965; J. R. Dorfman and H. van Beijeren, 1977).
Although the divergences in the virial coefficients can be removed by resummations of the diverging terms in the viral expansion, the resulting structure of the density expansion of transport
coefficients is much more complicated than a simple power series, and little is known about all but the first few terms in this expansion. Combinations of powers and logarithmic terms in the density
certainly appear in the expansion. Moreover, the same kind of resummations that lead to the known finite results for three dimensional systems still lead to divergent transport coefficients for two
dimensional systems, via a different, but closely related mechanism, an effect of the presence of long-range dynamical correlations that exist in all non-equilibrium fluids. The structure of the
‘‘Navier-Stokes’’ hydrodynamics for two-dimensional systems is apparently inherently non-linear and non-analytic, due to these correlations. For the sake of brevity, we shall restrict this discussion
to classical systems, but quantum systems are similar in most essential features, except for systems at low temperatures or high densities.
The Boltzmann and Enskog theories
A systematic way of calculating the transport coefficients for a dilute gas is provided by the ‘‘Boltzmann transport equation’’. This is an equation for the single-particle distribution function, \(f
(\mathbf{r}, \mathbf{v}, t) \) for particles in the gas, defined so that, for a one-component gas, \(f (\mathbf{r}, \mathbf{v}, t)\delta\mathbf{r}\delta\mathbf{v}\) is the number of gas particles in
a small region of space and velocity about the point \((\mathbf{r}, \mathbf{v})\) at time t. For mixtures one defines a separate function for each species. By means of probabilistic and mechanical
arguments, Boltzmann derived an equation for this distribution function, which provides the foundation for the kinetic theory of dilute gases and allows for a derivation of the Navier-Stokes
equations together with expressions for all of the transport coefficients in terms of the gas density, temperature, and the forces between the particles. The derivation of the Navier-Stokes equations
requires an assumption that the gas is close to a ‘‘local equilibrium state’’, that is a ‘‘Maxwell-Boltzmann’’ equilibrium distribution with a local temperature, density, and mean velocity that
depend upon position and time. It is also assumed that the local variables vary on a scale that is long compared to the mean free path of the particles between collisions. These assumptions lead to
the ‘‘normal solution’’ of the Boltzmann equation that, in turn, leads to the Navier-Stokes, and higher order hydrodynamic equations together with explicit expressions for the associated transport
coefficients (S. Chapman and T. G. Cowling, 1971).
For many years it was assumed that one could expand the transport coefficients for moderately dense gases in a virial expansion in much the same way as one does for equilibrium properties of such
gases. This supposition was supported by the ‘‘Enskog theory’’ for transport coefficients for a ‘‘hard sphere gas’’ that does lead to a virial expansion of the transport coefficients. The Enskog
theory is constructed from Boltzmann's method by: (1) modifying the expression for the rate at which binary collisions take place in the gas by a factor of the local equilibrium two-particle
correlation function at contact which accounts for the fact that two particles cannot collide at points in space that are already occupied by other particles; (2) for hard spheres there are
additional mechanisms of momentum and energy transport, not included in the Boltzmann equation, due to the instantaneous, “collisional transfer” transport of energy and momentum across a distance of
a molecular diameter whenever two hard sphere particles collide. The Enskog theory takes these effects into account and leads to density expansions of transport coefficients about the Boltzmann
result in the form of a virial series in the density. However Enskog's method does not provide a systematic expansion of transport coefficients in powers of the density. It only applies to hard
sphere particles, for which simultaneous collisions of more than two particles have zero measure in phase space, and for which collisional transfer of momentum and energy are clear and well defined.
Further, the Enskog method approximates the dynamics of hard sphere collisions in terms of a simple excluded volume approximation
Time correlation functions
In addition to methods based upon a fundamental kinetic equation for distribution functions in a fluid, another approach to a microscopic theory for hydrodynamics is based upon ‘‘time correlation
functions’’ This method, first proposed by M. S. Green, R. Kubo, and others, is based upon the determination of solutions to the ‘‘Liouville equation’’ for many-particle systems that are close to a
state of local equilibrium, similar to the solution method used in the normal solutions of the Boltzmann equation, but adjusted to an N-particle distribution function (W. A. Steele, 1971). This
method also leads to the Navier-Stokes equations but with expressions for the associated transport coefficients that should be applicable to fluids at any density. These expressions are integrals of
time-correlation functions and, for a transport coefficient, \(\tau\ ,\) take the form \[ \tau = \int_{0}^{\infty} dt < J_{\tau}(\Gamma,t) J_{\tau}(\Gamma,0)>, \] where the angular brackets denote an
average over an equilibrium ensemble distribution function, where the positions and momenta of the \(N\) particles are denoted by \(\Gamma\ ,\) and \(J_{\tau}(\Gamma,0)\) is the initial value of a
microscopic current whose form depends on the phase space variables and upon the particular transport coefficient, \(\tau\ ,\) being considered, while \(J_{\tau}(\Gamma,t)\) is the value of this
current at a time \(t\) later as obtained by solving the equations of motion for the system and following the phase space trajectory for a time interval \(t\ ,\) starting at the phase point \(\Gamma\
Formal, non-equilibrium virial expansions
The problem of obtaining formal density expansions of the kinetic equation and time correlation functions was solved through the work of N. N. Bogoliubov, M. S. Green, E. G. D. Cohen and others (E.
G. D. Cohen, 1993, M. H. Ernst, 1998). Bogoliubov’s method was difficult to apply and not very transparent but it did stimulate other workers to look at the problem. Green and Cohen separately
realized that the same kind of cluster expansion methods used to obtain equilibrium virial expansions could be adapted and applied to non-equilibrium systems, as well. The procedure for obtaining a
density expansion for the corrections to the Boltzmann equation follows a few well-defined steps that can also be applied to a formal evaluation of the time correlation functions as a density
expansion about their low-density limit. One begins by integration the Liouville equation over the phase variables of all but one of the particles in order to obtain an exact equation for the
one-particle distribution function, \(F_{1}({\mathbf{r}}_{1}, {\mathbf {p}}_{1},t)\ .\) This leads to the so-called ‘‘first BBGKY hierarchy equation’’ that is an equation for the single-particle
distribution function, but there is an interaction term that necessarily involves the two-particle distribution function, \(F_{2} ({\mathbf{r}}_{1}, {\mathbf{p}}_{1}, {\mathbf{r}}_{2},{\mathbf{p}}_
{2},t)\ ,\) when the two particles are interacting with each other. Similarly one can obtain an equation for the two-particle distribution function, but it contains an interaction term that requires
knowledge of the three-particle distribution function, and so on.
This chain of equations can be made useful by means of non--equilibrium cluster expansion methods. The simplest cluster expansions, corresponding to the “Ursell expansions” for equilibrium systems,
are not useful for times long compared to the mean free time between collisions, since they each contain secular terms, that is, terms growing as powers of the time \(t\ .\) The secular terms reflect
the fact that the volumes of phase space where two particles collide at any time in the interval \([0,t]\) grow as powers of the time, \(t\ .\) Since the hydrodynamic equations are valid on time
scales long compared to the mean free time between collisions, these expansions, as they stand, are not appropriate for a derivation of a useful kinetic equation, valid on long time scales, thus not
useful for a derivation of the Navier-Stokes equations. This difficulty can be overcome by assuming that there are only short ranged correlations between the particles in the initial state of the gas
(cf. J. R. Dorfman and H. van Beijeren, 1977). The simplest assumption is that at \(t=0\ ,\) all of the \(n\)-particle distribution functions have a factorized form as products of one particle
distribution functions, or \(F_n(1,2,...,n,t=0) = \prod_{i=1}^{n}F_1(i,t=0)\ .\) One removes the secular terms in the cluster expansions by inverting the cluster expansion for \(F_1(i,t)\) in order
to express \(F_{1}(i,t=0)\) as an expansion in products of \(F_1(t)\ ,\) in much the same way that the fugacity is expressed as a series in the density in the equilibrium case. Then this expansion
can be inserted in the expansion for \(F_2(t)\) in a series involving products of one-particle functions, \(F_1(t)\ ,\) structurally similar to the ‘‘Husimi expansion’’ for equilibrium functions.
When this expansion is inserted in the first BBGKY hierarchy equation, one obtains the ‘‘generalized Boltzmann equation’’, a closed equation for \(F_1(t)\) where the interaction term is an expansion
in a power series in the density. The first term is the Boltzmann equation and the higher terms provide the sought after density corrections to it.
Using the resulting kinetic equation as a starting point for a derivation of the Navier-Stokes equations, one obtains explicit expressions for all the coefficients in the virial expansions of the
transport coefficients in terms of the dynamical events taking place among an isolated group of particles over some time \(t\) long compared to a typical microscopic time scale. One is then left with
the problem of determining what the relevant dynamical events are, and determining their contributions to the non-equilibrium virial coefficients. A very similar procedure can be used to evaluate the
time correlation function expressions for transport coefficients in the form of a virial expansion, with results that are identical with those resulting from the generalized Boltzmann equation.
Divergences in the virial expansion of the transport coefficients
The procedure just outlined solves the problem of generalizing the Boltzmann equation results for the transport coefficients as a virial expansion in the density, provided one can evaluate
expressions for the virial coefficients. As this involves the determination of the possible dynamical events taking place in a group, or cluster, of a given number of particles, the next problem to
be solved is to determine the dynamical events that contribute to each of the virial coefficients and to evaluate their contributions by carrying out the relevant integrals. It is here that a feature
of non-equilibrium virial expansions appears that is not present in equilibrium virial expansions. Unlike the equilibrium virial coefficients where the configurations of small groups of particles
contribute only when the particles are close to each other, the dynamical events that contribute to the non-equilibrium virial coefficients can take place over large distances and over large time
intervals. In Figure 1 we illustrate one of
the dynamical events that contributes to the three-body contribution to the non-equilibrium virial coefficients. In addition to simultaneous encounters of three or four particles that take place over
relatively short times, there are also contributions from correlated collision sequences that can take place over arbitrarily long time intervals. Both phase space estimates and explicit calculations
of the contributions of such correlated collision sequences to the non-equilibrium virial coefficients for transport coefficients reveal a new and deep problem. For two dimensional systems of
particles interacting with short ranged, repulsive forces the contributions to the three- and higher- body virial coefficients from correlated collision sequences ‘‘diverge’’ as the time \(t\)
approaches infinity, starting as \(\ln t \)for the three-body term, and growing as successively higher powers of \(t\) for successive virial coefficients. For systems in three dimensions, a \(\ln t\)
divergence appears in the four-body term, and the higher virial coefficients grow as successively higher powers of t. In other words, non-equilibrium virial expansions, unlike their equilibrium
counterparts do not provide a useful representation of the density dependence of the transport coefficients for a moderately dense gas (E. D. G. Cohen, 1993, M. H. Ernst, 1998). The source of the
problem can be easily identified. The non-equilibrium virial coefficients contain contributions from sequences of collisions among small groups of particles in which the particles can travel
arbitrarily large distances between its collisions in the sequence. In actuality such long paths occur with very small probability. Instead, a typical trajectory is interrupted by collisions with
particles in the gas. This means that the typical distance between collisions is a mean free path length. Virial expansions force us to ignore this ever-present collision damping of the probability
for a long, free trajectory between successive collisions. Nevertheless, virial expansions of the transport coefficients are directly related to virial expansions of time correlation functions, via
the Green-Kubo relations, and these expansions can be used to determine the short time behavior of the time correlation functions (de Schepper, Ernst and Cohen, 1981).
In order to obtain a well-behaved density expansion, not necessarily a virial expansion, of transport coefficients for a moderately dense gas, one must renormalize the virial expansion by summing the
most divergent contributions in each order of the density (K. Kawasaki and I. Oppenheim, 1965). This resummation, known as the ‘‘ring sum’’, introduces the mean free path damping of the trajectory
probabilities into now modified expressions for transport coefficients as functions of *The resummation has two important consequences:
• There are now logarithmic terms in the density expansion of the transport coefficients, of relative order \(\tilde{n}\ln{\tilde{n}}\) in two dimensions and \({\tilde{n}}^2\ln{\tilde{n}}\) in
three dimensions. Therefore the density expansion for transport coefficients, denoted by \(\tau\ ,\) for three dimensional gases is given by
\[ \frac{\tau(\tilde{n},T)}{\tau_0(\tilde{n},T)} = 1 + a_{1,\tau}\tilde{n} + a_{2,\tau}{\tilde{n}}^2 + a_{2,\tau}^{\prime}{\tilde{n}}^2\ln{\tilde{n}} + \cdots . \] where \(\tau_0\) is the Boltzmann
value. Terms beyond those explicitly above include higher powers and combinations of powers and logarithms of the reduced density \(\tilde{n}\ ,\) however little more than that is known. For hard
spheres particles the coefficients \(a_{1,\tau}\) and \(a_{2,\tau}^{\prime}\) are known exactly. There are reasonable estimates available for \(a_{2,\tau}\ ,\) also for hard spheres. The appearance
of the logarithmic terms can easily be understood as an effect of the mean free path damping, since it provides a cut-off on the order of a few mean free times on time integrals that would otherwise
be logarithmically divergent.
• The ring sum provides a microscopic explanation of the \(t^{-d/2}\ ,\) or ‘‘long time tails’’ in the time correlation functions appearing in the integrands of the Green-Kubo expressions for
transport coefficients, discovered by means of computer simulations by B. J. Alder and T. Wainwright (B. J. Alder and T, Wainwright, 1970). Kinetic theory provides a quantitative description of
this decay as resulting from the effects of the dynamical events included in the ring summation. For three-dimensional systems the long-time tail contribution to transport coefficients can be
large and are the dominant contribution for fluids that are near a critical points of a phase transition. The long time tail contributions to two-dimensional systems are divergent. The long time
tail decays of the Green-Kubo integrands and the resulting divergence of the Green-Kubo integrals in two dimensions requires a re-examination of the microscopic derivations of the Navier-Stokes
and higher order hydrodynamic equations. The conclusion reached from such an examination is that the familiar linear laws relating currents of mass, momentum and energy to gradients in
temperature, mean velocity, and pressure must be modified, at the Navier-Stokes equations for two dimensional fluids and at the higher order gradient terms beyond the Navier-Stokes order for
three dimensional fluids (M. H. Ernst and J. R. Dorfman, 1975).
Conclusion: Can this picture be checked?
In view of the many complications that arise when attempting to develop density expansions for a fundamental kinetic equation beyond the Boltzmann, dilute gas limit, or for transport coefficients
beyond their dilute gas limit, it is imperative to check the results obtained so far with experimental results or with the results of computer simulated molecular dynamics. Comparisons have been
carried out for three kinds of observations:
• Efforts to detect the presence of terms of order \({\tilde{n}}^2\ln {\tilde{n}}\) in the density expansions of transport coefficients in experiments on moderately dense gases at room temperature.
have been inconclusive. However comparisons of the theoretical results for hard-sphere systems, with exact values for \(a_{1,\tau}\),\(a_{2,\tau}\prime\) and an estimate of the coefficient \( a_
{2,\tau}\ ,\) in the expression above, with computer results of Alder, Gass, and Wainwright suggest good agreement over a range of densities beyond the dilute gas limit (J. R. Dorfman, T. R
Kirkpatrick and J. V. Sengers, 1994). This is illustrated in Figure 2, where the dashed line is the result obtained by keeping only the linear order term in the density.
The presence of logarithmic terms in the density expansion of the coefficient of diffusion has been unambiguously demonstrated by Bruin and van Leeuwen for a simplified model gas, the ‘‘Lorentz
gas’’, where a particle moves in a system of randomly placed, fixed scatterers with which it makes elastic, specular collisions (C. Bruin, 1974). For this simplified model it is possible to calculate
several in the expansion of the diffusion coefficient in powers, and products of powers and logarithms, of the scatterer density. The diffusion of electrons in helium gas at temperatures below \(4 K
\)has been studied using a quantum version of the theory described here. The diffusion coefficient for the electrons as a function of the helium density also has logarithmic terms, and the
theoretical results and experimental data are in good agreement
• As mentioned above, the long time tails were discovered by means of computer experiments (B. J. Alder and T. Wainwright, 1967). Theoretical values for the coefficients of these exponential decays
can be obtained by means of kinetic theory using the ring summation that was used to renormalize the divergent terms in the virial expansions of the transport coefficients. This resummation leads
both to the logarithmic terms in density expansions discussed above, as well as to the long time tails (Y. Pomeau, 1971; J. R. Dorfman and E. G. D. Cohen, 1970; see also Y. Pomeau and P.
Resibois,1975) The excellent agreement between theoretical and computer simulation values for the coefficients of the long time tail decays, over a range of densities, provides confirmation of
the basic soundness of the theory. One should also mention that the same theory can be generalized to cover the properties of transport coefficients near a critical point of a phase transition.
This generalization, known as ‘‘mode-coupling theory’’ also leads to results that are in good agreement with experiments.
• The ring resummation can be applied to calculation of the spectrum of light scattered by a fluid maintained in a non-equilibrium stationary state. Kirkpatrick, and co-workers have shown that this
leads to a significant enhancement of the central, or Rayleigh peak, of the scattered light for a fluid that has a stationary temperature gradient (T. R. Kirkpatrick, E. G. D. Cohen, and J. R.
Dorfman, 1982). Sengers and co-workers have measured this enhancement and demonstrated the excellent agreement of the theoretical and experimental results (Ortiz de Zarate and J. V. Sengers,
• Alder(1967). Velocity Correlations for Hard Spheres. Physical Review Letters 18: 988. doi:10.1103/physrevlett.18.988.
• Bruin, C (1974). A Computer Experiment on Diffusion in the Lorentz Gas. Physica 72: 261. doi:10.1016/0031-8914(74)90029-9.
• Chapman, S and Cowling, T G (1971). The Mathematical Theory of Non-uniform Gases. Third edition. Cambridge University Press, Cambridge.
• Cohen, E G D (1993). Fifty Years of Kinetic Theory. Physica A 194: 229. doi:10.1016/0378-4371(93)90357-a.
• Dorfman(1970). Velocity Correlations in Two and Three Dimensions. Physical Review Letters 25: 1257. doi:10.1103/physrevlett.25.1257.
• Dorfman, J R; Kirkpatrick, T R and Sengers, J V (1994). Generic Long Range Correlations in Molecular Fluids. Annual Review of Physical Chemistry 45: 213. doi:10.1146/annurev.pc.45.100194.001241.
• Dorfman(1977). Kinetic Theory of Gases Modern Theoretical Chemistry: Statistical Mechanics, B J Berne editor, Plenum New York 6B: 65.
• Ernst(1975). Non-analytic Dispersion Relations for Classical Fluids Journal of Statistical Physics 79: 4585. doi:10.1016/0375-9601(72)90074-6.
• Ernst, M H (1998). Bogoliubov Choh Uhlenbeck Theory: Cradle of Modern Kinetic Theory in Progress in Statistical Physics: Proceedings of the International Conference on Statistical Physics in
Memory of Soon-Takh Choh; Wokyoung Sung, editor World Scientific Publishing Company, Singapore.
• Hirschfelder, J O; Curtiss, C H and Bird, R B) (1964). Molecular Theory of Gases and Liquids John Wiley and Sons, New York.
• Kawasaki(1969). Logarithmic Term in the Density Expansion of Transport Coefficients Physical Review 139: 1763. doi:10.1103/physrev.139.a1763.
• Kirkpatrick, T R; Gohen, E G D and Dorfman, J R (1982). Light Scattering by a Fluid in a Non-equilibrium Steady State, II, Large Gradients. Physical Review A 26: 995. doi:10.1103/physreva.26.995.
• Kirkpatrick, T R; Belitz, D and Sengers, J V (2001). Long Time Tails, Weak Localization, and Classical and Quantum Critical Behavior Journal of Statistical Physics 109: 373.
• Ortiz de Zarate, J M and Sengers, J V (2006). Hydrodynamic Fluctuations in Fluids and Fluid Mixtures Elsevier, Amsterdam.
• Pomeau, Y (1971). Transport Theory for a Two Dimensional Gas Physical Review A 3: 1174. doi:10.1103/physreva.3.1174.
• Pomeau(1975). Time Dependent Correlation Functions and Mode-Coupling Theories Physics Reports 19: 63. doi:10.1016/0370-1573(75)90019-8.
• Reichl, L (1998). A Modern Course in Statistical Mechanics Second edition John Wiley and Sons, New York.
• Resibois, P M V and de Leener, M (1977). Classical Kinetic Theory of Fluids John Wiley and Sons, New York.
• de Schepper, I; Ernst, M H and Cohen, E G D (1981). The Incoherent Scattering Function and Related Correlation functions in Hard Sphere Fluids at Short Times. Journal of Statistical Physics 25:
321. doi:10.1007/bf01022190.
• Steele, W (1969). Time-Correlation Functions in Transport Phenomena in Fluids H J M Hanley, editor Marcel Dekker, New York. | {"url":"http://var.scholarpedia.org/article/Density_Expansions_of_Transport_Coefficients_for_Gases","timestamp":"2024-11-05T06:43:26Z","content_type":"text/html","content_length":"60670","record_id":"<urn:uuid:f510e745-810f-4fb6-96c5-7b7242f13589>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00575.warc.gz"} |
Optimizing Tuning Parameters for
Part of using balancing methods, including matching, weighting, and subclassification, involves specifying a conditioning method, assessing balance after applying that method, and repeating until
satisfactory balance is achieved. For example, in propensity score score subclassification, one needs to decide on the number of subclasses to use, and one way to do so is to try a number of of
subclasses, assess balance after subclassification, try another number of subclasses, assess balance, and so on. As another example, in estimating the propensity score model itself, one might decided
which covariates to include in the model (after deciding on a fixed set of covariates to balance), which covariates should have squares or interactions, and which link function (e.g., probit, logit)
to use. Or choosing the number of matches each unit should receive in k:1 matching, or which value of the propensity score should be used to trim samples, etc.
Essentially, these problems all involve selecting a specification by varying a number of parameters, which are sometimes called “tuning parameters”, in order to optimize balance. Many popular methods
adjust tuning parameters to optimize balance as inherent parts of the method, like genetic matching (Diamond and Sekhon 2013), which tunes variance importance in the distance matrix, or generalized
boosted modeling (McCaffrey, Ridgeway, and Morral 2004), which tunes the number of trees in the prediction model for propensity scores. This strategy tends to yield methods that perform better than
methods that don’t tune at all or tune to optimize a criterion other than balance (e.g., prediction accuracy) Pirracchio and Carone (2018).
bal.compute() and bal.init()
Broadly, these functions work by taking in the treatment, covariates for which balance is to be computed, and a set of balancing weights and return a scalar balance statistic that summarizes balance
for the sample. bal.compute() does the work of computing the balance statistic, and bal.init() processes the inputs so they don’t need to be processed every time bal.compute() is called with a new
set of weights.
For bal.init(), we need to supply the covariates, the treatment, the name of the balance statistic we wish to compute, sampling weights (if any), and any other inputs required, which depend on the
specific balance statistic requested. bal.init() returns a bal.init object, which is then passed to bal.compute() along with a set of balancing weights (which may result from weighting, matching, or
Below, we provide an example using the lalonde dataset. Our balance statistic will be the largest absolute standardized mean difference among the included covariates, which is specified as "smd.max".
We will first supply the required inputs to bal.init() and pass its output to bal.compute() to compute the balance statistic for the sample prior to weighting.
data("lalonde", package = "cobalt")
covs <- subset(lalonde, select = -c(treat, race, re78))
# Initialize the object with the balance statistic,
# treatment, and covariates
smd.init <- bal.init(covs,
treat = lalonde$treat,
stat = "smd.max",
estimand = "ATT")
# Compute balance with no weights
#> [1] 0.8263093
# Can also compute the statistic directly using bal.compute():
treat = lalonde$treat,
stat = "smd.max",
estimand = "ATT")
#> [1] 0.8263093
The largest absolute standardized mean difference with no weights is 0.8263, which we can verify and investigate further using bal.tab():
treat = lalonde$treat,
binary = "std",
estimand = "ATT",
thresholds = .05)
#> Balance Measures
#> Type Diff.Un M.Threshold.Un
#> age Contin. -0.3094 Not Balanced, >0.05
#> educ Contin. 0.0550 Not Balanced, >0.05
#> married Binary -0.8263 Not Balanced, >0.05
#> nodegree Binary 0.2450 Not Balanced, >0.05
#> re74 Contin. -0.7211 Not Balanced, >0.05
#> re75 Contin. -0.2903 Not Balanced, >0.05
#> Balance tally for mean differences
#> count
#> Balanced, <0.05 0
#> Not Balanced, >0.05 6
#> Variable with the greatest mean difference
#> Variable Diff.Un M.Threshold.Un
#> married -0.8263 Not Balanced, >0.05
#> Sample sizes
#> Control Treated
#> All 429 185
We can see that the largest value corresponds to the covariate married.
Now, lets estimate weights using probit regression propensity scores in WeightIt and see whether this balance statistic decreases after applying the weights:
w.out <- weightit(treat ~ age + educ + married + nodegree +
re74 + re75, data = lalonde,
method = "glm", estimand = "ATT",
link = "probit")
# Compute the balance statistic on the estimated weights
bal.compute(smd.init, get.w(w.out))
#> [1] 0.06936946
After weighting, our balance statistic is 0.0694, indicating a significant improvement. Let’s try again with logistic regression:
w.out <- weightit(treat ~ age + educ + married + nodegree +
re74 + re75, data = lalonde,
method = "glm", estimand = "ATT",
link = "logit")
# Compute the balance statistic on the estimated weights
bal.compute(smd.init, get.w(w.out))
#> [1] 0.04791925
This is better, but we can do even better with bias-reduced logistic regression (Kosmidis and Firth 2020):
w.out <- weightit(treat ~ age + educ + married + nodegree +
re74 + re75, data = lalonde,
method = "glm", estimand = "ATT",
link = "br.logit")
# Compute the balance statistic on the estimated weights
bal.compute(smd.init, get.w(w.out))
#> [1] 0.04392724
Instead of writing each complete call one at a time, we can do a little programming to make this happen automatically:
# Initialize object to compute the largest SMD
smd.init <- bal.init(covs,
treat = lalonde$treat,
stat = "smd.max",
estimand = "ATT")
# Create vector of tuning parameters
links <- c("probit", "logit", "cloglog",
"br.probit", "br.logit", "br.cloglog")
# Apply each link to estimate weights
# Can replace sapply() with purrr::map()
weights.list <- sapply(links, function(link) {
w.out <- weightit(treat ~ age + educ + married + nodegree +
re74 + re75, data = lalonde,
method = "glm", estimand = "ATT",
link = link)
}, simplify = FALSE)
# Use each set of weights to compute balance
# Can replace sapply() with purrr:map_vec()
stats <- sapply(weights.list, bal.compute,
x = smd.init)
# See which set of weights is the best
#> probit logit cloglog br.probit br.logit br.cloglog
#> 0.06936946 0.04791925 0.02726102 0.06444577 0.04392724 0.02457557
#> br.cloglog
#> 0.02457557
Interestingly, bias-reduced complimentary log-log regression produced weights with the smallest maximum absolute standardized mean difference. We can use bal.tab() to more finely examine balance on
the chosen weights:
treat = lalonde$treat,
binary = "std",
weights = weights.list[["br.cloglog"]])
#> Balance Measures
#> Type Diff.Adj
#> age Contin. -0.0089
#> educ Contin. -0.0246
#> married Binary -0.0012
#> nodegree Binary 0.0145
#> re74 Contin. -0.0209
#> re75 Contin. -0.0213
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 240.83 185
If balance is acceptable, you would move forward with these weights in estimating the treatment effect. Otherwise, you might try other values of the tuning parameters, other specifications of the
model, or other weighting methods to try to achieve excellent balance.
Balance statistics
Several balance statistics can be computed by bal.compute() and bal.init(), and the ones available depend on whether the treatment is binary, multi-category, or continuous. These are explained below
and on the help page ?bal.compute. A complete list for a given treatment type can be requested using available.stats(). Some balance statistics are appended with ".mean", ".max", or ".rms", which
correspond to the mean (or L1-norm), maximum (or L-infinity norm), and root mean square (or L2-norm) of the absolute univariate balance statistic computed for each covariate.
smd.mean, smd.max, smd.rms
The mean, maximum, and root mean square of the absolute standardized mean differences computed for the covariates using col_w_smd(). The other allowable arguments include estimand (ATE, ATC, or ATT)
to select the estimand, focal to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and pairwise to select whether mean differences should
be computed between each pair of treatment groups or between each treatment group and the target group identified by estimand (default TRUE). Can be used with binary and multi-category treatments.
ks.mean, ks.max, ks.rms
The mean, maximum, or root-mean-squared Kolmogorov-Smirnov statistic, computed using col_w_ks(). The other allowable arguments include estimand (ATE, ATC, or ATT) to select the estimand, focal to
identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and pairwise to select whether statistics should be computed between each pair of
treatment groups or between each treatment group and the target group identified by estimand (default TRUE). Can be used with binary and multi-category treatments.
ovl.mean, ovl.max, ovl.rms
The mean, maximum, or root-mean-squared overlapping coefficient compliment, computed using col_w_ovl(). The other allowable arguments include estimand (ATE, ATC, or ATT) to select the estimand, focal
to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and pairwise to select whether statistics should be computed between each pair of
treatment groups or between each treatment group and the target group identified by estimand (default TRUE). Can be used with binary and multi-category treatments.
The Mahalanobis distance between the treatment group means, which is computed as \[ \sqrt{(\mathbf{\bar{x}}_1 - \mathbf{\bar{x}}_0) \Sigma^{-1} (\mathbf{\bar{x}}_1 - \mathbf{\bar{x}}_0)} \] where \(\
mathbf{\bar{x}}_1\) and \(\mathbf{\bar{x}}_0\) are the vectors of covariate means in the two treatment groups, \(\Sigma^-1\) is the (generalized) inverse of the covariance matrix of the covariates (
Franklin et al. 2014). This is similar to "smd.rms" but the covariates are standardized to remove correlations between and de-emphasize redundant covariates. The other allowable arguments include
estimand (ATE, ATC, or ATT) to select the estimand, which determines how the covariance matrix is calculated, and focal to identify the focal treatment group when the ATT is the estimand. Can only be
used with binary treatments.
The total energy distance between each treatment group and the target sample, which is a scalar measure of the similarity between two multivariate distributions. See Huling and Mak (2022) for
details. The other allowable arguments include estimand (ATE, ATC, or ATT) to select the estimand, focal to identify the focal treatment group when the ATT is the estimand and the treatment has more
than two categories, and improved to select whether the “improved” energy distance should be used, which emphasizes difference between treatment groups in addition to difference between each
treatment group and the target sample (default TRUE). Can be used with binary and multi-category treatments.
The kernel distance between treatment groups, which is a scalar measure of the similarity between two multivariate distributions. See Zhu, Savage, and Ghosh (2018) for details. Can only be used with
binary treatments.
The median L1 statistic computed across a random selection of possible coarsening of the data. See Iacus, King, and Porro (2011) for details. The other allowable arguments include l1.min.bin (default
2) and l1.max.bin default (12) to select the minimum and maximum number of bins with which to bin continuous variables and l1.n (default 101) to select the number of binnings used to select the
binning at the median. covs should be supplied without splitting factors into dummies to ensure the binning works correctly. Can be used with binary and multi-category treatments.
r2, r2.2, r2.3
The post-weighting \(R^2\) of a model for the treatment given the covariates. Franklin et al. (2014) describe a similar but less generalizable metric, the “post-matching c-statistic”. The other
allowable arguments include poly to add polynomial terms of the supplied order to the model and int (default FALSE) to add two-way interactions between covariates into the model. Using "r2.2" is a
shortcut to requesting squares, and using "r2.3" is a shortcut to requesting cubes. Can be used with binary and continuous treatments. For binary treatments, the McKelvey and Zavoina \(R^2\) from a
logistic regression is used; for continuous treatments, the \(R^2\) from a linear regression is used.
p.mean, p.max, p.rms
The mean, maximum, or root-mean-squared absolute Pearson correlation between the treatment and covariates, computed using col_w_corr(). Can only be used with continuous treatments.
s.mean, s.max, s.rms
The mean, maximum, or root-mean-squared absolute Spearman correlation between the treatment and covariates, computed using col_w_corr(). Can only be used with continuous treatments.
The distance covariance between the scaled covariates and treatment, which is a scalar measure of the independence of two possibly multivariate distributions. See Huling, Greifer, and Chen (2023) for
details. Can only be used with continuous treatments.
Choosing a balance statistic
Given all these options, how should one choose? There has been some research into which yields the best results (Franklin et al. 2014; Griffin et al. 2017; Stuart, Lee, and Leacy 2013; Belitser et
al. 2011; Parast et al. 2017), but the actual performance of each depends on the unique features of the data and system under study. For example, in the unlikely case that the true outcome model is
linear in the covariates, using the "smd" or "mahalanobis" statistics will work well for binary and multi-category treatments. In more realistic cases, though, every measure has its advantages and
For binary and multi-category treatments, only "energy.dist", "kernel.dist", and "L1.med" reflect balance on all features of the joint covariate distribution, whereas the others summarize across
balance statistics computed for each covariate ignoring the others. Similarly, for continuous treatments, only "distance.cov" reflects balance on all features of the joint covariate distribution.
Given these advantages, "energy.dist" and "distance.cov" are my preferences. That said, other measures are better studied, possibly more intuitive, and more familiar to a broad audience. | {"url":"https://mirror.ibcp.fr/pub/CRAN/web/packages/cobalt/vignettes/optimizing-balance.html","timestamp":"2024-11-08T08:21:04Z","content_type":"text/html","content_length":"83126","record_id":"<urn:uuid:bd81e27c-7a94-46e0-ae86-44129671a732>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00622.warc.gz"} |
Box Method Division Worksheets - Divisonworksheets.com
Box Method Division Worksheets
Box Method Division Worksheets – Divide worksheets are a great way to help your child review and practice division. There are a variety of worksheets that you can choose from and you might even
create your own. They are great as you can download them and modify them to your preferences. They’re perfect for second-graders, kindergarteners, and first-graders.
Two people can make huge numbers
The child should work on worksheets, while subdividing massive numbers. There are often just two, three or four divisors in the worksheets. This doesn’t need children to worry about forgetting how to
divide huge numbers, or making mistakes in times tables. To help your child develop this math skill you can either download worksheets or find them on the internet.
Multi-digit Division worksheets let children to practice their skills and build their knowledge. This is an essential skill for complex mathematical topics as well as everyday calculations. By
offering an interactive set of questions and exercises that are based on division of multi-digit integers these worksheets aid in establishing the idea.
The task of dividing huge numbers can be quite difficult for students. The worksheets typically employ a common algorithm with step-by-step instructions. The students may not gain the understanding
they need from this. Teaching long division can be done using basic 10 blocks. Once you have learned the steps, long division will appear natural to students.
Pupils can practice the division of large numbers with a variety practice questions and worksheets. These worksheets cover fractional calculations in decimals. Worksheets for hundredths are even
available, which can be helpful for learning how to divide large sums of money.
Sort the numbers into compact groups.
Putting a number into small groups can be a challenge. It might look great on paper but many participants of small groups dislike the process. It’s a natural manifestation of the way that the body
develops and is a great way to aid in the Kingdom’s continual development. It motivates others, and motivates them to help those who are forgotten.
It is useful for brainstorming. You can create groups of people with similar qualities and experiences. This will allow you to think of new ideas. Once you’ve created groups, you’ll be able to
introduce yourself to all of them. It’s a good way to stimulate creativity and encourage innovative thinking.
The fundamental arithmetic operation of division is used to divide big amounts into smaller numbers. When you want to create an equal amount of things for several groups, it can be beneficial. A good
example is the large class which can be broken down into five groups. These groups can be added together to make the original 30 pupils.
If you are dividing numbers, there are two kinds of numbers you need to keep in mind: the divisor or the quotient. Dividing a number by another produces “ten/five,” whereas dividing two numbers with
two gives the exact same result.
Powers of ten should only be employed to solve huge numbers.
Splitting massive numbers into powers can help you compare them. Decimals are a standard element of the shopping process. They can be found on receipts as well as on price tags and food labels. To
display the price per gallon and the amount of gas that has been dispensed through a nozzle pumps use decimals.
There are two methods to divide huge numbers into powers of 10. The first is by shifting the decimal to the left, and then multiplying it by 10-1. The second option is to use the power of ten’s
association feature. Once you’ve learned the associative property of powers of 10, you are able to divide a large number into smaller powers that are equal to ten.
The first one involves mental computation. The pattern will be evident by dividing 2.5 times 10. The decimal point moves to the left as the power of 10 grows. This concept can be applied to tackle
any problem.
The second option involves mentally dividing massive numbers into powers of 10. Then, you may quickly write extremely large numbers using scientific notation. In scientific notation, large numbers
should be written in positive exponents. It is possible to shift the decimal point by five spaces on one side and then convert 450,000 into 4.5. You can either divide the large number into smaller
powers than 10, or split it into smaller powers of 10.
Gallery of Box Method Division Worksheets
Box Division Interactive Worksheet
Pin By Maureen C On Math Math Methods Teaching Math Learning Math
8 Helpful Tips For Teaching Long Division Krejci Creations In 2021
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multiple barplots in ggplot
# Thanks @woodward for the info. Hi, I want to draw a barplot with basic ggplot but I would like to color the bar in the plot (so genes) that have a specific value (e.g. ... (2, 2)) ggplot2 -
Multiple Plots. To arrange multiple ggplot2 graphs on the same page, the standard R functions – par() and layout() – cannot be used.. In this example, I construct the ggplot from a long data format.
And we will use gapminder data to make barplots and reorder the bars in both ascending and descending orders. The bar geometry defaults to counting values to make a … June 20, 2019, 6:36pm #1. It can
be difficult for a beginner to tie all this information together. There are two types of bar charts: geom_bar() and geom_col(). The width is the width of the “groups”. A simple plot: Customers per
Year. Dear All, I would like to plot a multiple bar plot as shown below against the same y axis. I produced plots close to my wish. Now, we can create two ggplots with the following R code: ggp1 <-
ggplot(data1, aes(x = x)) + # Create first plot # - cols: Number of columns in layout I would like to plot four barplots on a single graph in R. I have used the following code. This can be achieved
in this way. Frist, I have to tell ggplot what data frames and how the columns of the data frames are mapped onto the graph. © Copyright Statistics Globe – Legal Notice & Privacy Policy. data1 and
data2) contains the values for one plot. grid.arrange() and arrangeGrob() to arrange multiple ggplots on one page; marrangeGrob() for arranging multiple ggplots over multiple pages. In R, ggplot2
package offers multiple options to visualize such grouped boxplots. First, we need to create some example data for the creation of our plots. We can colors to the barplot in a few ways. If present,
'cols' is ignored. This function is from easyGgplot2 package. Now i have all legend bars in each plot, can i just have one legend based bar in each plot. Note that dose is a numeric column here; in
some situations it may be useful to convert it to a factor.First, it is necessary to summarize the data. In order to print several ggplot graphs side-by-side, we need to install and load the
gridExtra R package: install.packages("gridExtra") # Install gridExtra package Add multiple labels on ggplot2 boxplot, You should probably save the aggregate data on a separate object for clarity and
use position=position_dodge() in the geom_text() options: Add text labels with ggplot2 This document is dedicated to text annotation with ggplot2 . I hate spam & you may opt out anytime: Privacy
Policy. We can use this function to return our two example plots within the same plot window: grid.arrange(ggp1, ggp2, ncol = 2) # Apply grid.arrange function. Now let’s create these plots… We could
also specify ncol = 1 to return the two plots above each other. In this R programming tutorial you’ll learn how to draw multiple ggplots side-by-side. Arrange Plots Using the layout Function in R,
Set Legend Alpha of ggplot2 Plot in R (Example), Remove Legend in ggplot2 (3 Example Codes) | Delete One or All Legends, ggplot2 Error in R: Must be Data Frame not S3 Object with Class Uneval, Remove
NA Values from ggplot2 Plot in R (Example), Introduction to ggpattern Package in R (6 Examples) | ggplot2 Plots with Textures. One of the reasons you’d see a bar plot made with ggplot2 with no
ascending / descending order - ordering / arranged is because, By default, ggplot arranges bars in a bar plot alphabetically.. Barchart with Colored Bars. We can supply a vector or matrix to this
function. Note that we could store any type of graphic or plot in these data objects. However, ggplot2 is incredibly extensive in its customization capabilities. for example, just appearance based
bars in H1 plot, then just aroma based bars in second plot, and flavor, overall, aftertaste and texture. Get regular updates on the latest tutorials, offers & news at Statistics Globe. The boxplots
and barplots are created in single window basically creating a multi panel plots. With mosaic diagrams, the dimensions on both the x and y axis vary in order to reflect the different proportions.
Ggplot2 : Plotting Multiple Bar Against Same Y Axis. Before we can create plots with the ggplot2 package, we need to install and load the package to R: install.packages("ggplot2") # Install ggplot2
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Stem profile of red oaks in a bottomland hardwood restoration plantation forest in the Arkansas Delta (USA)
iForest - Biogeosciences and Forestry, Volume 15, Issue 3, Pages 179-186 (2022)
doi: https://doi.org/10.3832/ifor4057-015
Published: May 19, 2022 - Copyright © 2022 SISEF
Research Articles
Bottomland hardwoods are among the most diverse and productive forest ecosystems in the southeastern United States and are critically important for the provision of timber and non-timber ecosystem
services. Red oaks, the dominant species in this group of forests, are of high ecological and economic value. Stem profile models are essential for accurately estimating the merchantable volume of
oak trees, which is also closely indicative of total tree biomass and other ecosystem services given their allometric relationships. This study aims to develop and compare stem profiles among three
red oak species in an 18-year old plantation forest using destructive sampling. Sixty trees randomly selected from an oak restoration plantation in the Arkansas Delta were felled for measuring the
diameter-outside-bark (DOB) and diameter-inside-bark (DIB) at different stem heights. These sample composed of twenty trees from each of three species: cherry bark oak (CBO - Quercus pagoda Raf),
Nuttall oak (NUT - Quercus texana Buckley), and Shumard oak (SHU - Quercus shumardii Buckl). Multiple models, including the segmented-profile model, form-class profile model, and second-and
third-order polynomial models were fitted and compared. Results demonstrate that the form-class profile model was the best fitted for CBO and NUT, whereas the third-order polynomial model was the
best for SHU. CBO tends to grow taller and has a higher wood density than NUT and SHU. These findings will inform restoration and management decisions of bottomland hardwood forests, especially red
oaks in the region.
Cherry Bark Oak, Nuttall Oak, Shumard Oak, Taper Models, Wood Density, Southeastern United States
Bottomland hardwoods are among the most diverse and productive forest ecosystems in the southeastern US. They are critically important for providing many benefits including timber, water regulation,
wildlife habitats and biodiversity, natural sceneries, and carbon sequestration and storage, among others ([38], [18], [9]). This group of forests once covered nearly 12 million ha in the region;
however, sixty percent of bottomland hardwood forests have been lost in the past two hundred years mainly due to agricultural expansion ([9]). Extensive efforts have been undertaken to restore
bottomland hardwood forests in recent decades, mainly through plantations on cropland that was originally converted from forests ([2]). Among the most important tree species in this ecosystem are
oaks, particularly red oaks ([2], [18], [23]), which provide food for a variety of wildlife and lumber and veneer for the production of high quality furniture, flooring, and other products. Yet, the
availability of accurate growth and yield models for the forests in general and oaks in particular is limited compared to other commercial species like pines ([35]). Information on young oak stand
development is particularly limited. Such limitations have hindered the development and implementation of effective forest restoration and management decisions to enhance the provisions of timber and
non-timber ecosystem services from this group of forests ([25], [36]).
Stem profile studies can help improve the accuracy of growth and yield estimations and projections while providing valuable information about the characteristics of trees and forests. Tree profile
equations estimated statistically using empirical data predict the diameter at various heights along the bole of a tree ([28], [32]), improving volume and carbon estimates. Profile equations allow
for predicting volumes to merchantable top diameters, a fundamental basis for forest managers to compute volumes to any top diameter as well as diameter/height at a specific height/diameter along the
stem. However, there is a lack of published profile equations for oaks in the hardwood forests. Given the ecological importance of oaks in the forests and the high value of oak wood products, the
development of profile equations will aid in their volume and value estimation.
Profile modeling is an active area of research in forestry given the flexibility and utility afforded by the various existing models ([45], [42]). Taper equations represent the mathematical
expression of stem diameter change with respect to height on the basis of species, age, and stand condition ([16], [4]). A number of taper functions have been developed for different tree species
with various forms including simple taper functions, variable-form taper functions, and segmented polynomial taper functions ([28], [17]). Simple taper functions mainly define tree profiles with a
single continuous equation for the whole bole ([5], [19], [31], [15], [11], [47]). Though simple, they were unable to precisely describe the whole bole profile especially the base of the stem ([17]).
By contrast, the variable-form taper functions assume that the stem form/shape continuously changes with the height ([21]); therefore, those equations describe the stem profile with an exponent
variable to characterize the neiloid, paraboloid, and conic ([30], [20]). The advantages of the variable-form taper functions are that they describe the stem form through a continuous function and
can predict/estimate the upper-stem diameters more accurately. However, the disadvantage of these functions are their inability to be logically associated to estimating total volumes.
Segmented polynomial taper functions describe stem shape through fitting a distinct equation for each segment and then binding multiple equations together to produce and overall segmented structure
estimation ([27], [6], [8], [17]). Max & Burkhart ([27]) developed a well-known profile equation for loblolly pine (Pinus taeda) involving a segmented polynomial regression approach, which delineates
the bole of a tree into three sections based on the generalized relative geometric form. This approach is based on the assumption that in mathematical evaluation a segmented model will best
approximate the stem form. A segmented polynomial profile model can be created by grafting three sub-models at two joint points referred to as inflection points ([40]). Similar to Max & Burkhart (
[27]) model, Cao ([7]) segmented profile model also consists of three sub-models that are joined at two points; each sub-model in Cao ([7]) has a form of the modified Goulding & Murray ([13]) model,
which can be inverted to predict height and volume. Clark et al. ([8]) proposed a form-class segmented profile model by incorporating the form quotient of the Girard form class to produce a more
robust model. Their model assesses stem structure partially based on the relative measurements involved with the quotient. They compared their model with Max & Burkhart ([27]) segmented polynomial
equation and found that the form-class segmented model provides more accurate volume estimations. The segmented-profile models including form-class segmented profile models were commonly employed in
fitting the stem of hardwood species. For example, Jiang et al. ([17]) employed the segmented-profile model by Max & Burkhart ([27]) and Clark et al. ([8]) in fitting the stem of yellow-poplar; Tian
et al. ([42]) fitted the invasive tallow stem using both Max & Burkhart ([27]) and Cao ([7]) as well as Clark et al. ([8]). In addition, Alkan & Ozçelik ([1]) tested nine taper equations for stem
profile and volume of Taurus fir in Taurus Mountains of Turkey and found that among segmented taper models, the Fang et al. ([10]) model was the most accurate taper equation for estimating diameter,
predicting merchantable, and total volume. Moreover, this model does not require numerical integration like the variable-form equations for volume estimations. Parker ([34]) suggested the use of a
non-segmented, third degree polynomial for small data sets with limited diameter and height ranges, and developed a computer program for calculating both non-segmented and single or multiple
segmented tree profile equations. His equations were based on non-destructive measurements of upper stem diameters of standing trees taken with an optical dendrometer. Overall, segmented polynomial
functions composed of a series of sub-models representing various sections of the stem perform better than simple or variable form taper functions and are widely used ([32]).
The main objective of this study is to develop and compare the stem profile models for three oak species planted in the Arkansas Delta, located in the lower Mississippi River alluvial valley (LMRAV).
Hardwood forests in the LMRAV are a typical example of bottomland hardwood forests in the southeastern US in terms of their ecological features, conversion to croplands, and restoration efforts ([24]
, [2]). The three oak species are cherry bark oak (CBO - Quercus pagoda Raf), Nuttall oak (NUT - Q. texana Buckley), and Shumard oak (SHU - Q. shumardii Buckl). Different profile models for each
species will be estimated and compared. Additionally, wood density will also be measured and compared among the three oak species, indicative of the physical property and carbon content of the oak
wood. As natural and primary hardwood forests become increasingly scarce and are protected for conservation purposes, a dominant amount of hardwood timber has come from second growth and plantations.
Plantations will become increasingly important in timber supply in the near future. Additionally, stem profile equations are fundamental to quantifying tree biomass which is closely related to the
provisions of non-timber ecosystem services. Hence, the results of this study will inform decision making in oak plantation design and management to restore the bottomland hardwood forests for both
timber and non-timber objectives while enriching the existing literature in oak stem profile, volume, and wood density.
Study site
The data for this study were collected from an oak restoration plantation established at the University of Arkansas Division of Agriculture Pine Tree Research Station (AgPTRS) in the Arkansas Delta
(35° 07′ 48″ N, 90° 55′ 45″ W). The 12.15 ha hardwood plantation was on a prior flood prone and low productive bean field that was enrolled in the Conservation Reserve Program. The oak trees involved
in this study were planted in February 2004 after the site was ripped to a depth of 0.91 m in November 2003. Seedlings were hand planted at a 3.05 × 3.05 m spacing in 12 plots with 4 replications.
Each of the three red oak species (cherry bark oak, CBO - Quercus pagoda Raf; Nuttall oak, NUT - Quercus texana Buckley; and Shumard oak, SHU - Quercus shumardii Buckl) were randomly assigned to
68.58 × 68.58 m plots. (Fig. 1). No further management was applied to the plots in the 18-year interval between planting and data collection for this study.
Data collection
Destructive sampling procedures are a standard practice in tree profile data collection because stem diameters including diameter-outside-bark (DOB) and diameter-inside-bark (DIB) can easily be
measured on felled trees. In addition, stem profile data measured along a felled tree provides reliable data, representing one of the most accurate methods for volume quantification. We employed
destructive sampling for accurate and reliable profile data measurements of diameter and height value pairs along the entire length of the tree stem.
A total of 60 oak trees were randomly selected (20 for each species of CBO, NUT, and SHU) equally from all each of the four replications in the study (5 trees from each replication). Prior to
felling, total height of the standing sample trees was measured with a clinometer; meanwhile, diameter at breast height (DBH) were also measured for all the 60 selected trees. They were then felled
at approximately 0.15 m from the ground, and the bole was cleared of branches to facilitate measurements. Merchantable stem length was recorded for each tree by running a tape measurement along the
length of the main apical branch until a 10.16 cm DOB was reached. The stem was divided into 1.83 m sections in the field and disks with 2.5-5.0 cm in thickness were then extracted from the midpoint
of each section; DOB and DIB were measured at the lower and upper ends of each section. These collected disks were marked with species, tree number as well as disk number. They were sealed in
pre-weighed and marked plastic bags to preserve their green weight. In addition, both DOB and DIB at a selected height (~5.27 m) was also recorded. As a result, DOB and DIB were recorded along the
bole at relative heights of 0.15 m, 1.37 m, 1.83 m, 3.66 m, 5.27 m, 5.49 m, 7.32 m etc. until the point of a 10.16 cm diameter outside bark was reached. At each recorded height along the felled stem,
both DOB and DIB were measured twice in two different directions for accuracy.
The tree disks were returned to the laboratory and weighed in the pre-weighed bags to determine green weight. The disks were then oven dried at 80 °C and reweighed to determine moisture content. A
wedge of wood was then separated from the oven dry disk and the weight of that wedge was recorded. The volume of the wedge was then determined by the displacement method in water at 3 °C after
coating the wedge in paraffin wax. The wood density was computed using the dry weight of the wedge divided by the volume of the same wedge, denoted by grams per cubic centimeter (g cm^-3).
Profile models
A total of seven profile models were fitted in this study including the segmented polynomial models of Max & Burkhart ([27]), Cao ([7]), Clark et al. ([8]) and optimized Cao ([7]); and the form-class
models of second- and third-order polynomial equations. To be specific, those seven profile models were: (MB) Max & Burkhart ([27]); (CAO) Cao ([7]); (CSS) Clark et al. ([8]), optimized Cao ([7]);
(OCAO_F) modified MB to pass through DBH and FC at FC height, optimized Cao ([7]); (OCAO_R) modified MB to pass through DBH and FC at relative height; (POLY3) 3^rd-order polynomial joined at breast
height to 5th order polynomial; and (POLY2) 2^nd-order polynomial joined at breast height to 5^th-order polynomial. Specific mathematic expressions of these models are shown in the equations below.
The notations associated with the seven taper equations are listed in Tab. 1.
Tab. 1 - Notations associated with stem profile models.
Notations Definition
TH The total height of a tree
h [D] Breast height (1.37 m)
D Diameter at breast height (DBH)
d Diameter at specific height of h, including diameter-outside-bark (DOB) and diameter-inside-bark (DIB)
h Height above the ground to the measurement point
d ^* Calibrated diameter
F diameter at 5.27 m, including DOB and DIB
The widely used MB profile function is expressed in eqn. 1 and b[1], b[2], b[3], b[4], α, and β are parameters to be estimated, α and β represent the joint points as showed in the constraint
conditions function. This segmented polynomial model describes a tree stem with three sections (eqn. 1):
$$\frac{d}{D} = \left [ b_1 (S -1) + b_2 (S_2 -1) + b_3 (\alpha- S)^2 I_1 + b_4 (\beta- S)^2 I_2 \right ]^{0.5}$$
where S = (h/TH) and S2 = (h)^2/(TH)^2. The constraint conditions for this model are (eqn. 2):
$$\cases { \text{if}\; h=H & \text{then}\; d=0 \\ \text{if}\; h=h_{D} & \text{then}\; d=D \\ \text{if}\; S \le \alpha &\text{then}\; I_1 =1 \;\text{else}\; I_1 =0 \\ \text{if}\; S \le \beta & \text
{then}\; I_2 =1\; \text{else}\; I_2 =0 }$$
The profile model of CAO including the optimized OCAO_F and OCAO_R ([7]) is shown in eqn. 3-5 with parameter b[1] being modified to parameter b[1]^* for OCAO_R (eqn. 3, eqn. 4, eqn. 5):
$$y^{\text{*}} (1-S) = b_1^{ \text{*}} (1- S) + b_2 (1- S)^2 +b_3 I_1 (1- S - \alpha)^2 + b_4 I_2 (1- S -\beta)^2$$
$$y^{ \text{*}} = \frac{(d^{ \text{*}})^2} {D^2}$$
$$b_1^{ \text{*}} = \frac{y (1- S_3) - \hat{y} (1- S_3) + b_1 (1- S_3)}{1- S_3}$$
where S[3] = 4.5/TH.
The segmented model of CSS describes the stem with four sections composed of the butt section (from stump to breast height of 1.37 m), lower stem (between 1.37 m and 5.27 m), middle stem (from 5.27 m
to 40-70% of total height), and upper stem (from 40% to 70% of total height to the tip of the tree - [17], [33]). The mathematic expression is presented in eqn. 9, where b[1], b[2], b[3] are the
parameters for the butt section; b[4] is the parameter for the lower stem; b[5], b[6] are the coefficients for height above 5.27 m; Z[1] = 1-(h/TH) = 1 - S; Z[2] = 1-(4.5/TH) = 1 - S[3], Z[3] = (h
$$\omega_1 = D^2 \left [1+{\frac{ \left (b_2+\frac{b_3}{D^3} \right ) \left (Z_1^{b_1} - Z_2^{b_1} \right )}{1-Z_2^{b_1}}} \right ]$$
$$\omega_2 = D^2-\frac{(D^2 - F^2) (Z_2^{b_4} - Z_1^{b_4})} { Z_2^{b_4} - \left (1-\frac{17.3}{TH} \right)^{b_4} }$$
$$\omega_3 = \alpha (Z_3 - 1)^2 +I_{M} {\frac{1-\alpha}{b_5^2}} (b_5 - Z_3)^2$$
$$d=(I_{s} \omega_1 +I_{B} \omega_2+I_{T} F^2 \omega_3)^{0.5}$$
Hence, the constraints for this model are (eqn. 10):
$$\cases{ \text{if} \; h \le 1.37 & \text{then} \; I_{S} =1 \; \text{else} \; I_{S} =0\\ \text{if} \; 1.37 \le h \le 5.27 & \text{then} \; I_{B} =1 \; \text{else} \; I_{B} =0\\ \text{if} \; 5.27 \le
h & \text{then} \; I_{T} =1 \; \text{else} \; I_{T} =0\\ \text{if} \; h \le \left (5.27+ b_5 \left (H-5.27 \right ) \right ) & \text{then} \; I_{M} =1 \; \text{else } \;I_{M} =0 }$$
By contrast, the form-class models of second- and third-order polynomial models are shown in eqn. 11 and eqn. 12, respectively:
$$d= b_{1} + b_{2} {h} ^ {2} + b_{3} h+ b_{4}$$
$$d= b_{1} + b_{2} {h} ^ {3} + b_{3} {h} ^ {2} + b_{4} h+ b_{5}$$
where b[1], b[2], b[3], b[4], b[5] are the parameters to be estimated.
Statistical analysis
To test for differences in DBH and total height of CBO, NUT, and SHU at a 0.05 significance level, analysis of variance (ANOVA) was employed.
The measurements of DOB, DIB, and height pairs provided the inputs for TProfile ([26]) to estimate the parameters of each profile equation described above. All profile equations fitted in this study
were calculated using Tprofile which can fit commonly used profile functions for a variety of hardwood species throughout the Southeast US ([26], [12]). One potential problem in fitting the stem
profile model is autocorrelation ([43], [46]); to address this concern, Durbin-Watson test for both DOB and DIB data of three oak species were performed. The test statistic values were all around 1.4
/1.5 (p-value ~ 0.09), indicating that autocorrelation was not a major concern in the data.
Those fitted equations were evaluated and compared using mean absolute error (MAE), root mean square error (RMSE), and the index of fit (FI). Specifically, the MAE (eqn. 13) was computed along the
bole as the average absolute difference between observed and predicted diameter values. The RMSE (eqn. 14) was calculated by the square root of the average of the absolute value of the squared biases
whereas FI (eqn. 15) was calculated as one minus the sum of squared errors divided by the total sum of squares.
$$MAE= \frac{\sum_{i=1}^{n} | Y_{i} - \hat{Y_i} | } {n}$$
$$RMSE= \sqrt {\frac{\sum_{i=1}^{n} {{(Y_{i} - \hat {Y_{i}} )} ^ {2}}}{n}}$$
$$FI=1- \frac{\sum_{i=1}^{n} {{ \left (Y_{i} - { \hat {Y}}_{i} \right )} ^ {2}}} {\sum_{i=1}^{n} {{ \left (Y_{i} - \bar {Y} \right )} ^ {2}}}$$
where Y[i] denotes the observations/measurements, Ŷ[i] represents the predictions, bar{Y} is the mean of Y[i], and n is the number of observations in the dataset.
Furthermore, we plotted the residuals (computed by observed values minus predicted values) of DOB and DIB against relative height to diagnose how well each model predicted the stem profile data.
Together with the computed RMSE and FI values, we determined the best fit profile equation for each oak species.
Descriptive statistics for three oak species
The descriptive statistics of DBH, total height, and wood density for all three oak species of CBO, NUT, and SHU are summarized in Tab. 2. Average DBH for CBO, NUT, and SHU were 18.44 cm, 16.51 cm,
and 16.61 cm, respectively. Total height, on average, was 17.75 m, 15.56 m, and 13.58 m respectively for CBO, NUT, and SHU. The results of ANOVA tests indicated that there was no distinct difference
in DBH among CBO, NUT, and SHU; however, a significant difference was found in total height among them. To be specific, CBO was significantly taller than NUT which was taller than SHU, though they
were planted in a similar environment at the same time. NUT and SHU have a similar wood density, which is significantly lower than that of CBO.
Tab. 2 - Descriptive statistics of DBH and total tree height for CBO, SHU, and NUT. Different letters in each column indicate statistically different values (p<0.05). (CBO): cherry bark oak (Quercus
pagoda Raf); (NUT): Nuttall oak (Quercus texana Buckley); (SHU): Shumard oak (Quercus shumardii Buckl); (n): sample size; (SD) standard deviation.
Oak species (n) Statistics DBH Total Height Wood density
(cm) (m) (g cm^-3)
Average 18.44^a 17.75^a 0.73^a
CBO Min 13.72 15.24 0.65
Max 23.11 19.81 0.86
SD 2.67 1.09 0.05
Average 16.51^a 15.56^b 0.68^b
NUT Min 11.68 12.19 0.49
Max 20.60 18.29 0.83
SD 2.59 1.48 0.07
Average 16.61^a 13.58^c 0.68^b
SHU Min 11.05 11.28 0.59
Max 26.67 16.46 0.77
SD 4.39 1.34 0.04
Profile equation parameters and comparisons
Tprofile estimated parameters for all seven profile models for the three oak species. Tab. S1 (Supplementary material) summarizes parameter estimations of all fitted profile models for both DOB and
DIB of each oak species and the number of observations for each model. According to the MAE (lower), RMSE (lower), and FI (higher - Tab. 3) along with graphical evaluation of residual plots (Fig.
S1a-S1c in Supplementary material), the CSS profile model was the best fitted for both CBO and NUT; however, the third-order polynomial model (POLY3) was the best fitted for SHU, especially in terms
of DIB. Specifically, the FI, MAE, and RMSE for both DOB and DIB of CBO were FI[OB/IB ]= 0.987, MAE[OB/IB] = 0.025, and RMSE[OB/IB] = 0.094 with the CSS model; whereas for NUT with the CSS model, the
FI, MAE, and RMSE were estimated at FI[OB] = 0.973, MAE[OB] = 0.033, and RMSE[OB] = 0.145 for DOB and at FI[IB] = 0.967, MAE[IB] = 0.033, and RMSE[IB] = 0.150 for DIB, respectively. By contrast, the
FI, MAE, and RMSE for SHU with the POLY3 model were estimated at FI[OB] = 0.976, MAE[OB] = 0.041, and RMSE[OB] = 0.140 for DOB, and FI[IB] = 0.968, MAE[IB] = 0.064, and RMSE[IB] = 0.152 for DIB. To
be noted, the MAE of CSS for both DOB and DIB profile of CBO species was slightly higher than the MB model (MAE[MB_OB] = 0.023, MAE[MB_IB] = 0.015) while it was slightly higher than the CAO model
(MAE[CAO_OB] = 0.028) for DOB profile of NUT. In addition, there was no difference in FI and RMSE for SHU between the POLY3 and CSS models for DOB. Yet, there was a slight difference in FI and RMSE
for the DIB profile of SHU between the two models with FI[IB] = 0.966 and RMSE[IB] = 0.155 for the CSS model. Tab. S1 (Supplementary material) summarizes the parameters estimated for all seven models
for both DOB and DIB profiles together with the total number of observations utilized in the model fitting; Tab. 3summarized all the statistics of model evaluation.
Tab. 3 - Model evaluation statistics of MAE, EMSE, and FI of seven profile models for oak species of CBO, NUT, and SHU. (CBO): cherry bark oak (Quercus pagoda Raf); (NUT): Nuttall oak (Quercus texana
Buckley); (SHU): Shumard oak (Quercus shumardii Buckl); (MB): Max & Burkhart ([27]); (CAO); Cao ([7]); (CSS) Clark et al. ([8]); (OCAO_F): optimized Cao ([7]) modified MB to pass through DBH and FC
at FC height; (OCAO_R): optimized Cao ([7]) modified MB to pass through DBH and FC at relative height; (POLY3): 3^rd-order polynomial joined at breast height to 5^th-order polynomial; (POLY2): 2^
nd-order polynomial joined at breast height to 5^th-order polynomial; (DIB): diameter-inside-bark; (DOB): diameter-inside-bark.
CBO NUT SHU
Model Oak species Statistics
DOB DIB DOB DIB DOB DIB
RMSE 0.117 0.109 0.234 0.229 0.183 0.188
MB MAE 0.023 0.015 0.033 0.036 0.089 0.089
Index of Fit 0.979 0.979 0.930 0.925 0.958 0.951
RMSE 0.135 0.122 0.224 0.244 0.196 0.028
CAO MAE 0.071 0.056 0.028 0.117 0.140 0.254
Index of Fit 0.973 0.975 0.934 0.914 0.953 0.891
RMSE 0.094 0.094 0.145 0.150 0.140 0.155
CSS MAE 0.025 0.025 0.033 0.033 0.089 0.091
Index of Fit 0.987 0.987 0.973 0.967 0.976 0.966
RMSE 0.109 0.104 0.201 0.236 0.157 0.221
OCAO_F MAE 0.051 0.043 0.074 0.178 0.079 0.157
Index of Fit 0.982 0.981 0.947 0.920 0.970 0.932
RMSE 0.099 0.086 0.173 0.191 0.140 0.208
OCAO_R MAE 0.036 0.028 0.048 0.071 0.081 0.160
Index of Fit 0.985 0.986 0.958 0.945 0.974 0.934
RMSE 0.102 0.104 0.191 0.229 0.140 0.152
POLY3 MAE 0.041 0.030 0.036 0.074 0.041 0.064
Index of Fit 0.984 0.984 0.953 0.940 0.976 0.968
RMSE 0.104 0.109 0.188 0.206 0.140 0.152
POLY2 MAE 0.043 0.036 0.036 0.079 0.046 0.064
Index of Fit 0.984 0.982 0.953 0.939 0.976 0.967
In addition, model performance was further evaluated using average residual plots for all seven models of three oak species (see Fig. S1-S3 in Supplementary material). In these plots, the y-axis
represents the average residues and x-axis represents the relative height, h/TH. By comparing seven average residual curves for each species, we found that the best fitted model for both CBO and NUT
was the CSS model while POLY3 is the best for SHU.
Finally, graphical profile comparisons were also made to visualize the difference between the average observed and predicted stem forms with 0.1 relative height interval (Fig. 2, Fig. 3, Fig. 4). We
constructed the stem profile curves for both DOB and DIB for each species using its best fitted profile equations and the measurement data. Those curves also demonstrated that there was a good fit
between the observed and predicted stem diameter values along the stem, which in turn confirmed that those profile equations were able to accurately predict the stem profile.
Fig. 2 - DOB and DIB profiles for cherry bark oak (CBO) using the best fitted model of CSS ([8]).
Fig. 3 - DOB and DIB profiles for Nuttall oak (NUT) using the best fitted model of CSS ([8]).
Fig. 4 - DOB and DIB profiles for Shumard oak (SHU) using the best fitted model of POLY3 (3^rd-order polynomial joined at breast height to 5^th-order polynomial).
As explained by Avery & Burkhart ([3]), numerous factors could affect the performance of profile equations, including species, region, number of sample trees, data collection method among others.
Those factors were all carefully considered and taken into account in the development of oak profile equations in this study. As reported by Saarinen et al. ([37]), the sample size of destructive
stem analysis could affect parametrizing the stem profile equations. A total sample of 60 oak trees, with 20 trees for each species, is considered to be an appropriate sample size for this study
given that all oak trees were felled on the same even-aged restoration plantation site. However, since the data comes from a single site, the effect of geographical variability is not addressed.
Variables including total height, DBH, various heights along the stem, and DOB and DIB at different heights were measured using the destructive sampling method. This procedure provided reliable data
for fitting the oak profile models. Seven different profile models were estimated and compared for each oak species. Multiple indicators including RMSE, index of fit, and residual plots were used to
diagnose the fitted models, representing a statistically sound approach for model evaluation.
Our results show that the best fitted model was CSS for both CBO and NUT, whereas it was POLY3 for SHU. Although no difference was found between CSS and POLY3 for fitting the DOB profile of SHU, for
the DIB profile of SHU the FI of the CSS model was slightly lower than that of POLY3, while the RMSE of the CSS model was slightly higher than that of POLY3. As reported by Muhairwe et al. ([29]),
species could be an influencing factor for stem form variation. The MB model did not exhibit better performance than CSS. This may be partly because the MB model is constrained by a lack of
representation of upper stem diameter, which is evidenced by an insufficient number of observations in the upper crown area. Moreover, the most important advantage of Clark et al. ([8]) is using
extra upper diameter measurement at 5.27 m which also contributed to the model performance. Leites & Robinson ([22]) reported that crown dimension is another potential factor that contributes to the
differences of stem profile. Therefore, future work can incorporate crown length and crown ratio into the profile models. In addition, utilization of modern regression techniques such as the
mixed-effects modeling approach can also improve the profile model performance. For example, Gómez-García et al. ([14]) developed a mixed-effect variable exponent taper equation for birch trees in
northwestern Spain which produced the best fitting statistics. The profile equations that express mathematical relationships between tree height and diameter are essential for quantifying stem volume
as well as total tree biomass and carbon content. Based on the profile equations, stem volume can then be calculated ([44]). With properly estimated allometric relationships between the volumes of
stem and other parts of the tree, total tree biomass and carbon content can also be derived. Besides, the similarities and differences in DBH and total heights among the three species can aid in
selecting a proper species mix to achieve different management objectives. For example, Cherry bark oak is a better species choice than Nuttall or Shumard oak if the management objective is for
timber, biomass, or carbon because cherry bark oak grows taller and has a higher wood density than Nuttall or Shumard oak, although they have similar DBH at the same age ([41]). However, because of
their different heights, the mix of the three species can create more diverse stand structure and enhance biodiversity. Thus, our results provide a base-station for guiding the restoration and
management of bottomland hardwood plantation forests to achieve both timber and non-timber objectives.
We estimated stem profile models for three red oak species (cherry bark, Nuttall, and Shumard oaks) with data collected from 60 felled trees randomly selected from a plantation in the Arkansas Delta
in the southern United States. We found the best fitted model was Clark et al. ([8]) profile equation for cheery bark and Nuttall oaks and the third-order polynomial model for Shumard oak.
Additionally, there is no statistically significant difference in DBH among the three oak species while cherry bark oak is the tallest and has the highest wood density of the three species. The
fitted profile equations in this study can be implemented into inventory software such as TCruise ([26]) to improve the stem volume estimates of young oak trees in Arkansas Mississippi Delta. Given
that an increasing amount of hardwood timber is supplied from second growth and plantations, our findings will guide decision making in red oaks restoration and management and their wood product
marketing. Moreover, our estimated stem profile models and wood density can help quantify total tree biomass and associated carbon content. This, however, will entail the development of allometric
relationships between the volumes or biomass quantities of stem and other parts of trees such as branches, leaves, and roots. Future studies on this site will determine if the stem profiles change
over time as this plantation matures.
A few caveats of this study should be noted. First, while the results of fitted profile models provide a guideline for estimating the stem volume of oak species, we do not include comparisons with
oak volume models given that the focus of this study was to compare stem profiles among different oak species grown on the same site instead of volume prediction. Second, the data were collected from
an even-aged plantation and could not symbolize to other natural/planted mixed species and uneven-aged bottomland hardwood forests in this region, further study is needed to assess the viability of
widespread application.
We are thankful for the support provided by the College of Forestry, Agriculture & Natural Resources at the University of Arkansas at Monticello and the Arkansas Center for Forest Business in
completing this study. We also thank Ouname Mhotsha for helping data collection, and Ruxin Tao for his suggestion in making the study site map. Additionally, we are grateful for all field supports
including technical and equipment support provided by the Division of Agriculture Pine Tree Research Station.
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Paper Info
Tian N, Gan J, Pelkki M (2022). Stem profile of red oaks in a bottomland hardwood restoration plantation forest in the Arkansas Delta (USA). iForest 15: 179-186. - doi: 10.3832/ifor4057-015
Academic Editor
Agostino Ferrara
Paper history
Received: Jan 05, 2022
Accepted: Apr 05, 2022
First online: May 19, 2022
Publication Date: Jun 30, 2022
Publication Time: 1.47 months
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Multiply 3-digit by 1-digit Worksheets (answers, printable, online, grade 3)
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IB Math IA: Modelling a Skateboard Ramp
Mr. Flynn IB
24 Apr 202115:39
TLDRThis video tutorial demonstrates how to model skateboard ramps using GeoGebra, a mathematical software. It guides viewers through the process of importing a photo, scaling it accurately, and
fitting mathematical functions like lines or curves to represent the ramp's shape. The video also explores different ramp models, including straight lines, parabolas, and exponential functions, and
discusses the importance of calculating gradients for understanding ramp steepness. The host encourages viewers to apply these techniques to any photo for various applications beyond skateboarding.
• 🛹 The video is about using GeoGebra to model skateboard ramps for an IB Math IA project.
• 📐 The process involves importing a photo of a ramp into GeoGebra and using it as a reference for modeling.
• 📏 It's important to measure the dimensions of the ramp to ensure the model is to scale.
• 📉 The video demonstrates how to fit a straight line to represent a simple ramp and adjust parameters for accuracy.
• 📈 The concept of gradients and derivatives is introduced, showing how to calculate the gradient of ramps using calculus.
• 🔍 The presenter suggests considering different mathematical models for more complex ramps, such as parabolas or exponential functions.
• 🔧 GeoGebra's functionality for adjusting the fit of curves and lines to the ramp in the photo is highlighted.
• 📈 The video also covers how to find the derivative of a function to determine the gradient at various points on the ramp.
• 🤔 The presenter encourages viewers to think about the 'goodness of fit' and the mathematical justification for the chosen model.
• 🛠️ The possibility of creating a custom ramp design using a combination of mathematical functions is discussed.
• 🎓 The video concludes by emphasizing the importance of understanding the gradient in the context of skateboarding and its impact on ramp performance.
Q & A
• What is the main purpose of the video?
-The main purpose of the video is to demonstrate how to use GeoGebra to model different skateboard ramps and to explain how this can be applied to any photo for mathematical analysis,
particularly using calculus to find gradients of ramps.
• Why is the skateboard ramp a good topic for an IB Math IA?
-The skateboard ramp is a good topic for an IB Math IA because it allows students to apply mathematical concepts such as gradients, derivatives, and calculus to real-world scenarios, making it
both practical and interesting.
• What is GeoGebra and how does it relate to the video?
-GeoGebra is a dynamic mathematics software that combines geometry, algebra, spreadsheets, graphing, statistics, and calculus. In the video, it is used to model and analyze the shape and
gradients of skateboard ramps.
• How does one begin to model a skateboard ramp in GeoGebra?
-To begin modeling a skateboard ramp in GeoGebra, one should first import a photo of the ramp, adjust its scale to fit the workspace, and then use the software's tools to draw lines or curves
that represent the ramp's shape.
• What is the significance of measuring the ramp in the video?
-Measuring the ramp is significant because it allows for accurate scaling of the photo within GeoGebra, ensuring that the mathematical model reflects the real-world dimensions of the ramp.
• How can one adjust the transparency of the photo in GeoGebra for easier modeling?
-In GeoGebra, one can adjust the transparency of the photo by clicking on the photo, selecting 'Color and Line' properties, and then reducing the opacity to see through the image for easier
• What mathematical concepts are used to analyze the gradients of the ramps?
-The mathematical concepts used to analyze the gradients of the ramps include finding the derivatives of the modeled functions to determine the gradients at different points along the ramp.
• What types of mathematical functions are discussed in the video for modeling ramps?
-The video discusses several mathematical functions for modeling ramps, including linear functions, parabolas, exponential functions, and even circular functions.
• How does the video suggest improving the fit of a modeled ramp?
-The video suggests improving the fit of a modeled ramp by adjusting the parameters of the function, using more points to fit the curve, and considering different types of functions that might
better represent the shape of the ramp.
• What additional topic is briefly mentioned in the video for future discussion?
-The video briefly mentions the topic of least square regression and the goodness of fit as an additional subject for future discussion, which can be used to evaluate how well a curve fits a set
of points.
• What is the potential aim of an IB Math IA project based on the video?
-A potential aim of an IB Math IA project based on the video could be to determine which ramp is best suited for a beginner skateboarder based on the gradient analysis, or to design and model a
custom skateboard ramp.
🛹 Introducing Skateboard Ramp Modeling with GeoGebra
The speaker begins by sharing an experience at a skate park with their kids, which inspired the idea of using GeoGebra to model skateboard ramps. The video aims to demonstrate how to utilize GeoGebra
to create models of different ramps, a skill applicable to any photo-based modeling. The speaker emphasizes the usefulness of this technique for skateboarding enthusiasts and others interested in
photo modeling. The video encourages viewers to support the channel and navigates to GeoGebra Classic for the demonstration, highlighting the importance of accurate measurements and scaling for
realistic ramp modeling.
📐 Adjusting and Fitting Ramp Models in GeoGebra
This paragraph delves into the technical process of adjusting the ramp models within GeoGebra. The speaker explains how to manipulate the sliders for parameters 'a' and 'b' to fit a straight line to
the ramp's image. Tips are given on how to adjust the step size and range for more precise fitting. The speaker also discusses the importance of zooming in for accuracy and the process of setting the
domain of the function to match the ramp's dimensions. The paragraph concludes with a mention of the potential for more complex ramp modeling using calculus and derivatives to find gradients of
🔍 Exploring Complex Ramp Modeling with Different Functions
The speaker moves on to model more complex ramps, such as those with curved surfaces. They discuss the importance of understanding the type of mathematical function that best fits the ramp, whether
it be a parabola, exponential function, or a circle. The process of adjusting the parameters of a parabolic function is demonstrated, along with the consideration of exponential functions as
potential models for the ramps. The speaker also touches on the possibility of using a circle's equation for higher-level mathematics, suggesting that viewers could explore these different
mathematical approaches in their own projects.
📈 Creating Custom Skateboard Ramps and Analyzing Gradients
In the final paragraph, the speaker shares a personal project of creating a custom skateboard ramp using a combination of different mathematical functions. They discuss the importance of finding the
derivative of the function to determine the gradient at various points on the ramp, which is crucial for understanding the ramp's steepness and how it affects skateboarding. The speaker concludes by
encouraging viewers to apply these modeling techniques to their own projects, whether it be for skateboarding or other photo-based curve fitting applications, and hints at future videos that might
delve deeper into the topic.
GeoGebra is a dynamic mathematics software that combines geometry, algebra, spreadsheets, graphing, statistics, and calculus. In the context of the video, it is used to model different skateboard
ramps by incorporating a photograph and mathematical functions to fit the shape of the ramps. The script mentions using GeoGebra Classic for its favored features among the different versions
💡Skateboard Ramp
A skateboard ramp is a structure used in skateboarding to perform aerial tricks and stunts. In the video, the ramp serves as the central object of study, with the aim of modeling its shape and
calculating its gradients using mathematical tools within GeoGebra.
In the context of the video, gradient refers to the slope or steepness of the skateboard ramp, which is crucial for determining the speed and trajectory of a skateboarder. The script discusses
finding the gradient of ramps using calculus, which involves taking derivatives to understand how the slope changes at different points on the ramp.
The derivative in calculus is a measure of how a function changes as its input changes. In the video, derivatives are used to find the gradients of the skateboard ramps at various points. The script
mentions using the derivative to analyze more complicated ramps that are not straight lines.
💡Straight Line
A straight line in geometry is the simplest form of a ramp, which has a constant gradient. The video script uses the concept of a straight line to demonstrate how to find the gradient easily and
serves as a starting point for more complex ramp modeling.
Calculus is a branch of mathematics that deals with rates of change and accumulation. In the video, calculus is applied to find the gradients of the skateboard ramps, particularly through the use of
derivatives to analyze the slopes of the ramps at different points.
Modeling in the video refers to the process of creating a mathematical representation of a real-world object, such as a skateboard ramp, using functions and equations. The script describes using
various mathematical functions to fit the shape of the ramp from a photograph.
A photograph is an image captured by a camera. In the context of the video, a photograph of a skateboard ramp is imported into GeoGebra to provide a visual reference for modeling the ramp's shape
Scale in the video refers to the proportional representation of the skateboard ramp's size in GeoGebra. The script emphasizes the importance of scaling the photograph accurately to ensure that the
mathematical model corresponds correctly to the actual dimensions of the ramp.
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video script discusses using
different types of functions, such as linear, parabolic, and exponential, to model the shape of skateboard ramps.
💡Exponential Function
An exponential function is a mathematical function where the variable is in the exponent. In the video, an exponential function is one of the options considered for modeling the curved part of a
skateboard ramp, as it can represent growth or decay at a rate proportional to the current value.
A parabola is a conic section, the intersection of a right circular conical surface and a plane, and is used in the video to model the curved shape of a skateboard ramp. The script mentions using the
parabolic function y = a(x - h)^2 + k to fit the ramp's shape.
💡Quarter Circle
A quarter circle is a segment of a circle that represents one-fourth of its circumference. In the video, it is mentioned that many skateboard ramps are modeled as quarter circles for simplicity in
construction, although a parabola might provide a better fit.
💡Fitting Curve
Fitting a curve refers to the process of adjusting a mathematical curve to a set of data points, such as the shape of a skateboard ramp in the video. The script describes using GeoGebra to fit
exponential and parabolic curves to points on the ramp's photograph.
💡Least Square Regression
Least square regression is a statistical method used to find the line of best fit for a set of data points. Although not deeply explored in the script, it is mentioned as a method to discuss the
goodness of fit for the curves modeled on the skateboard ramp.
Using GeoGebra to model different skateboard ramps for an IB Math IA project.
The potential of applying these modeling skills to any photo for various applications.
Finding gradients of ramps using basic calculus for straight lines.
The process of importing and scaling a photo in GeoGebra to match a real-world measurement.
Adjusting the opacity of the photo for easier grid alignment in GeoGebra.
Creating a straight line model for a ramp using the equation y = ax + b.
Fine-tuning the model parameters for a better fit using sliders in GeoGebra.
The importance of accurately measuring and scaling the ramp in the modeling process.
Analyzing the curvature of the ramp and considering different mathematical models like parabolas or exponential functions.
Researching common ramp designs, such as quarter circles or parabolic shapes.
Experimenting with different functions to find the best fit for a curved ramp model.
Using the derivative of a function to find the gradient at different points on the ramp.
The practical application of modeling in determining the steepness and usability of a ramp for skateboarding.
The educational value of this modeling process for students interested in skateboarding or mathematics.
Creating a custom ramp model combining different mathematical elements for a unique design.
The possibility of using least square regression to discuss the goodness of fit in the model.
Encouraging viewers to explore the modeling of their own skateboard ramps as a project aim.
The broader application of this modeling technique beyond skateboarding to other areas.
The final encouragement for viewers to apply these skills in their own IB Math IA projects. | {"url":"https://math.bot/blog-IB-Math-IA-Modelling-a-Skateboard-Ramp-38046","timestamp":"2024-11-06T23:49:14Z","content_type":"text/html","content_length":"132532","record_id":"<urn:uuid:8ddafaac-7395-46da-b46d-9e3c1284a30a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00786.warc.gz"} |
30 Years after Reincarnation, it turns out to be a Romance Fantasy Novel
30 Years after Reincarnation, it turns out to be a Romance Fantasy Novel
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Status: Ongoing Type: Web Novel Author: 쏘쏘라 Released: 2024 Native Language: Korean Posted on: Updated on:
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Synopsis 30 Years after Reincarnation, it turns out to be a Romance Fantasy Novel
After being reborn and living for 30 years, I realized that my life was actually a romance fantasy novel?
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This is the story of a man whose life spans across different genres by himself. | {"url":"https://cpunovel.com/series/30-years-after-reincarnation-it-turns-out-to-be-a-romance-fantasy-novel/","timestamp":"2024-11-02T23:23:47Z","content_type":"text/html","content_length":"260474","record_id":"<urn:uuid:25dbd844-6bb9-48e8-a340-476db2f3b839>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00434.warc.gz"} |
Techniques for Solving the Traveling Salesperson Problem | Cratecode
Techniques for Solving the Traveling Salesperson Problem
Note: this page has been created with the use of AI. Please take caution, and note that the content of this page does not necessarily reflect the opinion of Cratecode.
The Traveling Salesperson Problem (TSP) is a classic conundrum that has tickled the brains of computer scientists and mathematicians alike for centuries. The challenge: given a list of cities and the
distances between each pair of cities, find the shortest possible route that visits every city exactly once and returns to the origin city. Doesn't sound too tough, right? Well, buckle up, because
the TSP is a notorious example of an NP-hard problem, meaning it can be quite the computational doozy.
Fear not! Even though the TSP is a challenging problem, there are several techniques to tackle it, ranging from exact solutions to approximation algorithms. Let's dive into some of these strategies
and see how they fare in finding the shortest path for our enthusiastic salesperson.
Brute Force
When in doubt, brute force it out! A brute-force algorithm generates all possible permutations of the cities, calculates the total distance for each permutation, and then selects the shortest path.
This approach guarantees we'll find the optimal solution, but it comes with a hefty price tag: its time complexity is O(n!), where n is the number of cities. As the number of cities increases, the
computational time required explodes, rendering this method impractical for large instances of TSP.
Backtracking is a refined version of brute force that seeks to reduce the search space by eliminating partial solutions that clearly won't lead to an optimal result. The algorithm proceeds
depth-first through the decision tree (formed by the cities) and backtracks whenever a partial solution is deemed infeasible. While the worst-case time complexity still hovers around O(n!),
backtracking is generally more efficient than brute force, especially for small to medium-sized instances of TSP.
Dynamic Programming
Enter the world of dynamic programming, where we take advantage of overlapping subproblems to optimize our solution. The Held-Karp algorithm is a dynamic programming method that solves the TSP in O(n
^2 * 2^n) time, a significant improvement over brute force and backtracking for larger instances. The algorithm's trade-off is its O(n * 2^n) space complexity, meaning it requires a substantial
amount of memory.
Approximation Algorithms
When exact solutions become too computationally expensive, we can turn to approximation algorithms. These techniques provide near-optimal solutions with a fraction of the computation time. One such
method is the Christofides algorithm, which guarantees a solution within 3/2 times the optimal route. By combining minimum spanning trees, perfect matchings, and Eulerian circuits, the Christofides
algorithm provides a relatively efficient and effective approach to tackling larger instances of TSP.
In conclusion, the TSP is a fascinating problem with a variety of solution techniques, each with its own strengths and weaknesses. Depending on the problem size and specific requirements, one may
choose between brute force, backtracking, dynamic programming, or approximation algorithms to help our salesperson find the most efficient path. Happy traveling!
Hey there! Want to learn more? Cratecode is an online learning platform that lets you forge your own path. Click here to check out a lesson: What Programming Means (psst, it's free!).
What is the Traveling Salesperson Problem?
The Traveling Salesperson Problem (TSP) is a classic optimization challenge in computer science. It involves a salesperson who needs to visit a set of cities, with the goal of finding the shortest
possible route that visits each city exactly once and returns to the starting city. TSP is known as an NP-hard problem, meaning it is computationally complex, and finding an optimal solution becomes
increasingly difficult as the number of cities increases.
What are some common techniques to solve the Traveling Salesperson Problem?
There are various strategies and techniques for solving the TSP, including:
• Brute Force: This involves checking all possible routes and selecting the shortest one. However, this method becomes impractical as the number of cities increases, due to the computational
• Greedy Algorithm: This technique involves selecting the nearest unvisited city for each step, which might not always result in the optimal solution.
• Dynamic Programming: Using a bottom-up approach, it breaks the problem into smaller overlapping subproblems, and stores the results to avoid redundant calculations.
• Genetic Algorithm: Inspired by natural selection, this method uses a population of candidate solutions and evolves them through selection, crossover, and mutation to find an optimal solution.
• Simulated Annealing: A probabilistic optimization algorithm that simulates the annealing process in metallurgy to find an optimal solution.
Can the Traveling Salesperson Problem be solved optimally in polynomial time?
Currently, there is no known algorithm that can solve TSP optimally in polynomial time. TSP is an NP-hard problem, which means that it is computationally complex and finding an optimal solution
becomes increasingly difficult as the number of cities increases. However, research continues to discover new algorithms and techniques that can potentially solve TSP faster or with better
How does the Greedy Algorithm work for solving the Traveling Salesperson Problem?
The Greedy Algorithm is a simple technique for solving the TSP that follows a "greedy" approach. The algorithm starts at a particular city, and for each step, it selects the nearest unvisited city as
the next destination. Once all cities are visited, the algorithm returns to the starting city. While it can provide a quick solution, the Greedy Algorithm might not always result in the optimal
route, as it makes local decisions without considering the global implications.
What are some real-world applications of the Traveling Salesperson Problem?
Although the TSP is often framed as a salesperson visiting cities, its applications extend far beyond that. Real-world applications of the TSP include:
• Vehicle routing for delivery services or garbage collection
• Scheduling and routing for airline or transportation networks
• Circuit board drilling optimization in electronics manufacturing
• DNA sequencing and protein folding in bioinformatics
• Tour planning for sightseeing or field service technicians Understanding and solving the TSP can lead to significant efficiency improvements and cost savings in various industries. | {"url":"https://cratecode.com/info/traveling-salesperson-problem","timestamp":"2024-11-14T10:16:13Z","content_type":"text/html","content_length":"99198","record_id":"<urn:uuid:dff93cbd-d091-4648-a520-9a68b381b9b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00293.warc.gz"} |
Concept information
isopropyl nitrite mass concentration
• Mass concentration means mass per unit volume and is used in the construction mass_concentration_of_X_in_Y, where X is a material constituent of Y. A chemical species denoted by X may be
described by a single term such as 'nitrogen' or a phrase such as 'nox_expressed_as_nitrogen'. The chemical formula for isopropyl nitrite is C3H7NO2. Isopropyl nitrite is a member of the group of
nitrogeneous organics. The IUPAC name for isopropyl nitrite is propan-2-yl nitrite.
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{{/if}} {{/each}} | {"url":"https://vocabulary.actris.nilu.no/skosmos/actris_vocab/en/page/isopropylnitritemassconcentration","timestamp":"2024-11-02T18:00:51Z","content_type":"text/html","content_length":"20747","record_id":"<urn:uuid:b62690e1-6d14-4be5-b465-6b22e598d8a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00250.warc.gz"} |
Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?
Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?
1363 views
Answer to a math question Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?
94 Answers
Let's denote the length of the complete roll of tape as x meters. According to the given information, \frac{3}{5}\text{{ of the roll measures 2 meters. We can set up the following equation:}} \frac
{3}{5}x = 2 To find the length of the complete roll, we can solve this equation for x. \text{{Multiplying both sides of the equation by}} \frac{5}{3} x = \frac{2 \times 5}{3} Simplifying: x = \frac
{10}{3} \text{{Therefore, the length of the complete roll of tape is }}\frac{10}{3}\text{{ meters or approximately 3.33 meters.}}
Frequently asked questions (FAQs)
What is the value of x in log(base 2) of x = 5?
What is the equation of a circle with center (5, -3) and radius 8?
What is the volume of a rectangular solid with length 10, width 5, and height 8? | {"url":"https://math-master.org/general/solve-this-mathematical-problem-if-3-5-of-a-roll-of-tape-measures-2m-how-long-is-the-complete-roll","timestamp":"2024-11-07T16:48:18Z","content_type":"text/html","content_length":"241490","record_id":"<urn:uuid:b32428ae-7d22-4a95-8178-a550b625ca41>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00654.warc.gz"} |
Start learning
• Start: You can start learning about or reviewing mathematics
Would you like to start learning about or reviewing mathematics? The information here might help you. The focus here is on mathematics that might help you understand basic electronics. This
information is a part of the Electronics Centre at the Equinet Broadcasting Network.
• Introduction: What is electronics, and what electronics jobs exist?
The Electronics Centre at the Equinet Broadcasting Network provides an introduction. That introduction might help you understand information here.
• Learning mathematics: How might I learn about mathematics?
You might use the information here to help you begin understanding mathematics. You might want both an understanding of mathematics ideas and appropriate mathematics skills. Try doing some of the
mathematics exercises and activities. You might get some mathematics knowledge and skills via an appropriate educational institution. You might look for an instructor whose style matches your
learning styles appropriately. If appropriate, you might find an appropriate study-partner.
Mathematics is a changing field. New information and discoveries in the field of mathematics may affect other fields. You might use various resources to continue improving and updating your
knowledge and skills.
□ Studying and learning
□ Mathematics (Math)
You might find that learning or reviewing some mathematics helps you understand electronics. You might find that knowing some Arithmetic, Algebra, and Trigonometry helps. (To understand
Trigonometry, you might want to know some basic Geometry).
☆ Especially important topics
Mathematical subject areas like Arithmetic, Algebra, and Trigonometry include many interesting topics. To help you understand basic electronics, begin by understanding these specific
mathematical topics:
○ Arithmetic
■ Whole numbers (with Addition, Subtraction, Multiplication, Division)
■ Fractions
★ Decimals (Decimal fractions)
★ Percents
■ Signed numbers (including negative numbers)
○ Algebra
■ Algebraic expressions
★ Exponents, powers, and roots
★ Powers of 10 (with Scientific notation and Engineering notation)
★ Logarithms
★ Polynomials (including Factoring)
■ Equations
★ Methods of solving equations
■ Linear equations
★ Systems of linear equations (Simultaneous linear equations)
○ Graphs
○ Trigonometry (including basic Geometry for the understanding of Trigonometry)
○ Metric system
○ Computer mathematics
■ Binary, octal, and hexadecimal numbers
■ Binary logic (and Boolean algebra)
☆ Mathematics resources
○ Internet-based resources
○ Other resources
■ Public library
You might access some of these resources via your public library. Ask the librarians for more information or for help. If the material is unavailable locally, you might be able to
use an "inter-library loan". You might also ask the librarians about equivalent alternative resources. If English is not your first language, you might begin with children's
materials about mathematics.
■ Books
★ Basic mathematics
These books present basic mathematics. These books do not necessarily cover all of the mathematics for basic electronics.
◎ Zeman, Anne and Kelly, Kate. Everything you need to know about math homework. New York: Scholastic Reference, 1997.
"Everything you need to know about math homework" presents some of the basic mathematics. The book uses pictures. EAL/ESL women might find the book helpful. ("EAL" means
"English as an Additional Language"; "ESL" means "English as a Second Language").
◎ Study guides for the GED examination (with various titles and by various publishers/producers).
The General Educational Development (GED) examination is a test. Some people may not have graduated from high school. They want to show that their abilities are at least
equivalent to those of high school graduates. Some people write the GED to try to show this equivalence (to employers and others). Various publishers publish study guides
to help people study for the GED. You might find the Mathematics section of these study guides to help you with basic mathematics. (The study guides might not present all
of the mathematics used in basic electronics). EAL/ESL women might find the Mathematics and the English (Grammar and Writing) sections to be helpful. GED study guide books
and videos may be accessible in libraries.
★ Mathematics for introductory electronics
These books present much or all of the mathematics needed for basic electronics.
■ Videos
You might use mathematics video cassette tapes to help you study mathematics. One source of basic mathematics videos and higher mathematics videos is Video Aided Instruction Inc..
■ Adult Basic Education and other courses
You might contact your local educational institutions, like schools and colleges. Ask about mathematics courses - such as high school mathematics courses. The tuition for courses
for high school completion might be free. If the courses are not free, and if appropriate, ask what financial assistance is available. If appropriate, ask about child care.
■ Computer software
You might be interested in the "Are You Ready...?" series of mathematics software. You might obtain the software free from University of Arizona Software.
Suggestions? Questions? Having trouble with a link here?
Feel free to e-mail the Equinet Broadcasting Network at ebn@excite.com.
This page was updated on August 20, 2004.
Barry G. Wong
Equinet Broadcasting Network
E-mail: ebn@excite.com
World Wide Web: https://mythanks.tripod.com/
Copyright © 1998-2004 by Barry G. Wong. All rights reserved. | {"url":"https://mythanks.tripod.com/enable/elnmath.htm","timestamp":"2024-11-10T12:53:01Z","content_type":"text/html","content_length":"29604","record_id":"<urn:uuid:36667733-2974-4d7f-a4af-ebbba41ba4df>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00051.warc.gz"} |
Multidimensional superstring vibrations
There are infinite numbers so that means there are infinite frequencys
This means there are infinite frequencys a superstring(infinite tetrahedron grid) can have
This means there are infinite dimensions
The superstrings that make up the dimension you are in are 7 dimensions higher(the 3 dimension is made out of superstring vibrating in the 10 dimension )
There are 12 vibrational dimensions with infinite dimensions in between
After the 12 vibrational dimensions there are infinite sub multidimensional levels | {"url":"https://www.64tge8st.com/post/2017/06/26/untitled","timestamp":"2024-11-01T19:07:53Z","content_type":"text/html","content_length":"1050486","record_id":"<urn:uuid:1b5ea03f-27cc-4bb1-8c65-7743c51338ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00871.warc.gz"} |
Math Story : Introduction to subtraction - Fun2Do LabsMath Story : Introduction to subtraction
Math Story : Introduction to subtraction
Fun With Ms Subtract
One day, Cirha was playing in the garden. Suddenly, she spots a bright and shining object in the bushes.
Cirha is curious. She runs near the bushes to find out what it is. “Wow! It looks like a gadget”, she says.
As she grabs the mysterious object, it vanishes in the air and says, “I am Ms Subtract – a Math gadget. I make things disappear by
taking them away. Remember, to make anything disappear, you need to tell me the correct answer of how many will remain in the end”, and it settles back in Cirha’s hand.
Cirha always wanted to make things disappear without touching them. “Now that I have Ms Subtract, I can fulfil my wish”, she claims. She is delighted and decides to try the gadget on some stones.
She rolls the gadget in the air and says, “Ms Subtract! Take away 6 stones from the group of 10, only 1 stone remains. Go!”
But nothing happens! She counts the stones to cross-check whether the gadget worked or not. But they are still the same
10 stones.
Cirha does not give up. She tries again, “Ms Subtract! Take away 6 stones from the bunch of 10, only 2 stones remain. Go!”
None of the stones disappears. She is disappointed.
Uncle Math is stressed. He is searching for something. Suddenly, he spots Cirha and screams “My gadget! My gadget! I have been
looking for it since morning. Where did you find it?” She quickly narrates the morning scene and her struggles with the gadget.
Uncle Math decides to help. “Ms Subtract is not working because your answer is wrong Cirha”, he says. “Here are your 10 stones. If
I take away 6, how many remain?” he questions. “4” answers Cirha.
“Correct! This method of taking away one or more things is called Subtraction. We show subtraction using the “-” minus
symbol. So, we say 10 stones minus 6 stones is equal to 4 stones”, explains Uncle Math.
“Ohh! Now I get it”, says Cirha. She understands subtraction and is ready to use the gadget. This time she tries it on some
empty pots. “Ms Subtract! Take away 4 pots out of 7, only 3 remain. Go!” The pots disappear. Cirha jumps out of joy.
“Well done Cirha! Let us make people disappear now”, suggests Uncle Math. “No! People might get hurt. How can I hurt anyone?”
she says. “I do not get you. If your gadget works, then everybody will like you. You will be famous. How is this hurting anyone?” he
“Definitely. But if I try this gadget on innocent people for fun, then it would be selfish”, she answers. Uncle Math proudly
says, “I wanted to test you to see if you would misuse any of your superpowers, and I got my answer”. Cirha smiles and
continues her experiments.
When the gadget became Cirha’s superpower, she decided to use it only for learning without hurting anyone. With great
power comes great responsibility. Indeed beautiful gesture, beautiful subtraction and beautiful Cirha.
We Learnt That…
• The method of taking away one or more things is called Subtraction.
• Subtraction is denoted using the “ – “ Minus symbol.
• We should never misuse our powers.
• We should not be selfish, but empathetic towards others.
• With great power comes great responsibility.
Let’s Discuss
• What was the name of the gadget?
• How did Uncle Math help Cirha in using the gadget?
• Cirha did not try the gadget on people. Does this make her good or bad? Why? Explain.
• Ms Subtract became Cirha’s superpower. What type of superpower would you wish for? Why?
Experience the immersive video version of this captivating story by clicking the link below, and let the excitement come to life :
Please refer this guide by Fun2Do Labs for teaching subtraction to kids : | {"url":"https://fun2dolabs.com/math-story-introduction-to-subtraction/fun-with-ms-subtract/","timestamp":"2024-11-07T12:32:54Z","content_type":"text/html","content_length":"47276","record_id":"<urn:uuid:aac2b82e-d46f-4ccd-a4e8-3af910aae716>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00318.warc.gz"} |
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Gallon (imperial)
⇅ Switch toGallon (imperial) to Load Converter
How to use this Load to Gallon (imperial) Converter 🤔
Follow these steps to convert given volume from the units of Load to the units of Gallon (imperial).
1. Enter the input Load value in the text field.
2. The calculator converts the given Load into Gallon (imperial) in realtime ⌚ using the conversion formula, and displays under the Gallon (imperial) label. You do not need to click any button. If
the input changes, Gallon (imperial) value is re-calculated, just like that.
3. You may copy the resulting Gallon (imperial) value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Load to Gallon (imperial)?
The formula to convert given volume from Load to Gallon (imperial) is:
Volume[(Gallon (imperial))] = Volume[(Load)] × 311.44177295214126
Substitute the given value of volume in load, i.e., Volume[(Load)] in the above formula and simplify the right-hand side value. The resulting value is the volume in gallon (imperial), i.e., Volume
[(Gallon (imperial))].
Calculation will be done after you enter a valid input.
Consider that a truck carries a load of 10 units.
Convert this load from units to Gallon (imperial).
The volume in load is:
Volume[(Load)] = 10
The formula to convert volume from load to gallon (imperial) is:
Volume[(Gallon (imperial))] = Volume[(Load)] × 311.44177295214126
Substitute given weight Volume[(Load)] = 10 in the above formula.
Volume[(Gallon (imperial))] = 10 × 311.44177295214126
Volume[(Gallon (imperial))] = 3114.4177
Final Answer:
Therefore, 10 is equal to 3114.4177 gal.
The volume is 3114.4177 gal, in gallon (imperial).
Consider that a construction site receives a load of 15 units of bricks.
Convert this load from units to Gallon (imperial).
The volume in load is:
Volume[(Load)] = 15
The formula to convert volume from load to gallon (imperial) is:
Volume[(Gallon (imperial))] = Volume[(Load)] × 311.44177295214126
Substitute given weight Volume[(Load)] = 15 in the above formula.
Volume[(Gallon (imperial))] = 15 × 311.44177295214126
Volume[(Gallon (imperial))] = 4671.6266
Final Answer:
Therefore, 15 is equal to 4671.6266 gal.
The volume is 4671.6266 gal, in gallon (imperial).
Load to Gallon (imperial) Conversion Table
The following table gives some of the most used conversions from Load to Gallon (imperial).
Load () Gallon (imperial) (gal)
0.01 3.1144 gal
0.1 31.1442 gal
1 311.4418 gal
2 622.8835 gal
3 934.3253 gal
4 1245.7671 gal
5 1557.2089 gal
6 1868.6506 gal
7 2180.0924 gal
8 2491.5342 gal
9 2802.976 gal
10 3114.4177 gal
20 6228.8355 gal
50 15572.0886 gal
100 31144.1773 gal
1000 311441.773 gal
The load is a unit of measurement used to quantify large volumes of material, particularly in agriculture and transport. It is a somewhat informal unit and can vary in definition depending on the
context and region. Historically, the load was used to describe the capacity of carts, wagons, or other vehicles for carrying goods, such as grain or coal. Today, it is often used in contexts where
precise volume measurements are less critical, and the term provides a practical understanding of how much material can be moved or stored in one instance.
Gallon (imperial)
The Imperial gallon is a unit of measurement used to quantify liquid volumes, primarily in the UK and countries using the Imperial system. It is defined as 4.54609 liters, making it slightly larger
than the US gallon. Historically, the Imperial gallon was used for various liquids, including water and fuel, and was essential for standardizing measurements in trade and commerce. Today, it remains
in use in the UK and some other countries for measuring liquids, particularly in contexts like fuel consumption and beverage volumes.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Load to Gallon (imperial) in Volume?
The formula to convert Load to Gallon (imperial) in Volume is:
Load * 311.44177295214126
2. Is this tool free or paid?
This Volume conversion tool, which converts Load to Gallon (imperial), is completely free to use.
3. How do I convert Volume from Load to Gallon (imperial)?
To convert Volume from Load to Gallon (imperial), you can use the following formula:
Load * 311.44177295214126
For example, if you have a value in Load, you substitute that value in place of Load in the above formula, and solve the mathematical expression to get the equivalent value in Gallon (imperial). | {"url":"https://convertonline.org/unit/?convert=load-gallon_imperial","timestamp":"2024-11-09T21:06:46Z","content_type":"text/html","content_length":"93070","record_id":"<urn:uuid:14dc918c-5f34-4fe2-adcb-a1585c19b0d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00307.warc.gz"} |
model = sqw_acoustopt(p, h,k,l,w, {signal}) : acoutsic/optic dispersion(HKL) with DHO(energy)
function signal=sqw_acoustopt(varargin)
model = sqw_acoustopt(p, h,k,l,w, {signal}) : acoutsic/optic dispersion(HKL) with DHO(energy)
iFunc/sqw_acoustopt: a 4D S(q,w) with a 3D HKL dispersion with
quadratic dependency, and a DHO line shape.
This dispersion corresponds with a local description of an excitation,
with its minimum around an (H0,K0,L0,E0) point.
The model requires to define a direction corresponding with a Slope1 dependency
as well as a second direction. An ortho-normal coordinate basis is then derived.
All HKL coordinates are in rlu, and energies are in meV.
The dispersion has the form:
w(q) = sqrt(E0^2 + Slope^2*(q-HKL0)^2)
so that when the dispersion is linear for E0=0 or far from HKL0, else it is
To define a pure acoustic mode, use (and adjust slopes):
sqw_acoustopt(0) % minimum is E0=0 at q=0
To define an optical mode with energy E0 at Q=0, use:
sqw_acoustopt([ E0 ]) % minimum is E0 at q=0
To define a mode which has its minimum E0 at a given HKL location, use:
sqw_acoustopt([ H K L E0 ])
When creating the Model, the following syntax is possible:
sqw_acoustopt(E0) centers the excitation at q=0 with energy E0
sqw_acoustopt([ h k l E0 ]) centers the excitation at q=[H K L] and energy E0
You can of course tune other parameters once the model object has been created.
WARNING: Single intensity and line width parameters are used here.
To model more than one branch, just add these models together.
s=sqw_acoustopt(5); qh=linspace(0,.5,50);qk=qh; ql=qh'; w=linspace(0.01,10,50);
f=iData(s,s.p,qh,qk,ql,w); plot3(log(f(:,1,:,:)));
Reference: https://en.wikipedia.org/wiki/Phonon
input: p: sqw_acoustopt model parameters (double)
p(1) = DC_Hdir1 Slope1 dispersion direction, H [rlu]
p(2) = DC_Kdir1 Slope1 dispersion direction, K [rlu]
p(3) = DC_Ldir1 Slope1 dispersion direction, L [rlu]
p(4) = DC_Hdir2 Slope2 dispersion direction, H (transverse) [rlu]
p(5) = DC_Kdir2 Slope2 dispersion direction, K (transverse) [rlu]
p(6) = DC_Ldir2 Slope2 dispersion direction, L (transverse) [rlu]
p(7) = DC_Slope1 Dispersion slope along 1st axis [meV/rlu]
p(8) = DC_Slope2 Dispersion slope along 2nd axis (transverse) [meV/rlu]
p(9) = DC_Slope3 Dispersion slope along 3rd axis (vertical) [meV/rlu]
p(10)= Ex_H0 Minimum of the dispersion, H [rlu]
p(11)= Ex_K0 Minimum of the dispersion, K [rlu]
p(12)= Ex_L0 Minimum of the dispersion, L [rlu]
p(13)= Ex_E0_Center Minimum of the dispersion, Energy [meV]
p(14)= DHO_Amplitude
p(15)= DHO_Damping Excitation damping, half-width [meV]
p(16)= DHO_Temperature Temperature [K]
p(17)= Background
or p='guess'
qh: axis along QH in rlu (row,double)
qk: axis along QK in rlu (column,double)
ql: axis along QL in rlu (page,double)
w: axis along energy in meV (double)
signal: when values are given, a guess of the parameters is performed (double)
output: signal: model value
Version: Nov. 26, 2018
See also iData, iFunc/fits, iFunc/plot, gauss, sqw_phonons, sqw_cubic_monoatomic, sqw_vaks
<a href="matlab:doc(iFunc,'Models')">iFunc:Models</a>
(c) E.Farhi, ILL. License: EUPL.
This function calls: This function is called by: Generated on Mon 26-Nov-2018 15:08:42 by m2html © 2005. iFit (c) E.Farhi/ILL EUPL 1.1 | {"url":"http://ifit.mccode.org/techdoc/Scripts/Models/Specialized/sqw_acoustopt.html","timestamp":"2024-11-11T19:20:23Z","content_type":"text/html","content_length":"6442","record_id":"<urn:uuid:4f241128-52fc-4246-bcb8-a2599bdd5391>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00515.warc.gz"} |
Fast Bayesian Methods for AB Testing
bayesAB provides a suite of functions that allow the user to analyze A/B test data in a Bayesian framework. bayesAB is intended to be a drop-in replacement for common frequentist hypothesis test such
as the t-test and chi-sq test.
Bayesian methods provide several benefits over frequentist methods in the context of A/B tests - namely in interpretability. Instead of p-values you get direct probabilities on whether A is better
than B (and by how much). Instead of point estimates your posterior distributions are parametrized random variables which can be summarized any number of ways.
While Bayesian AB tests are still not immune to peeking in the broadest sense, you can use the ‘Posterior Expected Loss’ provided in the package to draw conclusions at any point with respect to your
threshold for error.
The general bayesAB workflow is as follows:
• Decide how you want to parametrize your data (Poisson for counts of email submissions, Bernoulli for CTR on an ad, etc.)
• Use our helper functions to decide on priors for your data (?bayesTest, ?plotDistributions)
• Fit a bayesTest object
□ Optional: Use combine to munge together several bayesTest objects together for an arbitrary / non-analytical target distribution
• print, plot, and summary to interpret your results
□ Determine whether to stop your test early given the Posterior Expected Loss in summary output
Optionally, use banditize and/or deployBandit to turn a pre-calculated (or empty) bayesTest into a multi-armed bandit that can serve recipe recommendations and adapt as new data comes in.
Note, while bayesAB was designed to exploit data related to A/B/etc tests, you can use the package to conduct Bayesian analysis on virtually any vector of data, as long as it can be parametrized by
the available functions.
Get the latest stable release from CRAN:
Or the dev version straight from Github:
devtools::install_github("frankportman/bayesAB", build_vignettes = TRUE)
Some useful links from my blog with bayesAB examples (and pictures!!):
For a more in-depth look please check the package vignettes with browseVignettes(package = "bayesAB") or the pre-knit HTML version on CRAN here. Brief example below. Run the following code for a
quick overview of bayesAB:
A_binom <- rbinom(100, 1, .5)
B_binom <- rbinom(100, 1, .55)
# Fit bernoulli test
AB1 <- bayesTest(A_binom,
priors = c('alpha' = 1, 'beta' = 1),
distribution = 'bernoulli')
Distribution used: bernoulli
Using data with the following properties:
A B
Min. 0.00 0.00
1st Qu. 0.00 0.00
Median 1.00 0.00
Mean 0.55 0.44
3rd Qu. 1.00 1.00
Max. 1.00 1.00
Priors used for the calculation:
[1] 1
[1] 1
Calculated posteriors for the following parameters:
Monte Carlo samples generated per posterior:
[1] 1e+05
Quantiles of posteriors for A and B:
0% 25% 50% 75% 100%
0.3330638 0.5159872 0.5496165 0.5824940 0.7507997
0% 25% 50% 75% 100%
0.2138149 0.4079403 0.4407221 0.4742673 0.6369742
P(A > B) by (0)%:
[1] 0.93912
Credible Interval on (A - B) / B for interval length(s) (0.9) :
5% 95%
-0.01379425 0.58463290
Posterior Expected Loss for choosing A over B:
[1] 0.03105786 | {"url":"https://cloud.r-project.org/web/packages/bayesAB/readme/README.html","timestamp":"2024-11-01T23:41:11Z","content_type":"application/xhtml+xml","content_length":"14230","record_id":"<urn:uuid:b4f23074-e57b-4e54-bc5b-ecd288f6eedd>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00279.warc.gz"} |
A tensor approach to double wave vector diffusion-weighting experiments on restricted diffusion
Previously, it has been shown theoretically that in case of restricted diffusion, e.g. within isolated pores or cells, a measure of the pore size, the mean radius of gyration, can be estimated from
double wave vector diffusion-weighting experiments. However, these results are based on the assumption of an isotropic orientation distribution of the pores or cells which hampers the applicability
to samples with anisotropic or unknown orientation distributions, such as biological tissue. Here, the theoretical considerations are re-investigated and generalized in order to describe the signal
dependency for arbitrary orientation distributions. The second-order Taylor expansion of the signal delivers a symmetric rank-2 tensor with six independent elements if the two wave vectors are
concatenated to a single six-element vector. With this tensor approach the signal behavior for arbitrary wave vectors and orientation distributions can be described as is demonstrated by numerical
simulations. The rotationally invariant trace of the tensor represents a pore size measure and can be determined from three orthogonal directions with parallel and antiparallel orientation of the two
wave vectors. Thus, the presented tensor approach may help to improve the applicability of double wave vector diffusion-weighting experiments to determine pore or cell sizes, in particular in
biological tissue. | {"url":"https://research.uni-luebeck.de/en/publications/a-tensor-approach-to-double-wave-vector-diffusion-weighting-exper","timestamp":"2024-11-13T13:21:42Z","content_type":"text/html","content_length":"44705","record_id":"<urn:uuid:1aa94b2b-1b20-4d58-be32-7c4f209bcea7>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00217.warc.gz"} |
RSICC Home Page
RSICC CODE PACKAGE PSR-464
1. NAME AND TITLE
SOLA-LOOP: Nonequilibrium, Drift-Flux Code System for Two-Phase Flow Network Analysis.
Los Alamos National Laboratory, NM, through the Energy Science and Technology Software Center, Oak Ridge, TN.
FORTRAN IV; CDC7600 (P00464760000).
4. NATURE OF PROBLEM SOLVED
SOLA-LOOP is designed for the solution of transient two-phase flow in networks composed of one-dimensional components. The fluid dynamics is described by a nonequilibrium, drift-flux formulation of
the fluid conservation laws. Although developed for nuclear reactor safety analysis, SOLA-LOOP may be used as the basis for other types of special-purpose network codes. The program can accommodate
almost any set of constitutive relations, property tables, or other special features required for different applications.
5. METHOD OF SOLUTION
The drift-flux equations are formulated as continuity equations, the momentum equation, and the internal energy equation. The mixture density, the macroscopic vapor density, the center of mass
velocity, and the mixture specific internal energy are chosen as dependent variables, and time and axial position are the independent variables. Constitutive relations and exchange rates are
determined by the intended use of the code. The calculation cycle used to solve by point relaxation methods the finite difference formulation of the flow equations in a single one-dimensional
component is made up of four tasks. First, the momentum equation is advanced explicitly using the values from the previous cycle for all contributions. Next, an iteration is made to replace the
pressure with advanced time values. This pressure iteration scheme is a variant of the Implicit Continuous fluid Eulerian (ICE) technique. Then, all other dependent variables are updated, and in the
fourth task data output, time-step control, and housekeeping operations are performed. Various boundary conditions may be applied at the ends of the one-dimensional component meshes to represent
inlet and exit conditions including prescribed velocities or pressures, uniform or gradient-free outflow, and periodic boundaries in which the bottom and top of a component are joined. Where two or
more components are coupled, special coupling equations are solved to obtain the appropriate boundary conditions for each. Different time-steps can be used in various components. The time-steps are
determined by numerical stability requirements and other user-specified conditions.
Current dimensioning in the SOLA-LOOP program allows maxima of 10 components, 8 segments per component, 200 junctions, 6 time levels, pressure groups, and vapor production rates per cell and 5
boundary data sets.
7. TYPICAL RUNNING TIME
The time required is highly problem-dependent. NESC executed the sample problem in less than 3 CP minutes on a CDC7600.
SOLA-LOOP was developed on a CDC 7600 computer.
A Fortran IV compiler is required to compile the code which ran under the SCOPE operating system.
10. REFERENCES
a) included in documentation:
C.W. Hirt, T.A. Oliphant, W.C. Rivard, N.C. Romero, and M.D. Torrey, "SOLA-LOOP: A Nonequilibrium, Drift-Flux Code for Two-Phase Flow in Networks," NUREG/CR-0626, LA-7659 (June 1979).
NESC No. 859.7600, "SOLA-LOOP Title Input, NESC Note 80-29," (December 17, 1979).
Sample problem output from CDC.
b) background information:
C.W. Hirt, N.C. Romero, M.D. Torrey, and J.R. Travis, "SOLA-DF: A Solution Algorithm for Nonequilibrium Two-Phase Flow," NUREG/CR-0690, LA-7725-MS (June 1979).
C.W. Hirt, B.D. Nichols, and N.C. Romero, "SOLA, A Numerical Solution Algorithm for Transient Fluid Flow," LA-5852 (April 1975).
11. CONTENTS OF CODE PACKAGE
Included are the referenced documents in (10.a) and one DS/HD diskette which includes the Fortran source and sample problem input.
12. DATE OF ABSTRACT
August 2000. | {"url":"https://rsicc.ornl.gov/codes/psr/psr4/psr-464.html","timestamp":"2024-11-14T01:58:37Z","content_type":"text/html","content_length":"6619","record_id":"<urn:uuid:25961153-18d7-4242-85a4-85c4fbe58e76>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00634.warc.gz"} |
Graph each function and identify the horizontal and vertical as-Turito
Are you sure you want to logout?
Graph each function and identify the horizontal and vertical asymptotes
A rational function is a function that is the ratio of polynomials. Any function of one variable, x, is called a rational function if, it can be represented as f(x) = p(x)/q(x), where p(x) and q(x)
are polynomials such that q(x) ≠ 0.
Rational functions are of the form y=f(x)y=fx , where f(x)fx is a rational expression .
• If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote.
• If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
• If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote.
The correct answer is: From the graph we can analyze that the vertical asymptote of the rational function is x = 2 and horizontal asymptote is y = 0
1. Find the asymptotes of the rational function, if any.
2. Draw the asymptotes as dotted lines.
3. Find the x -intercept (s) and y -intercept of the rational function, if any.
4. Find the values of y for several different values of x .
5. Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.
The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x – 2 = 0
x = 2
The vertical asymptote of the rational function is x= 2.
This function has no x -intercept and has y -intercept at (0,-1.5) . We will find more points on the function and graph the function
From the graph we can analyze that the vertical asymptote of the rational function is x= 2 and horizontal asymptote is y = 0
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/graph-each-function-and-identify-the-horizontal-and-vertical-asymptotes-f-x-3-x-2-q2e2d294a","timestamp":"2024-11-11T01:36:59Z","content_type":"application/xhtml+xml","content_length":"740022","record_id":"<urn:uuid:aab9d367-19e0-47fc-916a-d99b23aefa38>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00082.warc.gz"} |
profile -
Thank-you for the quick answer.
frac() gives a function which is periodic on positive reals. I just adapted your solution to get it periodic on \RR.
f = lambda x: 1 if (x - RR(x).floor()) < 1/2 else 0
I would have liked to be able to define a symbolic function though. Is it doable?
I was also trying to get a function whose plot is correct (without asking it to be pointwise) as it is the case for Piecewise().
Below are some of the things (not chronologically ordred though) I had tried (just for completness as I guess they are just full of beginer's classical mistakes).
which does not evaluate
returns ValueError: cannot convert float NaN to integer.
returns ValueError: Value not defined outside of domain.
also returns ValueError: Value not defined outside of domain.
I also tried to redefine unit_step:
def echelon_unite(x):
if x<0:
return hres
problem integral(echelon_unite(x),x,-10,3) returns 13
numerical integral returns a coherent result.
Other tentative with incoherent result (still with integrate and not numerical_integral)
def Periodisation_int(f,a,b):
x = var('x')
h0(x) = (x-b)-(b-a)*floor((x-b)/(b-a))
hres = compose(f,h0)
return hres
sage: g=Periodisation_int(sin,0,1)
sage: integrate(g(x),x,0,2)
-cos(1) + 2
sage: integrate(g(x),x,0,1)
-cos(1) + 1
sage: integrate(g(x),x,1,2)
-1/2*cos(1) + 1
My guess is I was using integrate on inappropriate objects. I would still like to know how to define corresponding symbolic function (if it is possible).
Thanks again,
best regards. | {"url":"https://ask.sagemath.org/users/2013/lc/?sort=recent","timestamp":"2024-11-14T16:44:57Z","content_type":"application/xhtml+xml","content_length":"21407","record_id":"<urn:uuid:ca44839c-4ae5-4272-8e4d-d2550a7b4f46>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00163.warc.gz"} |
How many slices is a large pizza at Pizza Hut? - Cooking Brush
How many slices is a large pizza at Pizza Hut?
How many slices is a large pizza at Pizza Hut?
A large-sized Pizza Hut pie is 14 inches in diameter and yields 12 slices enough to feed four to six people. We’ll take you to the detailed answer and give you tips on ordering pizzas from the Pizza
What’s bigger 2 medium pizzas or 1 large Pizza Hut?
If you’re planning on ordering a takeaway pizza this week, you might want to rethink your order. Mathematicians have revealed that getting one large pizza is a better idea than ordering two medium
pizzas. Surprisingly, one 18-inch pizza actually has more ‘pizza’ than two 12-inch pizzas.
What are pizza Huts sizes?
Pizza Hut has three pizza sizes: small, medium, and large. The small pizza, also called the personal pizza, is great for a single serving. This 8” pizza has 4-6 slices and is good enough to serve one
to two people.
What size is the $10 pizza at Pizza Hut?
How many slices are in a large pizza Pizza Hut?
So, what is the number of slices in a large pizza? A large-sized Pizza Hut pie is 14 inches in diameter and yields 12 slices enough to feed four to six people.
How large is a large Pizza Hut pizza?
14 inches
How many slices is large pizza?
10 slices
What are Pizza Hut pizza sizes?
Our new Regular base is 7.5 inches, our large is 10.75 inches and our Extra Large is an impressive 14 inches. This may vary slightly due to the rising agents, oils etc in our bases.
Does 2 medium pizzas equal large?
That large pizza has the same area as 2 medium pizzas (14 inches) and 6.3 smalls (8 inches). To equal the same amount of pizza as the large, it would set you back an additional $8.82 and $30.79
respectively. For an even more drastic example, a large 30-inch pizza will run you about $31.24.
Is a large or 2 medium pizzas bigger?
Mathematicians state that ordering one large pizza is a better deal than ordering two medium-sized pizzas. Surprisingly, one 18-inch pizza has more pizza than two 12-inch pizzas. Just to confirm the
number of slices per pizza by size: Small pizza: 8-10 inches, six slices.
Are 2 small pizzas bigger than a large?
It has to do with the mathematical nature of the area of circles. When you increase the width of your pizza it actually adds to the total size of your pie exponentially. For example, a 16-inch large
might seem twice as big as an 8 inch small but it’s actually four times as much pizza
How much bigger is a large pizza than a medium Pizza Hut?
A medium pizza at Pizza Hut is 12 inches in diameter and costs $9.99. A large pizza at Pizza Hut is 14 inches in diameter and costs $11.99.
What size is Pizza Hut small?
How big is a Pizza Hut large?
A large Pizza Hut pizza is 14-inches in size with 12 slices. If you talk about the area, a large pizza is about 190 square inches.
What size is a Pizza Hut individual pizza?
However, a small is supposed to be 9 inches across. Joe: In the US, at a Pizza Hut, the only small pizza you can get is one with gluten-free crust. This one here is 10 inches. Harry: The next size we
have in the UK is the medium pizza.
What is the 7.99 deal at Pizza Hut?
$7.99 Large 3-Topping Pizza | Pickup Deal Pick up a Large 3-Topping Pizza for only $7.99 at Pizza Hut! Order Now!
What size are Pizza Hut pizzas?
A personal pan at Pizza Hut is 6 inches in diameter and costs $4.50. A medium pizza at Pizza Hut is 12 inches in diameter and costs $9.99. A large pizza at Pizza Hut is 14 inches in diameter and
costs $11.99.
What is a tastemaster pizza?
The Tastemaker Pizza is a large pizza that comes with your choice of up to three toppings. It’s essentially the same as the recently launched $9.99 large 3-topping pizza deal, but with a cool name
attached to it.
How big is a 12 inch pizza from Pizza Hut?
The medium pizza, which is about 12” in size, has about 8 slices and should be enough to serve two to four people. Lastly, the large pizza is about 14” in size and has 12 slices. It is good to serve
about 4-6 people.
How many slices does a Pizza Hut pizza have?
6 Slices
What size is a Pizza Hut large pizza?
A large pizza at Pizza Hut is 14 inches in diameter and costs $11.99.
What is bigger 2 medium pizzas or 1 large pizza Pizza Hut?
10 slices
How many slices is a large Pizza Hut pizza?
12 slices
How big is a large at pizza pizza?
Our new Regular base is 7.5 inches, our large is 10.75 inches and our Extra Large is an impressive 14 inches. This may vary slightly due to the rising agents, oils etc in our bases.
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Normal distribution
Normal distribution
Background to the schools Wikipedia
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Probability density function
The green line is the standard normal distribution
Cumulative distribution function
Colors match the image above
Parameters $\mu$ location (real)
$\sigma^2>0$ squared scale (real)
Support $x \in\mathbb{R}\!$
PDF $\frac1{\sigma\sqrt{2\pi}}\; \exp\left(-\frac{\left(x-\mu\right)^2}{2\sigma^2} \right) \!$
CDF $\frac12 \left(1+\mathrm{erf}\,\frac{x-\mu}{\sigma\sqrt2}\right) \!$
Mean $\mu$
Median $\mu$
Mode $\mu$
Variance $\sigma^2$
Skewness 0
Ex. kurtosis 0
Entropy $\ln\left(\sigma\sqrt{2\,\pi\,e}\right)\!$
MGF $M_X(t)= \exp\left(\mu\,t+\frac{\sigma^2 t^2}{2}\right)$
CF $\chi_X(t)=\exp\left(\mu\,i\,t-\frac{\sigma^2 t^2}{2}\right)$
The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by
two parameters, location and scale: the mean ("average", μ) and variance (standard deviation squared) σ^2, respectively. The standard normal distribution is the normal distribution with a mean of
zero and a variance of one (the green curves in the plots to the right). Carl Friedrich Gauss became associated with this set of distributions when he analyzed astronomical data using them, and
defined the equation of its probability density function. It is often called the bell curve because the graph of its probability density resembles a bell.
The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioural sciences is due to the central limit theorem. Many psychological measurements and
physical phenomena (like noise) can be approximated well by the normal distribution. While the mechanisms underlying these phenomena are often unknown, the use of the normal model can be
theoretically justified by assuming that many small, independent effects are additively contributing to each observation.
The normal distribution also arises in many areas of statistics. For example, the sampling distribution of the sample mean is approximately normal, even if the distribution of the population from
which the sample is taken is not normal. In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of
underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are
based on the assumption of normality. In probability theory, normal distributions arise as the limiting distributions of several continuous and discrete families of distributions.
The normal distribution was first introduced by Abraham de Moivre in an article in 1733, which was reprinted in the second edition of his The Doctrine of Chances, 1738 in the context of approximating
certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the theorem of de Moivre-Laplace.
Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method
since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors.
The name "bell curve" goes back to Jouffret who first used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently
by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875. Despite this terminology, other probability distributions may be more appropriate in some contexts; see the discussion of
occurrence, below.
There are various ways to characterize a probability distribution. The most visual is the probability density function (PDF). Equivalent ways are the cumulative distribution function, the moments,
the cumulants, the characteristic function, the moment-generating function, the cumulant- generating function, and Maxwell's theorem. See probability distribution for a discussion.
To indicate that a real-valued random variable X is normally distributed with mean μ and variance σ² ≥ 0, we write
$X \sim N(\mu, \sigma^2).\,\!$
While it is certainly useful for certain limit theorems (e.g. asymptotic normality of estimators) and for the theory of Gaussian processes to consider the probability distribution concentrated at μ
(see Dirac measure) as a normal distribution with mean μ and variance σ² = 0, this degenerate case is often excluded from the considerations because no density with respect to the Lebesgue measure
The normal distribution may also be parameterized using a precision parameter τ, defined as the reciprocal of σ². This parameterization has an advantage in numerical applications where σ² is very
close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution.
Probability density function
The continuous probability density function of the normal distribution is the Gaussian function
$\varphi_{\mu,\sigma^2}(x) = \frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x- \mu)^2}{2\sigma^2}} = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right),\quad x\in\mathbb{R},$
where σ > 0 is the standard deviation, the real parameter μ is the expected value, and
$\varphi(x)=\varphi_{0,1}(x)=\frac{1}{\sqrt{2\pi\,}} \, e^{-\frac{x^2}{2}},\quad x\in\mathbb{R},$
is the density function of the "standard" normal distribution, i.e., the normal distribution with μ = 0 and σ = 1. The integral of $\varphi_{\mu,\sigma^2}$ over the real line is equal to one as shown
in the Gaussian integral article.
As a Gaussian function with the denominator of the exponent equal to 2, the standard normal density function $\scriptstyle\varphi$ is an eigenfunction of the Fourier transform.
The probability density function has notable properties including:
• symmetry about its mean μ
• the mode and median both equal the mean μ
• the inflection points of the curve occur one standard deviation away from the mean, i.e. at μ − σ and μ + σ.
Cumulative distribution function
The cumulative distribution function (cdf) of a probability distribution, evaluated at a number (lower-case) x, is the probability of the event that a random variable (capital) X with that
distribution is less than or equal to x. The cumulative distribution function of the normal distribution is expressed in terms of the density function as follows:
\begin{align} \Phi_{\mu,\sigma^2}(x) &{}=\int_{-\infty}^x\varphi_{\mu,\sigma^2}(u)\,du\\ &{}=\frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x \exp \Bigl( -\frac{(u - \mu)^2}{2\sigma^2} \ \Bigr)\, du
\\ &{}= \Phi\Bigl(\frac{x-\mu}{\sigma}\Bigr),\quad x\in\mathbb{R}, \end{align}
where the standard normal cdf, Φ, is just the general cdf evaluated with μ = 0 and σ = 1:
$\Phi(x) = \Phi_{0,1}(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\Bigl(-\frac{u^2}{2}\Bigr) \, du, \quad x\in\mathbb{R}.$
The standard normal cdf can be expressed in terms of a special function called the error function, as
$\Phi(x) =\frac{1}{2} \Bigl[ 1 + \operatorname{erf} \Bigl( \frac{x}{\sqrt{2}} \Bigr) \Bigr], \quad x\in\mathbb{R},$
and the cdf itself can hence be expressed as
$\Phi_{\mu,\sigma^2}(x) =\frac{1}{2} \Bigl[ 1 + \operatorname{erf} \Bigl( \frac{x-\mu}{\sigma\sqrt{2}} \Bigr) \Bigr], \quad x\in\mathbb{R}.$
The complement of the standard normal cdf, $1 - \Phi(x)$, is often denoted $Q(x)$, and is sometimes referred to simply as the Q-function, especially in engineering texts. This represents the tail
probability of the Gaussian distribution. Other definitions of the Q-function, all of which are simple transformations of $\Phi$, are also used occasionally.
The inverse standard normal cumulative distribution function, or quantile function, can be expressed in terms of the inverse error function:
$\Phi^{-1}(p) = \sqrt2 \;\operatorname{erf}^{-1} (2p - 1), \quad p\in(0,1),$
and the inverse cumulative distribution function can hence be expressed as
$\Phi_{\mu,\sigma^2}^{-1}(p) = \mu + \sigma\Phi^{-1}(p) = \mu + \sigma\sqrt2 \; \operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).$
This quantile function is sometimes called the probit function. There is no elementary primitive for the probit function. This is not to say merely that none is known, but rather that the
non-existence of such an elementary primitive has been proved. Several accurate methods exist for approximating the quantile function for the normal distribution - see quantile function for a
discussion and references.
The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions.
Strict lower and upper bounds for the cdf
For large x the standard normal cdf $\scriptstyle\Phi(x)$ is close to 1 and $\scriptstyle\Phi(-x)\,{=}\,1\,{-}\,\Phi(x)$ is close to 0. The elementary bounds
$\frac{x}{1+x^2}\varphi(x)<1-\Phi(x)<\frac{\varphi(x)}{x}, \qquad x>0,$
in terms of the density $\scriptstyle\varphi$ are useful.
Using the substitution v = u²/2, the upper bound is derived as follows:
\begin{align} 1-\Phi(x) &=\int_x^\infty\varphi(u)\,du\\ &<\int_x^\infty\frac ux\varphi(u)\,du =\int_{x^2/2}^\infty\frac{e^{-v}}{x\sqrt{2\pi}}\,dv =-\biggl.\frac{e^{-v}}{x\sqrt{2\pi}}\biggr|_{x^2/
2}^\infty =\frac{\varphi(x)}{x}. \end{align}
Similarly, using $\scriptstyle\varphi'(u)\,{=}\,-u\,\varphi(u)$ and the quotient rule,
\begin{align} \Bigl(1+\frac1{x^2}\Bigr)(1-\Phi(x)) &=\int_x^\infty \Bigl(1+\frac1{x^2}\Bigr)\varphi(u)\,du\\ &>\int_x^\infty \Bigl(1+\frac1{u^2}\Bigr)\varphi(u)\,du =-\biggl.\frac{\varphi(u)}u\
biggr|_x^\infty =\frac{\varphi(x)}x. \end{align}
Solving for $\scriptstyle 1\,{-}\,\Phi(x)\,$ provides the lower bound.
Generating functions
Moment generating function
The moment generating function is defined as the expected value of exp(tX). For a normal distribution, the moment generating function is
\begin{align} M_X(t) & {} = \mathrm{E} \left[ \exp{(tX)} \right] \\ & {} = \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi} } \exp{\left( -\frac{(x - \mu)^2}{2 \sigma^2} \right)} \exp{(tx)} \,
dx \\ & {} = \exp{ \left( \mu t + \frac{\sigma^2 t^2}{2} \right)} \end{align}
as can be seen by completing the square in the exponent.
Cumulant generating function
The cumulant generating function is the logarithm of the moment generating function: g(t) = μt + σ²t²/2. Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.
Characteristic function
The characteristic function is defined as the expected value of $\exp (i t X)$, where $i$ is the imaginary unit. So the characteristic function is obtained by replacing $t$ with $it$ in the
moment-generating function.
For a normal distribution, the characteristic function is
\begin{align} \chi_X(t;\mu,\sigma) &{} = M_X(i t) = \mathrm{E} \left[ \exp(i t X) \right] \\ &{}= \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left(- \frac{(x - \mu)^2}{2\sigma^2} \
right) \exp(i t x) \, dx \\ &{}= \exp \left( i \mu t - \frac{\sigma^2 t^2}{2} \right). \end{align}
Some properties of the normal distribution:
1. If $X \sim N(\mu, \sigma^2)$ and $a$ and $b$ are real numbers, then $a X + b \sim N(a \mu + b, (a \sigma)^2)$ (see expected value and variance).
2. If $X \sim N(\mu_X, \sigma^2_X)$ and $Y \sim N(\mu_Y, \sigma^2_Y)$ are independent normal random variables, then:
□ Their sum is normally distributed with $U = X + Y \sim N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ ( proof). Interestingly, the converse holds: if two independent random variables have a
normally-distributed sum, then they must be normal themselves — this is known as Cramér's theorem.
□ Their difference is normally distributed with $V = X - Y \sim N(\mu_X - \mu_Y, \sigma^2_X + \sigma^2_Y)$.
□ If the variances of X and Y are equal, then U and V are independent of each other.
□ The Kullback-Leibler divergence, $D_{\rm KL}( X \| Y ) = { 1 \over 2 } \left( \log \left( { \sigma^2_Y \over \sigma^2_X } \right) + \frac{\sigma^2_X}{\sigma^2_Y} + \frac{\left(\mu_Y - \mu_X\
right)^2}{\sigma^2_Y} - 1\right).$
3. If $X \sim N(0, \sigma^2_X)$ and $Y \sim N(0, \sigma^2_Y)$ are independent normal random variables, then:
□ Their product $X Y$ follows a distribution with density $p$ given by
$p(z) = \frac{1}{\pi\,\sigma_X\,\sigma_Y} \; K_0\left(\frac{|z|}{\sigma_X\,\sigma_Y}\right),$ where $K_0$ is a modified Bessel function of the second kind.
□ Their ratio follows a Cauchy distribution with $X/Y \sim \mathrm{Cauchy}(0, \sigma_X/\sigma_Y)$. Thus the Cauchy distribution is a special kind of ratio distribution.
4. If $X_1, \dots, X_n$ are independent standard normal variables, then $X_1^2 + \cdots + X_n^2$ has a chi-square distribution with n degrees of freedom.
Standardizing normal random variables
As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.
If $X$ ~ $N(\mu, \sigma^2)$, then
$Z = \frac{X - \mu}{\sigma} \!$
is a standard normal random variable: $Z$ ~ $N(0,1)$. An important consequence is that the cdf of a general normal distribution is therefore
$\Pr(X \le x) = \Phi \left( \frac{x-\mu}{\sigma} \right) = \frac{1}{2} \left( 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right) .$
Conversely, if $Z$ is a standard normal distribution, $Z$ ~ $N(0,1)$, then
$X = \sigma Z + \mu$
is a normal random variable with mean $\mu$ and variance $\sigma^2$.
The standard normal distribution has been tabulated (usually in the form of value of the cumulative distribution function Φ), and the other normal distributions are the simple transformations, as
described above, of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.
The first few moments of the normal distribution are:
Number Raw moment Central moment Cumulant
1 $\mu$ 0 $\mu$
2 $\mu^2 + \sigma^2$ $\sigma^2$ $\sigma^2$
3 $\mu^3 + 3\mu\sigma^2$ 0 0
4 $\mu^4 + 6 \mu^2 \sigma^2 + 3 \sigma^4$ $3 \sigma^4$ 0
5 $\mu^5 + 10 \mu^3 \sigma^2 + 15 \mu \sigma^4$ 0 0
6 $\mu^6 + 15 \mu^4 \sigma^2 + 45 \mu^2 \sigma^4 + 15 \sigma^6$ $15 \sigma^6$ 0
7 $\mu^7 + 21 \mu^5 \sigma^2 + 105 \mu^3 \sigma^4 + 105 \mu \sigma^6$ 0 0
8 $\mu^8 + 28 \mu^6 \sigma^2 + 210 \mu^4 \sigma^4 + 420 \mu^2 \sigma^6 + 105 \sigma^8$ $105 \sigma^8$ 0
All cumulants of the normal distribution beyond the second are zero.
Higher central moments (of order $2k$ with $\mu=0$) can be obtained using the formula
$E\left[x^{2k}\right]=\frac{(2k)!}{2^k k!} \sigma^{2k}.$
Generating values for normal random variables
For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods and the most basic is to invert the standard normal cdf. More efficient
methods are also known, one such method being the Box-Muller transform. An even faster algorithm is the ziggurat algorithm.
The Box-Muller algorithm says that, if you have two numbers a and b uniformly distributed on (0, 1], (e.g. the output from a random number generator), then two standard normally distributed random
variables are c and d, where:
$c = \sqrt{- 2 \ln a} \cdot \cos(2 \pi b)$
$d = \sqrt{- 2 \ln a} \cdot \sin(2 \pi b)$
This is because the chi-square distribution with two degrees of freedom (see property 4 above) is an easily-generated exponential random variable.
The central limit theorem
Under certain conditions (such as being independent and identically-distributed with finite variance), the sum of a large number of random variables is approximately normally distributed — this is
the central limit theorem.
The practical importance of the central limit theorem is that the normal cumulative distribution function can be used as an approximation to some other cumulative distribution functions, for example:
• A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 (some books recommend using this approximation only if np and n(1 − p) are both
at least 5; in this case, a continuity correction should be applied).
The approximating normal distribution has parameters μ = np, σ^2 = np(1 − p).
• A Poisson distribution with parameter λ is approximately normal for large λ.
The approximating normal distribution has parameters μ = σ^2 = λ.
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such
approximations are less accurate in the tails of the distribution. A general upper bound of the approximation error of the cumulative distribution function is given by the Berry–Esséen theorem.
Infinite divisibility
The normal distributions are infinitely divisible probability distributions: Given a mean μ, a variance σ ^2 ≥ 0, and a natural number n, the sum X[1] + . . . + X[n] of n independent random variables
$X_1,X_2,\dots,X_n \sim N(\mu/n, \sigma^2\!/n)\,$
has this specified normal distribution (to verify this, use characteristic functions or convolution and mathematical induction).
The normal distributions are strictly stable probability distributions.
Standard deviation and confidence intervals
About 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie
within three standard deviations. This is known as the " 68-95-99.7 rule" or the " empirical rule."
To be more precise, the area under the bell curve between μ − nσ and μ + nσ in terms of the cumulative normal distribution function is given by
\begin{align}&\Phi_{\mu,\sigma^2}(\mu+n\sigma)-\Phi_{\mu,\sigma^2}(\mu-n\sigma)\\ &=\Phi(n)-\Phi(-n)=2\Phi(n)-1=\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr),\end{align}
where erf is the error function. To 12 decimal places, the values for the 1-, 2-, up to 6-sigma points are:
$n\,$ $\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr)\,$
1 0.682689492137
2 0.954499736104
3 0.997300203937
4 0.999936657516
5 0.999999426697
6 0.999999998027
The next table gives the reverse relation of sigma multiples corresponding to a few often used values for the area under the bell curve. These values are useful to determine (asymptotic) confidence
intervals of the specified levels for normally distributed (or asymptotically normal) estimators:
$\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr)$ $n\,$
0.80 1.28155
0.90 1.64485
0.95 1.95996
0.98 2.32635
0.99 2.57583
0.995 2.80703
0.998 3.09023
0.999 3.29052
where the value on the left of the table is the proportion of values that will fall within a given interval and n is a multiple of the standard deviation that specifies the width of the interval.
Exponential family form
The Normal distribution is a two-parameter exponential family form with natural parameters μ and 1/σ^2, and natural statistics X and X^2. The canonical form has parameters ${\mu \over \sigma^2}$ and
${1 \over \sigma^2}$ and sufficient statistics $\sum x$ and $-{1 \over 2} \sum x^2$.
Complex Gaussian process
Consider complex Gaussian random variable,
where X and Y are real and independent Gaussian variables with equal variances $\scriptstyle \sigma_r^2\,$. The pdf of the joint variables is then
$\frac{1}{2\,\pi\,\sigma_r^2} e^{-(x^2+y^2)/(2 \sigma_r ^2)}$
Because $\scriptstyle \sigma_z\, =\, \sqrt{2}\sigma_r$, the resulting pdf for the complex Gaussian variable Z is
$\frac{1}{\pi\,\sigma_z^2} e^{-|z|^2/\sigma_z^2}.$
Related distributions
• $R \sim \mathrm{Rayleigh}(\sigma^2)$ is a Rayleigh distribution if $R = \sqrt{X^2 + Y^2}$ where $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are two independent normal distributions.
• $Y \sim \chi_{u}^2$ is a chi-square distribution with $u$ degrees of freedom if $Y = \sum_{k=1}^{u} X_k^2$ where $X_k \sim N(0,1)$ for $k=1,\dots,u$ and are independent.
• $Y \sim \mathrm{Cauchy}(\mu = 0, \theta = 1)$ is a Cauchy distribution if $Y = X_1/X_2$ for $X_1 \sim N(0,1)$ and $X_2 \sim N(0,1)$ are two independent normal distributions.
• $Y \sim \mbox{Log-N}(\mu, \sigma^2)$ is a log-normal distribution if $Y = e^X$ and $X \sim N(\mu, \sigma^2)$.
• Relation to Lévy skew alpha-stable distribution: if $X\sim \textrm{Levy-S}\alpha\textrm{S}(2,\beta,\sigma/\sqrt{2},\mu)$ then $X \sim N(\mu,\sigma^2)$.
• Truncated normal distribution. If $X \sim N(\mu, \sigma^2),\!$ then truncating X below at $A$ and above at $B$ will lead to a random variable with mean $E(X)=\mu + \frac{\sigma(\varphi_1-\
varphi_2)}{T},\!$ where $T=\Phi\left(\frac{B-\mu}{\sigma}\right)-\Phi\left(\frac{A-\mu}{\sigma}\right), \; \varphi_1 = \varphi\left(\frac{A-\mu}{\sigma}\right), \; \varphi_2 = \varphi\left(\frac
{B-\mu}{\sigma}\right)$ and $\varphi$ is the probability density function of a standard normal random variable.
• If $X$ is a random variable with a normal distribution, and $Y=|X|$, then $Y$ has a folded normal distribution.
Descriptive and inferential statistics
Many scores are derived from the normal distribution, including percentile ranks ("percentiles"), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, a number of behavioural
statistical procedures are based on the assumption that scores are normally distributed; for example, t-tests and ANOVAs (see below). Bell curve grading assigns relative grades based on a normal
distribution of scores.
Normality tests
Normality tests check a given set of data for similarity to the normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small
P-value indicates non-normal data.
• Kolmogorov-Smirnov test
• Lilliefors test
• Anderson-Darling test
• Ryan-Joiner test
• Shapiro-Wilk test
• Normal probability plot ( rankit plot)
• Jarque-Bera test
Estimation of parameters
Maximum likelihood estimation of parameters
are independent and each is normally distributed with expectation μ and variance σ² > 0. In the language of statisticians, the observed values of these n random variables make up a "sample of size n
from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on the observed values of this sample. The continuous joint
probability density function of these n independent random variables is
\begin{align}f(x_1,\dots,x_n;\mu,\sigma) &= \prod_{i=1}^n \varphi_{\mu,\sigma^2}(x_i)\\ &=\frac1{(\sigma\sqrt{2\pi})^n}\prod_{i=1}^n \exp\biggl(-{1 \over 2} \Bigl({x_i-\mu \over \sigma}\Bigr)^2\
biggr), \quad(x_1,\ldots,x_n)\in\mathbb{R}^n. \end{align}
As a function of μ and σ, the likelihood function based on the observations X[1], ..., X[n] is
$L(\mu,\sigma) = \frac C{\sigma^n} \exp\left(-{\sum_{i=1}^n (X_i-\mu)^2 \over 2\sigma^2}\right), \quad\mu\in\mathbb{R},\ \sigma>0,$
with some constant C > 0 (which in general would be even allowed to depend on X[1], ..., X[n], but will vanish anyway when partial derivatives of the log-likelihood function with respect to the
parameters are computed, see below).
In the method of maximum likelihood, the values of μ and σ that maximize the likelihood function are taken as estimates of the population parameters μ and σ.
Usually in maximizing a function of two variables, one might consider partial derivatives. But here we will exploit the fact that the value of μ that maximizes the likelihood function with σ fixed
does not depend on σ. Therefore, we can find that value of μ, then substitute it for μ in the likelihood function, and finally find the value of σ that maximizes the resulting expression.
It is evident that the likelihood function is a decreasing function of the sum
$\sum_{i=1}^n (X_i-\mu)^2. \,\!$
So we want the value of μ that minimizes this sum. Let
be the "sample mean" based on the n observations. Observe that
\begin{align} \sum_{i=1}^n (X_i-\mu)^2 &=\sum_{i=1}^n\bigl((X_i-\overline{X}_n)+(\overline{X}_n-\mu)\bigr)^2\\ &=\sum_{i=1}^n(X_i-\overline{X}_n)^2 + 2(\overline{X}_n-\mu)\underbrace{\sum_{i=1}^n
(X_i-\overline{X}_n)}_{=\,0} + \sum_{i=1}^n (\overline{X}_n-\mu)^2\\ &=\sum_{i=1}^n(X_i-\overline{X}_n)^2 + n(\overline{X}_n-\mu)^2. \end{align}
Only the last term depends on μ and it is minimized by
That is the maximum-likelihood estimate of μ based on the n observations X[1], ..., X[n]. When we substitute that estimate for μ into the likelihood function, we get
$L(\overline{X}_n,\sigma) = \frac C{\sigma^n} \exp\biggl(-{\sum_{i=1}^n (X_i-\overline{X}_n)^2 \over 2\sigma^2}\biggr), \quad\sigma>0.$
It is conventional to denote the "log-likelihood function", i.e., the logarithm of the likelihood function, by a lower-case $\ell$, and we have
$\ell(\overline{X}_n,\sigma)=\log C-n\log\sigma-{\sum_{i=1}^n(X_i-\overline{X}_n)^2 \over 2\sigma^2}, \quad\sigma>0,$
and then
\begin{align} {\partial \over \partial\sigma}\ell(\overline{X}_n,\sigma) &=-{n \over \sigma} +{\sum_{i=1}^n (X_i-\overline{X}_n)^2 \over \sigma^3}\\ &=-{n \over \sigma^3}\biggl(\sigma^2-{1 \over
n}\sum_{i=1}^n (X_i-\overline{X}_n)^2 \biggr), \quad\sigma>0. \end{align}
This derivative is positive, zero, or negative according as σ² is between 0 and
$\hat\sigma_n^2:={1 \over n}\sum_{i=1}^n(X_i-\overline{X}_n)^2,$
or equal to that quantity, or greater than that quantity. (If there is just one observation, meaning that n = 1, or if X[1] = ... = X[n], which only happens with probability zero, then $\hat\sigma{}
_n^2=0$ by this formula, reflecting the fact that in these cases the likelihood function is unbounded as σ decreases to zero.)
Consequently this average of squares of residuals is the maximum-likelihood estimate of σ², and its square root is the maximum-likelihood estimate of σ based on the n observations. This estimator $\
hat\sigma{}_n^2$ is biased, but has a smaller mean squared error than the usual unbiased estimator, which is n/(n − 1) times this estimator.
Surprising generalization
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is subtle. It involves the spectral theorem and the reason it can be better to view a
scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices.
Unbiased estimation of parameters
The maximum likelihood estimator of the population mean $\mu$ from a sample is an unbiased estimator of the mean, as is the variance when the mean of the population is known a priori. However, if we
are faced with a sample and have no knowledge of the mean or the variance of the population from which it is drawn, the unbiased estimator of the variance $\sigma^2$ is:
$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2.$
This "sample variance" follows a Gamma distribution if all X[i] are independent and identically-distributed:
$S^2 \sim \operatorname{Gamma}\left(\frac{n-1}{2},\frac{2 \sigma^2}{n-1}\right).$
Approximately normal distributions occur in many situations, as a result of the central limit theorem. When there is reason to suspect the presence of a large number of small effects acting
additively and independently, it is reasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption, for example the Kolmogorov-Smirnov test.
Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is
normally distributed. The distribution of the directly observed variable is then called log-normal.
Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the
external variable is held constant, the resulting marginal distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is
related to the theory of errors (see below).
To summarize, here is a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.
• In counting problems (so the central limit theorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as
• In physiological measurements of biological specimens:
□ The logarithm of measures of size of living tissue (length, height, skin area, weight);
□ The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
□ Other physiological measures may be normally distributed, but there is no reason to expect that a priori;
• Measurement errors are often assumed to be normally distributed, and any deviation from normality is considered something which should be explained;
• Financial variables
□ Changes in the logarithm of exchange rates, price indices, and stock market indices; these variables behave like compound interest, not like simple interest, and so are multiplicative;
□ Other financial variables may be normally distributed, but there is no reason to expect that a priori;
• Light intensity
□ The intensity of laser light is normally distributed;
□ Thermal light has a Bose-Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.
Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.
Photon counting
Light intensity from a single source varies with time, as thermal fluctuations can be observed if the light is analyzed at sufficiently high time resolution. The intensity is usually assumed to be
normally distributed. Quantum mechanics interprets measurements of light intensity as photon counting. The natural assumption in this setting is the Poisson distribution. When light intensity is
integrated over times longer than the coherence time and is large, the Poisson-to-normal limit is appropriate.
Measurement errors
Normality is the central assumption of the mathematical theory of errors. Similarly, in statistical model-fitting, an indicator of goodness of fit is that the residuals (as the errors are called in
that setting) be independent and normally distributed. The assumption is that any deviation from normality needs to be explained. In that sense, both in model-fitting and in the theory of errors,
normality is the only observation that need not be explained, being expected. However, if the original data are not normally distributed (for instance if they follow a Cauchy distribution), then the
residuals will also not be normally distributed. This fact is usually ignored in practice.
Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that
the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not
been taken into account. Whether this assumption is valid is debatable. A famous and oft-quoted remark attributed to Gabriel Lippmann says: "Everyone believes in the [normal] law of errors: the
mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics."
Physical characteristics of biological specimens
The sizes of full-grown animals is approximately lognormal. The evidence and an explanation based on models of growth was first published in the 1932 book Problems of Relative Growth by Julian
Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further make the distribution of sizes deviate from lognormality.
The assumption that linear size of biological specimens is normal (rather than lognormal) leads to a non-normal distribution of weight (since weight or volume is roughly proportional to the 2nd or
3rd power of length, and Gaussian distributions are only preserved by linear transformations), and conversely assuming that weight is normal leads to non-normal lengths. This is a problem, because
there is no a priori reason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem"
goes away if lognormality is assumed.
On the other hand, there are some biological measures where normality is assumed, such as blood pressure of adult humans. This is supposed to be normally distributed, but only after separating males
and females into different populations (each of which is normally distributed).
Financial variables
Already in 1900 Louis Bachelier proposed representing price changes of stocks using the normal distribution. This approach has since been modified slightly. Because of the exponential nature of
inflation, financial indicators such as stock values and commodity prices exhibit "multiplicative behaviour". As such, their periodic changes (e.g., yearly changes) are not normal, but rather
lognormal - i.e. returns as opposed to values are normally distributed. This is still the most commonly used hypothesis in finance, in particular in asset pricing. Corrections to this model seem to
be necessary, as has been pointed out for instance by Benoît Mandelbrot, the popularizer of fractals, who observed that the changes in logarithm over short periods (such as a day) are approximated
well by distributions that do not have a finite variance, and therefore the central limit theorem does not apply. Rather, the sum of many such changes gives log-Levy distributions.
Distribution in testing and intelligence
Sometimes, the difficulty and number of questions on an IQ test is selected in order to yield normal distributed results. Or else, the raw test scores are converted to IQ values by fitting them to
the normal distribution. In either case, it is the deliberate result of test construction or score interpretation that leads to IQ scores being normally distributed for the majority of the
population. However, the question whether intelligence itself is normally distributed is more involved, because intelligence is a latent variable, therefore its distribution cannot be observed
Diffusion equation
The probability density function of the normal distribution is closely related to the (homogeneous and isotropic) diffusion equation and therefore also to the heat equation. This partial differential
equation describes the time evolution of a mass-density function under diffusion. In particular, the probability density function
$\varphi_{0,t}(x) = \frac{1}{\sqrt{2\pi t\,}}\exp\left(-\frac{x^2}{2t}\right),$
for the normal distribution with expected value 0 and variance t satisfies the diffusion equation:
$\frac{\partial}{\partial t} \varphi_{0,t}(x) = \frac{\partial^2}{\partial x^2} \varphi_{0,t}(x).$
If the mass-density at time t = 0 is given by a Dirac delta, which essentially means that all mass is initially concentrated in a single point, then the mass-density function at time t will have the
form of the normal probability density function with variance linearly growing with t. This connection is no coincidence: diffusion is due to Brownian motion which is mathematically described by a
Wiener process, and such a process at time t will also result in a normal distribution with variance linearly growing with t.
More generally, if the initial mass-density is given by a function φ(x), then the mass-density at time t will be given by the convolution of φ and a normal probability density function.
Numerical approximations of the normal distribution and its cdf
The normal distribution is widely used in scientific and statistical computing. Therefore, it has been implemented in various ways.
The GNU Scientific Library calculates values of the standard normal cdf using piecewise approximations by rational functions. Another approximation method uses third-degree polynomials on intervals .
The article on the bc programming language gives an example of how to compute the cdf in Gnu bc.
Generation of deviates from the unit normal is normally done using the Box-Muller method of choosing an angle uniformly and a radius exponential and then transforming to (normally distributed) x and
y coordinates. If log, cos or sin are expensive then a simple alternative is to simply sum 12 uniform (0,1) deviates and subtract 6 (half of 12). This is quite usable in many applications. The sum
over 12 values is chosen as this gives a variance of exactly one. The result is limited to the range (-6,6) and has a density which is a 12-section eleventh-order polynomial approximation to the
normal distribution .
A method that is much faster than the Box-Muller transform but which is still exact is the so called Ziggurat algorithm developed by George Marsaglia. In about 97% of all cases it uses only two
random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases where the combination of those two falls outside the "core of the ziggurat" a
kind of rejection sampling using logarithms, exponentials and more uniform random numbers has to be employed.
There is also some investigation into the connection between the fast Hadamard transform and the normal distribution since the transform employs just addition and subtraction and by the central limit
theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn
arbitrary data sets into a normally distributed data.
In Microsoft Excel the function NORMSDIST() calculates the cdf of the standard normal distribution, and NORMSINV() calculates its inverse function. Therefore, NORMSINV(RAND()) is an accurate but slow
way of generating values from the standard normal distribution, using the principle of Inverse transform sampling.
• The last series of the 10 Deutsche Mark banknotes featured Carl Friedrich Gauss and a graph and formula of the normal probability density function. | {"url":"https://ftp.worldpossible.org/endless/eos-rachel/RACHEL/RACHEL/modules/wikipedia_for_schools/wp/n/Normal_distribution.htm","timestamp":"2024-11-12T16:17:26Z","content_type":"text/html","content_length":"75268","record_id":"<urn:uuid:894ee9ec-ab2c-4f3a-b9d0-53f8bbb5229c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00849.warc.gz"} |
Mechanisms of color vision in cortex have not been as well characterized as those in sub-cortical areas, particularly in humans. We used fMRI in conjunction with univariate and multivariate (pattern)
analysis to test for the initial transformation of sub-cortical inputs by human visual cortex. Subjects viewed each of two patterns modulating in color between orange-cyan or lime-magenta. We tested
for higher order cortical representations of color capable of discriminating these stimuli, which were designed so that they could not be distinguished by the postulated L–M and S–(L + M)
sub-cortical opponent channels. We found differences both in the average response and in the pattern of activity evoked by these two types of stimuli, across a range of early visual areas. This
result implies that sub-cortical chromatic channels are recombined early in cortical processing to form novel representations of color. Our results also suggest a cortical bias for lime-magenta over
orange-cyan stimuli, when they are matched for cone contrast and the response they would elicit in the L–M and S–(L + M) opponent channels.
Our rich experience of color includes the ability to discriminate and identify a diverse range of combinations of hue, saturation and luminance, yet our perceptual experience is based on the activity
of just three categories of cone photoreceptor and the transformation of these signals by sub-cortical and cortical areas. At the sub-cortical level, there exist chromatically opponent channels (L–M
and S–(L + M)) that carry information in parallel to visual cortex via the parvocellular and koniocellular layers of the LGN (Derrington, Krauskopf, & Lennie,
). Cortical mechanisms of color vision are generally less well understood, although psychophysical adaptation experiments indicate the existence of higher-order color mechanisms in the human visual
system, which receive input from both the opponent channels of sub-cortical areas (Krauskopf, Williams, Mandler, & Brown,
; Webster & Mollon,
; Zaidi & Shapiro,
). In macaque it has been demonstrated that there are cells as early as V1 which prefer chromatic directions away from the cardinal directions that isolate the L–M and S–(L + M) mechanisms (Conway,
; de Valois, Cottaris, Elfar, Mahon, & Wilson,
), implying a combination of information from the sub-cortical channels early in visual cortex. For cortical cells to have chromatic preference intermediate to the cardinal axes, there must be some
combination of the L–M and S–(L + M) channels. In a recent fMRI study, Brouwer and Heeger (
) found that the principal components of the response in V1 are consistent with a response dominated by an opponent coding of color, as found in sub-cortical areas, but that by hV4 and VO it more
closely resembles our perceptual color space. That is, V1 shows a differential response to variations in color but not a continuous representation of hue, while in higher areas colors of similar hue
evoke similar responses. This finding does not rule out the possibility that the signals of the fundamental pathways (L–M and the S–(L + M)) are combined in the early visual areas, such as V1, a
possibility we address specifically in this study.
We obtained high resolution functional images of the BOLD (blood-oxygen-level-dependent) response from subjects' occipital and parietal lobes while they viewed colored stimuli. Previous studies of
cortical chromatic mechanisms in humans have used perceptually relevant hues (Brouwer & Heeger,
; Parkes, Marsman, Oxley, Goulermas, & Wuerger,
). Our stimuli were not chosen for their perceptual relevance, but were designed to be metameric to the hypothesized sub-cortical chromatic mechanisms. Specifically, the stimuli were designed so as
to fulfill the following conditions: (1) to induce the same magnitude of activity in the L–M opponent channel; and (2) to induce the same magnitude of activity in the S–(L + M) opponent channel. This
was achieved by combining a given L–M modulation with a given S-cone isolating modulation in each of two different phases. When −S was in phase with M then the stimulus appeared lime-magenta; when +S
was in phase with M then it appeared orange-cyan. The lime-magenta and orange-cyan stimuli can only be distinguished by the BOLD signal if there are cells which receive a combination of inputs from
the L–M and the S–(L + M) pathways.
Univariate and multivariate analyses tested whether the BOLD response within each cortical visual area depended upon the color of the stimulus. Univariate analyses show what information about the
stimulus is detectable in the average activity across a region, while multivariate classifiers are capable of also learning differences in the pattern of activation between stimuli. Multivariate
classification analysis (for a review, see Haynes and Rees,
) is a useful tool to test for differences in the BOLD response of a visual area even where the mean activity of the area is not significantly different between stimuli, and has been used to infer
the selectivity of different early visual areas for a range of basic visual attributes and their combination (Haynes & Rees,
; Kamitani & Tong,
; Mannion, McDonald, & Clifford,
; Parkes et al.,
; Seymour, Clifford, Logothetis, & Bartels,
; Sumner, Anderson, Sylvester, Haynes, & Rees,
Both in the univariate and multivariate analyses employed here, an algorithm is trained to classify the stimulus from activity across a region, and tested on novel data. Above chance performance
indicates that the area contains information about the stimulus dimension that was varied. Here, our premise is that if we can use the activity across a visual area to discriminate between our
stimuli then that area contains a representation of color that could only be generated through a transformation of the signals from the sub-cortical L–M and S–(L + M) pathways.
Materials and methods
Color calibration procedures and display system
Stimuli were generated and displayed using Matlab (version 7) software on a Dell Latitude laptop computer driving an nVidia Quadro NVS 110M graphics card to draw stimuli to a 35 × 26 cm Philips LCD
monitor, with 60 Hz refresh rate, viewed from a distance of approximately 1.58 m. Scanning took place in a darkened room. Subjects, while lying in the scanner, viewed the monitor through a mirror
mounted above the head cage which reflected the image from the LCD monitor located behind the scanner. Stimuli were calibrated in situ for the LCD monitor and mirror arrangement, using measurements
obtained with a PR-655 SpectraScan spectrophotometer (by Photo Research Inc.).
Changes in both chromaticity and luminance of the screen with increasing R, G and B values were taken into account when generating the experimental stimuli. The CIE (xyY) coordinates measured for 16
values during calibration were interpolated to 255 values using the best-fitting spline, and these were used to calculate the luminance and chromaticity for each combination of R, G and B intensity
Data were collected on five subjects (three male), aged between 24 and 40 years, with normal or corrected to normal visual acuity and normal color vision, as tested using Ishihara plates (Ishihara,
). All subjects provided informed consent, and the entire study was carried out in accordance with guidelines of the University of Sydney Human Research Ethics Committee.
Chromatic, spatial and temporal stimulus properties
Example stimuli are shown in
Figure 1
. The stimulus was an annulus, centered on the fixation point, with an inner diameter of 0.8 degrees visual angle, and an outer diameter of 7.8 degrees. The remainder of the screen was held at the
mean luminance, which was 6.78 cd/m
[CIE (1931)
x, y
∼0.300, 0.337], and all stimuli were produced by spatiotemporal modulation around this point. The annulus contained a spatial pattern that counterphased sinusoidally at a temporal frequency of 1 Hz.
The spatial pattern was the multiplication of a radial and a concentric sinusoidal modulation, (the resultant plaid pattern is shown in
Figures 1A
). All these modulations can be represented in a three-dimensional color space described previously (Derrington et al.,
; DKL space). Along the L–M axis only the signals from L and M cones vary, in opposition, without variation in luminance. Along the orthogonal S-cone isolating axis there is no modulation of either
the L or M cones. The L–M and S axes define a plane in which only chromaticity varies. Normal to this plane is the luminance axis along which the signals from the L and M cones vary in proportion.
The axes were derived from the Stockman and Sharpe (
) 2-degree cone spectral sensitivities and adjusted individually for each observer (see below). The scaling of these axes is largely arbitrary; here we used modulations along the isoluminant axes
that were 90% of the maximum modulation achievable within the gamut of the monitor. Modulation along the L–M axis produced maximum cone contrasts of 15.4% in the L-cone and 17.8% in the M-cone; along
the S-cone axis the maximum S-cone contrast was 79.6%. Cone contrast values for all stimuli are listed in
Table 1
. Each frame of the stimulus was generated prior to the experiment as a bitmapped image, and then these images were drawn to the screen for each stimulus presentation using routines from PsychToolbox
3.0.8 (Brainard,
; Pelli,
Table 1
Stimulus Color L-cone Contrast M-cone Contrast S-cone Contrast
Dark Cyan −0.154 0.015 0.688
Light Cyan 0.031 0.178 0.796
Dark Orange −0.031 −0.178 −0.796
Light Orange 0.154 −0.015 −0.688
Dark Magenta −0.031 −0.178 0.688
Light Magenta 0.154 −0.015 0.796
Dark Lime −0.154 0.015 −0.796
Light Lime 0.031 0.178 −0.688
There were four stimuli, chosen such that over time all four would: (1) equally stimulate the L–M opponent channel; and (2) equally stimulate the S–(L + M) opponent channel. The angle of each
stimulus within the isoluminant plane was intermediate to that of the L–M and S–(L + M) axes, and was defined as a vector addition of modulations along those two axes. When the L–M modulation was in
phase with the S–(L + M) modulation the stimulus modulated between magenta and lime; when these pairings were reversed the modulation appeared orange-cyan. The point of subjective isoluminance (the
angle of the isoluminant plane from the luminance axis) was estimated separately for each observer using the minimum motion technique described by Anstis and Cavanagh (
), for the magenta-lime and for the orange-cyan modulations. A 25% luminance modulation was then added to the subjectively defined isoluminant modulation in one of two phases. The four stimuli
therefore appeared: light magenta–dark lime, dark magenta–light lime, light orange–dark cyan, and dark orange–light cyan (example stimuli in
Figure 1A
, and example cone contrast values in
Table 1
The luminance modulation was added so that if there were any residual differences in luminance between the lime-magenta and orange-cyan blocks, this difference should be masked by the much larger
luminance modulation. The contrast response of early visual areas to luminance defined stimuli is steeper at low than high contrast (for example, see Liu & Wandell,
). Thus a luminance artifact in our stimuli would result in a much smaller difference in the response to the two stimuli than if there was no luminance modulation (and hence lower luminance
contrast). The same rationale underlies the use of random luminance noise to mask potential luminance artifacts (Birch, Barbur, & Harlow,
; Kingdom, Moulden, & Collyer,
; Sumner et al.,
). For the analyses shown in
Figure 4
the classifier was trained and tested with two groups of blocks: one group included the two types of lime-magenta blocks (the two types differed in the relative phase of the luminance modulation) and
the other group included the two types of orange-cyan blocks. We also performed a control analysis, where the classifier was trained to discriminate lime-magenta vs. orange-cyan on blocks that had
only one luminance phase (one of the two types of lime-magenta blocks, and one of two types of orange-cyan blocks), and then tested on its ability to discriminate the other two types. The results of
this analysis are shown in
Experimental design
All experimental scans were completed during a single session for each subject. The session included ten functional scans, each lasting 4.5 minutes. During each scan the subject viewed 18 blocks of
the experimental stimulus, alternating between orange-cyan and lime-magenta blocks. Each block was 15 seconds long, and data from the first and last block were excluded from our analysis. In order to
change the color of the stimuli, either the L–M or the S–(L + M) modulation must change phase, as illustrated in
Figure 1C
. This phase reversal is likely to induce an increased response of a neural population which responds to the relevant cardinal color, as in the characteristic response rebound used in fMRI adaptation
(for example, see Engel & Furmanski,
; Kourtzi, Erb, Grodd, & Bülthoff,
). It is also possible that the response to the phase reversal may be evident in the BOLD signal, even when averaging across activity within a block, and could potentially be used to discriminate the
two types of stimuli (for example, if an orange-cyan block always commenced with a L–M phase reversal). In order to eliminate this potential source of information about the color of the stimuli, we
balanced the stimuli so that the pattern of phase reversals could not be used to predict the stimulus. We used one of two block orders for each run, where the second was a reversal of the first
To select those voxels in each visual area most responsive to the experimental stimuli, two additional localizer scans during the same session as the experimental scans for each subject. Using a
localizer scan that is separate from the experimental data avoids circularity that could otherwise be present (Kriegeskorte, Simmons, Bellgowan, & Baker,
). Localizer scans included a total of 17 blocks of 15 seconds each, comprised of stimulus blocks interleaved with blocks of fixation only. The stimulus blocks included 2 lime-magenta blocks and 2
orange-cyan blocks, in addition to 4 black-white blocks where the stimulus had the same spatial arrangement but was modulated between black and white.
Fixation task
Throughout experimental and localizer scans, subjects performed a task at fixation that was unrelated to the annular experimental stimulus. This task was designed to be attentionally demanding in
order to direct attention away from the experimental stimulus as much as possible. While this would have reduced the BOLD response, and so likely decreased the ratio of signal to noise in the data
which were input to the classifier, it greatly reduces the chance that subjects could have systematically directed more attention to one type of stimulus block.
Subjects were required to detect a conjunction of contrast polarity and number in a digit stream of the digits 0 to 9 inclusive, presented at fixation, updated at 3 Hz. Digits were either black or
white, against the mean gray of the background, as seen in
Figure 1B
, and the order was randomly generated for each run. Subjects responded with a button press to the onset of either of two target digits, one only when black and the other only when white (for example
a black 3 or a white 7). Responding to a conjunction of digit and contrast polarity made this a difficult task. Target digits were updated at the beginning of each run to increase task difficulty and
minimize practice effects.
All subjects performed the task significantly above chance (p < 0.01, permutation test), demonstrating that they were engaged in the task, but each subject also made errors, implying that the task
was not trivial and required attention. For no subject was there a significant difference in performance between lime-magenta and orange-cyan blocks, consistent with equal attentional resources being
devoted to the task in each case.
fMRI methods
fMRI data were collected using a 3T Philips scanner (Symbion Imaging Centre, Prince of Wales Medical Research Institute, Sydney, Australia), with a birdcage head coil.
Anatomical measurements and definition of gray matter
The anatomical image for each subject was generated from the average of three scans. Two of these were high resolution (1 × 1 × 1 mm) structural MR images of each subject's whole brain, acquired
using a Turbo Field Echo (TFE) protocol for enhanced gray–white contrast. A third, higher resolution (0.75 × 0.75 × 0.75 mm) scan of the caudal half of the head was also acquired in order to recover
more anatomical detail of the occipital lobes.
Using the Statistical Parametric Mapping (SPM) software package SPM5 (Frackowiak, Friston, Frith, Dolan, & Mazziotta,
), anatomical images were each reoriented to approximately the same space using anterior and posterior commissures as anatomical landmarks. Fine alignment of these anatomical images was carried out
using normalized mutual information based coregistration, and each of the anatomical images were resampled so that they were in the same voxel space with a resolution of 0.75 × 0.75 × 0.75 mm. From
each image we removed intensity inhomogeneities using a nonparametric inhomogeneity correction method (Manjón et al.,
), and normalized the images such that the white matter had an approximate intensity of 1. The coregistered, inhomogeneity corrected, normalized images were then averaged together to produce a mean
anatomical image for each subject.
ITKGray software (Yushkevich et al.,
) was used to define the white matter of each subject, initially using automatic segmentation tools, then using manual editing. The segmentation image was imported into mrGray, part of the mrVista
software package developed by the Stanford Vision and Imaging Lab (
). In mrGray, gray matter was ‘grown’ out from the white matter in a sheet with a maximum thickness of 4 voxels.
Functional measurements
fMRI data were acquired using a T2*-sensitive, FEEPI pulse sequence, with echo time (TE) of 32 ms; time to repetition (TR) of 3000 ms; flip angle 90; field of view 192 mm × 70.5 mm × 192 mm;
effective in-plane resolution 1.5 mm × 1.5 mm, and slice thickness 1.5 mm. 47 slices were collected in an interleaved, ascending order, in a coronal plane tilted such that the scan covered the whole
of the occipital lobe and the posterior part of the parietal and temporal lobes. Using SPM5, all functional data were preprocessed to correct for slice time and head motion before alignment to the
structural data. Data from functional scans were aligned to a whole head anatomical scan acquired in the same session, using normalized mutual information based coregistration. The functional data
were then aligned to the subject's average anatomical by first aligning the within session anatomical with the average anatomical scan, then applying the same transformation to the functional data.
Definition of retinotopic areas
For each subject, the precise anatomical locations of the early areas of visual cortex (V1, V2, V3, V3A/B, hV4, and VO) were found functionally using standard retinotopic mapping procedures (Engel,
Glover, & Wandell,
; Larsson & Heeger,
; see Wandell, Dumoulin, & Brewer,
, for a summary). Subjects were scanned while viewing first a rotating wedge then an expanding ring stimulus, overlaid on a fixation cross of light gray lines, as shown on the key above the maps in
Figure 2
(Schira, Tyler, Breakspear, & Spehar,
Averaged data from the wedge and ring stimuli were smoothed with a Gaussian kernel of half width 1.5 mm, then projected onto a computationally flattened representation of the cortex for each
hemisphere of each subject, using mrVista. Areas V1, V2, V3, V3A/B and hV4 were manually defined on the phase and eccentricity maps derived from the wedge and ring stimuli (shown for an example
subject in
Figures 2A
, respectively), using the conventions described by Larsson and Heeger (
). According to these definitions the foveal representation at the occipital pole is shared by V1, V2, V3, and hV4, while V3A and V3B, which border the dorsal part of V3, share a dorsal fovea. For
our analysis we did not attempt to separate V3A and V3B. We defined hV4 as a ventral hemifield representation that borders the ventral part of V3.
For area VO, the phase and eccentricity maps were considered in conjunction with a flattened map of those voxels that responded more to chromatic than achromatic stimuli in the localizer scan (as
shown for an example subject in
Figure 2C
). Where it existed, we used the hemifield representation from the phase and eccentricity maps to define VO. Where the retinotopic map from the wedge and ring stimuli was unclear, we tended towards a
liberal definition of VO, in order to avoid excluding any voxels in the region which showed a preference for chromatic stimuli in the localizer analysis. Each retinotopic area was defined on the
flattened map of a subject's cortex then transformed into the space of the subject's anatomical, smoothed (FWHM = 1.5 mm), and resliced to the resolution of the functional images using 4th degree
B-spline interpolation. Voxels assigned to each visual area were allocated a value reflecting the cumulative influence of such transformations. To prevent overlapping voxels between adjacent visual
areas, each voxel was assigned to the visual area for which it possessed the greatest value.
Area MT+ was defined on the basis of a separate localizer scan in which blocks of low contrast static and moving dots were interleaved with fixation only blocks. In SPM5, we specified a general
linear model of this data, and defined MT+ by finding lateral clusters of voxels that responded more to moving than to static dots. The definition of MT+ was projected onto the flattened map for the
purposes of illustration, as in
Figure 2
, but the original 3D definition of MT+ was used in the analysis of the functional data.
Functional preprocessing
Data from each of the two localizer scans were processed using the methods described above, then analyzed using a General Linear Model (GLM) in SPM5. We pooled responses to the luminance and
chromatically defined stimuli and contrasted these with the response to the fixation only blocks. The subsequent analyses included only those voxels that responded significantly more (
< 0.05, uncorrected) to the stimulus than fixation only. These voxels are shown on an example flattened map for one subject in
Figure 2D
For all functional scans the BOLD signal was labeled with the stimulus presented 2 images (6 seconds) previously, in order to compensate for the delayed hemodynamic response, then was highpass
filtered (cutoff 128 s, using filtering methods from SPM5) in order to remove low frequency confounds in the data, and finally converted into z-scores for each of the ten runs in order to reduce
variability from inter-run differences. Data from each voxel were z-scored separately. Within each 15 second block the BOLD response, normalized according to the procedures described above, was
averaged across the 5 measurements to give a single score for each block in each run.
Classifier analysis
Classifiers were restricted to each of several functionally defined visual areas for each subject and trained to discriminate the two patterns. We compared the performance of classifiers trained on
two types of data for each area: in the univariate case, the classifier was trained and tested on the average activation across voxels within an area (that is, 1 value per block), while in the
multivariate case the classifier was trained and tested on the pattern of activity across voxels within an area (n values per block).
We used a linear support vector machine (SVM) classification technique in our analysis. Support vector machines are powerful in their ability to learn a decision rule for multivariate data (Bennett &
): in our case, for
voxels with 144 data points each (72 from lime-magenta and 72 from orange-cyan blocks) they learn the hyperplane which best separates the data points in an n dimensional space, where each dimension
corresponds to the normalized response of one voxel (using linear SVMs, we require that the hyperplane's projection onto any two dimensions is linear). We evaluated classifier performance on its
ability to generalize, i.e. to correctly discriminate data that were not included in the training set. For the univariate classifier the technique was the same, but there was only one dimension along
which the 144 data points varied, so the power of the support vector machine was not utilized.
Cross-validation: Leave-one-out train and test
Classification analysis was performed using a Matlab (version 7.5) interface to SVM-light 6.01 (Joachims,
). The classifier was trained on the scores from 9 runs and tested on a tenth; this procedure was repeated 10 times. This leave-one-out train and test procedure resulted in the data from each run
being used as test data once, giving an average classifier performance (reported in the
section as a percentage correct) based on the accuracy across 160 classifications, while ensuring that the test data never included data that were used in training.
We repeated the classification analysis with increasing numbers of voxels (
) within each visual area, from
= 3 voxels, to the
= max
where max
is the total number of voxels that reached significance in the localizer analysis. Voxels in each area were ranked according to their
statistic from the localizer analysis, based on the separate localizer scans, in order to select voxels that responded to the area of visual field occupied by our experimental stimuli and exclude
those which represented areas of the visual field which were more foveal or peripheral than our annular stimuli. The top n most significant voxels were used in each case. Classifier performance
generally increased as more voxels were included in the analysis, but there was some variability around this trend. To summarize classification performance (as reported below) we fit the classifier
performance (
) as number of voxels (
) increased with an exponential growth function which reaches a limit (
), given by
$P=0.5+(L−0.5)(1− e − n / c),$
where 0.5 is chance performance (at
= 0), and
is a curvature parameter, specifying how many voxels the curve takes to reach the limit,
. When the curve fit the data, the classifier performance reported below corresponds to the limit (
) of the growth function. When the curve could not be fit to the data within 100 iterations of the Matlab function
(usually when classifier performance was low), the average classifier performance, rather than the limit of the curve, is reported as the summary statistic of classifier performance. Classifier
performance as a function of number of voxels, along with the best-fitting curve, is plotted in
Permutation analysis
To test the statistical significance of classifier performance, we ran a permutation analysis to estimate classification accuracy expected by chance alone. Permutation tests are non-parametric and so
do not include an assumption of normality, and such tests have previously been employed to evaluate classification analysis (Mourao-Miranda, Bokde, Born, Hampel, & Stetter,
). For each area in each subject we performed the same analysis as that described above, except that before training the classifier we randomly permuted the stimulus labels associated with each block
in the training data set. Using 1000 repetitions of this permutation analysis, we generated a population of 1000 estimates of the classifier accuracy that could be expected in cases where the data
did not contain any stimulus-related information. For each iteration of the permutation analysis we averaged these estimates across subjects for each area and compared these 1000 values with the
observed between-subject mean accuracy. In the statistics reported below for classifier performance,
-values were calculated by finding the proportion of these 1000 estimates which were greater than the observed classifier accuracy.
We tested for the presence of cortical representations of color capable of discriminating stimuli that cannot be distinguished by the L–M and S–(L + M) sub-cortical opponent channels. Our
lime-magenta and orange-cyan stimuli contain the same L–M and S–(L + M) modulations, varying only in the phase that these modulations are added together. For the lime-magenta stimuli, the L–M and S–
(L + M) modulations are the same phase, whereas for the orange-cyan stimuli the L–M and S–(L + M) modulations were in opposite phase. We used fMRI to measure changes in BOLD signal as an indirect
measure of neural activity, then asked the extent to which the visual stimulus could be discriminated from patterns of brain activity in a predefined visual area.
Below, we report stimulus related differences in both the mean activity and pattern of activity across a range of regions. There was a small but reliable bias across subjects for lime-magenta over
orange-cyan stimuli in the mean activity across each region, and we found that the difference in the mean activity was sufficient for a univariate classifier to learn to correctly discriminate the
stimuli. We also found evidence of additional stimulus-related information in the pattern of activity across V1 and V2, using multivariate classifiers.
Consistent bias for lime-magenta over orange-cyan stimuli in average activity
There was a significant difference in the response to lime-magenta vs. orange-cyan stimuli, impossible without combination of signals from the fundamental cone-opponent channels. For the univariate
analysis we averaged across those voxels within each area for which there was a significant difference in their response to the localizer stimulus versus fixation. The average
-scored activity across an area for each block was treated as a separate measure of the area's response to lime-magenta or orange-cyan stimuli, giving 80 measurements for each; the distributions of
these measurements for each subject are shown in
Figure 3
In all subjects each area that showed a significant bias for one color modulation showed the same bias; signal was greater for lime-magenta than for orange-cyan. As shown in
Figure 3
, the mean of the 80 lime-magenta blocks was significantly greater than the 80 orange cyan-blocks (
< 0.05, two-tailed
-test); in 25 areas across 5 subjects. The only area for which there was not a significant difference between the two types of stimuli for any subject was area MT+. We conclude that stimuli which
equally modulate the cardinal axes of color space are not equally represented in visual cortical areas, and this biased representation is seen as early as V1.
Consistent with the differences in the average response across each region, univariate classifiers performed significantly better than chance as early as V1, as shown in
Figure 4
(light gray bars). The average univariate classification performance across subjects in all areas was significantly above chance (
< 0.05, one-tailed permutation test). MT+ showed the lowest performance for 4 of 5 subjects.
For 4 of 5 subjects, the area with the best classifier performance was V2. The earlier cortical visual areas (V1, V2 and V3) generally outperformed the dorsal area V3A/B, as well as the ventral areas
hV4 and VO. V1, V2 and V3 generally had a greater number of voxels, which may account for their high performance. In order to test this, we repeated the classifier analysis on 100 voxels that were
randomly chosen from those voxels in the area that responded to the localizer stimulus. For this subset of 100 voxels, the classifier accuracy averaged across subjects in V1, V2 and V3 (61, 64 and
61%) was still better than in V3A/B, hV4 and VO (45, 56 and 52%). This suggests that differences in classifier performance cannot be accounted for by the generally greater number of voxels included
in the analysis for areas V1, V2 and V3. Area MT+ had fewer than 100 voxels that responded to the localizer in all subjects, and so was excluded from this reanalysis.
Additional information in the pattern of activity for areas V1 and V2
Multivariate pattern classifiers were also trained to discriminate the two types of stimulus, allowing us to test for additional stimulus related information in the pattern of activity across each
area. The univariate classifier was trained on a subset (9 of 10 runs) of the average data (as plotted in
Figure 3
), and tested on the remainder. The multivariate classifier was trained and tested in the same way on data which was not averaged across voxels, so that in addition to the average it can learn
differences in the pattern of activity across an area between blocks. This analysis allows a comparison of the results from the univariate and multivariate classification techniques, which gives an
indication of how the information in the pattern of activity across an area differs from information given by the mean response.
Multivariate classification performance across areas (see
Figure 4
) followed a similar trend to that found for the univariate classifiers; earlier visual areas (V1, V2 and V3) tended to outperform V3A/B, hV4 and VO. Classification accuracy was poorest in MT+, where
performance was not significantly different from chance. This trend was also found when the classifier was trained and tested on one hundred voxels, randomly chosen from those voxels in the area that
responded to the localizer stimulus: the average between-subjects classifier accuracy in V1, V2 and V3 (63, 67 and 59%) was still better than in V3A/B, hV4 and VO (57, 56 and 49%). Classification
performance was generally higher for multivariate classifiers, and this difference was significant (
< 0.01, two-tailed permutation test) for areas V1 and V2. In VO and MT+, the univariate classifier outperformed the multivariate classifier, but this difference was not significant.
Since there was no requirement on our classifiers to predict an equal number of orange-cyan and lime-magenta blocks, it was possible for classification performance to be better for one type of test
stimulus. For example, if classification of orange-cyan test stimuli were at chance but classification performance on lime-magenta blocks were perfect, this would give an overall performance of 75%.
We found that this was not the case; for both univariate and multivariate classifiers, classification performance for lime-magenta test stimuli and classification performance for orange-cyan stimuli
showed a positive linear correlation (univariate: slope = 0.33, R ^2 = 0.13, p < 0.05, multivariate: slope = 0.85, R ^2 = 0.61, p < 0.01).
Increased performance of the multivariate classifier compared with the univariate classifier in V1 and V2 indicates that there are reliable, stimulus-related patterns of activity in these areas. If
the pattern of activity across voxels were uninformative about the non-cardinal color of the stimulus, we would expect the multivariate classifier performance to be at best the same as the univariate
case (since if the classifier learnt a pattern that was not stimulus-related, performance could decrease).
We found evidence in human visual cortex for representations of color as early as V1 that combine information from the L–M and S–(L + M) opponent pathways hypothesized to carry information in
parallel from sub-cortical areas to cortex. The ability to use BOLD activity to discriminate stimuli matched for the postulated sub-cortical mechanisms demonstrates that the neural population must
include neurons modulated by signals from both the chromatically opponent pathways. Below we discuss the implications of these results for how color is represented in human visual cortex, in
particular the bias for lime-magenta over orange-cyan stimuli, and differences between visual areas in classifier performance.
Origin of asymmetry in the representation of two non-cardinal color modulations
We found a common bias across cortical visual areas for lime-magenta over orange-cyan blocks, even though our stimuli were matched for cone contrast, and for the response of sub-cortical pathways.
The consistency of this bias across subjects suggests that it reflects a typical asymmetry in cortical representations of color. Specifically, this finding implies that there is a more numerous or
more active population of neurons which respond to lime and/or magenta than to orange and/or cyan stimuli.
There is some evidence for a bias in the opposite direction in single-unit recordings in macaque V1, and from human psychophysics. Both Conway (
) and Solomon and Lennie (
) found a bias when testing the responses of macaque V1 cells to L, M and S-cone isolating stimuli. Of the 45 (Conway,
) and 19 (Solomon & Lennie,
) L–M color opponent cells that also responded to S-cone isolating stimuli, for 93% and 89% of cells (respectively) the response to the S-cone isolating stimulus had the same sign as the M-cone
isolating stimulus; that is, the cells preferred a color direction closer to orange-cyan than lime-magenta. Krauskopf and Gegenfurtner (
) report subtle psychophysical asymmetries for human observers between the non-cardinal axes in the effects of adaptation on discrimination threshold. Their data are consistent with a greater
prevalence of adaptable mechanisms tuned to orange-cyan than to lime-magenta. Our data imply a bias of the opposite direction in human visual cortex, suggesting further work is necessary to reconcile
these findings. In a recent study on human discrimination thresholds, Danilova and Mollon (
) found that discrimination thresholds were lowest along a line in chromaticity space connecting unique blue and unique yellow. The hypothetical channel Danilova and Mollon (
) propose to account for their results would lie closer to the lime-magenta modulation than the orange-cyan modulation, and these results could be a psychophysical correlate of the bias we observed
in our fMRI data.
Organization of color processing in early human cortical areas
While significant classifier performance indicates representations of non-cardinal colors, differences in classifier performance between areas are difficult to interpret. Brouwer and Heeger (
) found highest classifier performance in V1, yet their principal components analysis suggested that the representation of color in hV4 and VO more closely matches our perceptual experience.
Classifier performance depends not only on the presence of relevant information (here, non-cardinal representations of color) within an area, but also on the accessibility of this information at the
coarse spatial scale of our functional measurements.
For areas V1 and V2, multivariate classifiers significantly outperformed univariate classifiers, showing that there was stimulus related information in the pattern of activity. In macaque V1 and V2
there are orderly maps of hue selectivity (including both cardinal and non-cardinal colors) across the surface of the cortex (Xiao, Casti, Xiao, & Kaplan,
; Xiao, Wang, & Felleman,
). If similar maps exist early in human visual cortex, their existence may increase the chance of biased sampling of chromatic preferences across voxels. The size of these hue maps, which each
represent a large spectrum of hues, is only around 200 μm across the surface of the cortex in macaque V1, with individual maps separated by around 400 μm (Xiao et al.,
). If maps of approximately the same size exist in human V1, a single voxel would contain approximately 6 hue maps. It is unlikely that a single voxel would sample neurons whose preference included
only a narrow range of hues, but it could be that this map arrangement would make it more likely for biases between voxels to arise. Furthermore, in macaque, the hue maps in V2 are on average 2 to
2.5 times longer than the hue maps in V1 (Xiao et al.,
). Larger maps with the same voxel resolution should increase the likelihood of biased sampling of hue maps, and increase the magnitude of the biases, which could underlie the tendency in our results
for classifiers in V2 to outperform classifiers in V1.
Alternatively, it is possible that the stimulus related information in the pattern of activity is not due to qualitatively different patterns of response for lime-magenta and orange-cyan but instead
a single pattern of visually responsive voxels which respond more strongly in the lime-magenta case. Since there is a univariate bias, the multivariate classifier could potentially be learning the
difference between a strong signal in noise and a weaker version of the same signal (also in noise). The increased performance of the multivariate versus univariate classifiers might then be based
purely on the ability of the multivariate classifiers to ascribe more to weight individual voxels on the basis of their signal-to-noise ratio. Further empirical and theoretical work will be required
before it is possible to discriminate with certainty between these alternatives.
Response of dorsal visual areas
Poor classifier performance in MT+ is consistent with the classifier results of Brouwer and Heeger (
), as well as evidence from MT of rhesus monkey (Britten, Shadlen, Newsome, & Movshon,
; Dubner & Zeki,
) and human MT+ (Huk, Dougherty, & Heeger,
; Tootell et al.,
; Zeki et al.,
) that this area is not generally selective for the color of surfaces and is less responsive to chromatic than achromatic stimuli (Gegenfurtner et al.,
; Liu & Wandell,
; Wandell et al.,
), although sensitivity to chromatic motion (Barberini, Cohen, Wandell, & Newsome,
; Wandell et al.,
) has been reported. Likewise, the reduced classifier performance in V3A/B with respect to V1, V2 and V3 may reflect the general preference of this dorsal area for motion (Tootell et al.,
), and its reduced responsivity to chromatically defined stimuli (Liu & Wandell,
). Nevertheless, for each subject, MT+ had fewer voxels than any other area we defined, which alone may account for decreased classifier performance.
Response of ventral visual areas
Areas hV4 and VO are often thought to be specialized for the processing of color. In macaque V4 evidence from both single unit recordings (Zeki,
) and from neuroimaging (Conway & Tsao,
) implicate this area as a ‘color center’. In human, there is converging evidence from patients with cerebral achromatopsia (Zeki,
) and from PET (Lueck et al.,
) and fMRI (Bartels & Zeki,
; Hadjikhani, Liu, Dale, Cavanagh, & Tootell,
; Liu & Wandell,
; McKeefry & Zeki,
; Mullen, Dumoulin, McMahon, de Zubicaray, & Hess,
; Wade, Brewer, Rieger, & Wandell,
) neuroimaging studies that both hV4 and VO are involved in color vision. Additionally, there is evidence that the response properties of VO match color perception in showing weaker responses to high
than to low temporal frequencies (Jiang, Zhou, & He,
; Liu & Wandell,
), while V1 does not. It therefore might be expected that classifier performance would be greatest in these areas, but this was not the case for our stimuli, or for more perceptually relevant hues
(Brouwer & Heeger,
). We consider five possible reasons for this.
First, our definitions of hV4 and VO may not include the region of ventral visual cortex that is specialized for color processing; Hadjikhani et al. (
) reported color selectivity in area V8, but not hV4. We think this account of our results is unlikely because our definition of hV4, corresponding to that of Wandell et al. (
), would include part of the V8 described by Hadjikhani et al. (
), which showed color selectivity, with the remainder of their V8 corresponding to our VO.
Second, areas hV4 and VO may be more susceptible to task specific demands than other areas. We both asked subjects not to attend to the stimuli and required them to engage with a task at fixation. By
diverting attention from the experimental stimulus we aimed to avoid artifactual classifier performance that was based not on differences in the stimulus-driven response but on differences in
attention between the two conditions. However, when attention is directed to a task unrelated to the stimulus, the stimulus-driven BOLD response is suppressed, and this suppression increases with the
attentional load of the task (Rees, Frith, & Lavie,
; Schwartz et al.,
). Single-unit recordings in macaque and fMRI in humans show that V4 and hV4 show greater attentional modulation than earlier visual areas (Hansen, Kay, & Gallant,
; Reynolds & Chelazzi,
; Schwartz et al.,
Third, reduced performance of multivariate classifiers in hV4 and VO may result if our voxel size (1.5 mm each side) resulted in biases when sampling neural representations of color in V1 and V2, but
not hV4 or VO. The spatial arrangement of chromatic preferences may be either less ordered, ordered in a different way, or ordered on a smaller spatial scale than in the earlier visual areas.
Fourth, any nonlinearity in the signal which differs between areas may enhance or reduce the stimulus discriminability in the BOLD response. For example, the contrast response of VO to L–M and S–(L +
M) modulating stimuli is more nonlinear than V1 (Liu & Wandell,
). It is not clear how such nonlinearities should affect classifier performance, but it is possible that the reduced classifier performance in ventral areas were due to such nonlinearities.
Finally, there may be increased noise in our imaging results for these areas, which are typically located further from the surface of the head than the earlier visual areas and dorsal areas, and near
the transverse sinus, which can cause imaging artifacts (Winawer, Horiguchi, Sayres, Amano, & Wandell,
in press
). We think it likely that the reduced performance of the classifiers in hV4 and VO reflects some combination of the impact of attention, nonlinearities, imaging noise and (for multivariate
classifiers) the spatial arrangement of color processing within these areas, rather than implying their diminished selectivity for color.
Limitations of this study
All these conclusions are based on the assumption that our stimuli induced responses in sub-cortical pathways that are indistinguishable when considering the responses of each pathway independently.
We consider a number of reasons why this assumption may be invalid.
Macular pigmentation selectively attenuates shorter wavelengths in the central two degrees of the visual field (Hammond, Wooten, & Snodderly,
; Wyszecki & Stiles,
). When defining our stimuli we used the Stockman and Sharpe (
) 2 degree cone spectral sensitivities, which take into account the impact of macular pigmentation. Since macular pigmentation does not extend beyond the central 2 degrees of visual field (Hammond et
; Stringham, Hammond, Wooten, & Snodderly,
), it is possible that for the region peripheral to this, our stimuli were no longer balanced for responses they induce in the sub-cortical pathways. To rule out this potential artifact, we repeated
the classifier analysis, limiting the voxels included in the classifier to those within 2 degrees visual angle from fixation and thus excluding any voxels which respond to an area of visual field for
which the stimuli may not have been balanced. With this analysis, we found classifier performance was reduced, but remained significantly above chance in all areas except MT+ (data not shown). This
rules out the possibility that classifier performance in the original analysis was based on artifacts in the stimuli caused by macular pigmentation.
It is important also to emphasize the robustness of our conclusions to any inaccuracies in the determination of subjective equiluminance for each subject. For example, let us consider the situation
if there was a 1% artifactual modulation in the lime-magenta blocks and no artifact in orange-cyan blocks. The 25% luminance modulation was added in opposite phases for different lime-magenta blocks,
meaning that the effect of the luminance artifact would be to increase effective luminance contrast to 26% in half of the lime-magenta blocks and decrease it to 24% in the other half. For such a
bidirectional effect to introduce a bias in the univariate response between lime-magenta and orange-cyan blocks, the contrast response function in the vicinity of 25% luminance contrast would have to
be highly non-linear. Furthermore, any such bias would be unlikely to show the consistency between subjects observed here. In terms of the multivariate analysis, a classifier would have to learn a
disjunctive discrimination between (24 or 26%) vs. (25%) modulation in luminance in order to be able to classify the stimuli based on luminance alone. Our use of linear classifiers minimizes the
possibility that luminance artifacts were used as a cue to discrimination.
More fundamentally, it could be that the classic model of two chromatically opponent sub-cortical channels is insufficient to capture the selectivity of all neurons in the lateral geniculate nucleus
(LGN). The majority of observed LGN responses can be accounted for by the classic model, but some evidence suggests that the chromatic responses of the LGN may not be fully described by the two
channels. It is difficult to investigate physiological correlates of the higher-order color mechanisms in the LGN, since these mechanisms have primarily been revealed by psychophysical habituation
(Krauskopf et al.,
), and there is little or no habituation of cells in the LGN, yet these cells may contribute to the color tuning of the adaptable cortical cells (Tailby, Solomon, Dhruv, & Lennie,
). Signals tuned to color directions away from the two opponent mechanisms have also been proposed (Webster & Mollon,
; Zaidi & Shapiro,
) that could theoretically be inputs to cortical areas. Finally Tailby, Solomon, and Lennie (
) recently reported that in macaque LGN, a subset of neurons responded to modulations along both the S–(L + M) and L–M axes. Our present study does not address the question of whether the model of
two chromatically opponent sub-cortical channels is sufficient to describe the coding of color by the LGN. We plan to investigate this possibility further in future investigations using human fMRI.
Appendix A
Multivariate classifier performance as a function of the number of voxels
For each subject, the classifier analysis was repeated for increasing numbers of voxels from each area.
Figure A1
shows multivariate classifier performance as an increasing number of voxels were included in the classifier, on a semilogarithmic scale. It also shows the best fitting curve (see
Equation 1
), from which the limit was reported as the classifier performance in
Figure 4
, or the mean classifier performance in cases where the curve did not fit the data.
Appendix B
Generalization of classification across different luminance–color pairings
Our main analyses, grouping lime-magenta and orange-cyan blocks of different luminance pairings, show that the classifier generalizes across luminance levels. This is because both training and test
data contain an equal number of the two phases of luminance–color relations so the classifier has to generalize across different luminance levels to learn the stimuli. If the classifier can learn
this more general rule, it at first sight seems reasonable to test whether it can generalize from one level to the other.
However, there is a subtle but important point to make here. If we train on one luminance level and test on the other, the classifier may learn a different rule than lime-magenta vs. orange-cyan. If
the classifier is trained on dark lime-light magenta vs. dark orange-light cyan, the classifier could learn to separate the training data as dark green-light red vs. dark red-light green. If this
different rule were learned by the classifier then when tested with light lime-dark magenta vs. light orange-dark cyan it would give the opposite result to learning the non-cardinal color modulation.
Thus successful classification would provide evidence in support of the classifier learning the non-cardinal color modulation, and argue against the notion that the original classifier performance
could have been based on an artifact. But a negative result neither supports nor excludes the possibility that classifier performance was due to a luminance artifact. In short, in the original
analysis any small luminance artifact could have worked in favor of classifier performance but in this new analysis there is a 25% contrast luminance modulation working against classifier
The results of this analysis are shown in
Figure B1
. We found that in area MT+, classification performance was significantly below chance (
< 0.01, two-tailed
-test) across subjects, and for individual CC in areas V1, V2, V3 and V3A/B (
< 0.01, permutation test). Below chance performance is consistent with the classifier being significantly good at learning a decision rule based on the pairing of luminance and one of the cardinal
modulations (for example, successfully learning to separate the data as dark green-light red vs. dark red-light green). This below chance performance neither supports nor excludes the possibility
that the original classifier performance was based on a luminance artifact.
For subjects SM, DM and EG classification performance was significantly (p < 0.05, permutation test) above chance for V1 and V2, with SM and DM also significantly (p < 0.05, permutation test) above
chance in area V3. The fact that classifier performance is significantly above chance for some areas in some subjects gives evidence that in these cases the original classifier performance could not
have been based on a luminance artifact.
This work was supported by an Australian Postgraduate Award to E.G., an Australian Research Fellowship to C.C. and the Australian Centre of Excellence for Vision Science. We thank Kirsten Moffat and
the Prince of Wales Medical Research Institute for assistance with fMRI data collection, Mark Schira for assistance with retinotopic analysis, and Bill Levick for helpful comments regarding macular
Commercial relationships: none.
Corresponding author: Erin Goddard.
Email: erin.goddard@sydney.edu.au.
Address: Griffith Taylor Bldg A19, The University of Sydney, Sydney, New South Wales, Australia.
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Mums who had a C Section with their 1st baby
Mar 23, 2007
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Just thinking...
....yesterday was told that my baby was breech and if it didn't turn they would do a C section as it was my 1st. I know there is still plenty of time for it to turn so I'm not worried about that or
really the thought of having a C section as long as the baby is safely delivered.
What does bother me slightly is about my cousin's c section. Because her body didn't go through the natural birth process it took her body longer to produce milk. Is this normal? How long after your
C section did you start producing milk normally, like someone who had pushed the baby out? It would worry me if I didn't have milk to feed the baby and wouldn't particularly like to use formula.
Nov 28, 2006
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I had a caesarean and am successfully breastfeeding. I had colostrum, but only the tiniest amount at first. She was on a drip due to complications at her birth (emergency delivery) then we got her
onto formula while I started expressing to get my milk coming. By the time we went home a 5 days she was exclusively breastfeeding and is putting on plnty of weight.
Jun 28, 2006
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I had an emergancy c-section and although I had problems feeding at first it wasn't because of my milkn not coming in quickly enough but because Grace had trouble latching on.
You milk usually comes in day later with a c-section it isn't a problem for your baby...
Good luck - Breast feeding is the most wonderful feeling and although it can be difficult at times it is well worth it
I had an emergency section with my 1st baby and I successfully breastfed for 4 months and my body produced colostrum the day after I had him
Jun 16, 2007
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It took my milk about two days to come in,and before that i was producing loads of colostrum.If yours takes a while to come,you should maybe ask to use the milk bank at your hospital.
Aug 14, 2007
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I had a planned section, baby was transverse. My milk was very very slow at coming in, I took milk plus tablets that I got at the baby show, these did help a bit, i breast fed for 9 weeks, having
another section in february, all the best, x
Jul 24, 2007
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My milk came in after 3 days
Mar 23, 2007
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Becksss said:
My milk came in after 3 days
Did you bottle feed Riley during that time?
Jul 24, 2007
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Feb 1, 2005
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I had tons of colostrum before having Aaron but I hate to say it, my milk never came in. My boobs were a bit firm but that was it, I barely even leaked. I used about 4 breast pads after having Aaron
and they had a dot on it!
Jan 17, 2007
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i produced colostrum in good amounts at first but then it seemed to dry out...then came back...but unfortunately baby wouldn't latch on and no matter how many midwives and nurses tried to help...they
tired very quickly and they didn't seem as eager to help after a few tries...which upset me cos i really wanted to breast feed...ended up extracting milk for about a month.
then gave up
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Finding a particular angle withion Johnson Solid J2
• MHB
• Thread starter GreyArea
• Start date
In summary, Johnson Solid J2 is a pentagonal pyramid consisting of five equilateral triangles on a pentagonal pase, meeting at the apex. The angle formed by the junction of the two lines is 52.62263
MY apologies that I don't think this is particularly advanced; my algebra and trigonometry is quite good, but all seems to fall apart once i move from 2D to 3D environments.
Johnson Solid J2 is a pentagonal pyramid consisting of five equilateral triangles on a pentagonal pase, meeting at the apex. It's the shape you get if you slice off the top five triangular faces of
an icosahedron.
I can find a lot of info on this shape, height, surface area etc, but I need to know one specific angle that eludes me; the angle formed by the junction of;
1. A vertical line dropped from the apex of the solid to the base
2. a line that bisects one of the equilateral triangles, rising from the mid point of base of the triangle to its apex
If it can be calculated as a function of the height of the pyramid, or the height of the face that's fine, but I get the feeling it SHOULD be a constant value, similar to the dihedral angle of
(approx) 138 .2 degrees that is formed by the junction of any pair of triangular faces in this solid.
Thanks for your help!
As is often the case...I get stuck, post for help...then get lucky!
I've found the formula for the "slant height" (a term I did not previously know) which is the length of the bisecting line down the midpoint of the face.
Since I already have the formula for the height of the pyramid, I believe I have everything I need to find the angle as follows;
a = length of edge of face.
Slant height (s)
\(\displaystyle 0.5 * \sqrt{3} * a\)
Pyramid height (h)
\(\displaystyle sqrt{((5 - sqrt{5})/10)} * a\)
So if I'm right, s is my hypotenuse, h my adjacent...so I just need to use cosine?
I could probably substitute and simplify given time...but when Excel is so easy...the answer should anyone ever need it, is 52.62263 degrees and as I suspected does not vary with the length of the
triangular faces.
FAQ: Finding a particular angle withion Johnson Solid J2
1. How do you find a particular angle within Johnson Solid J2?
The particular angle within Johnson Solid J2 can be found by using the formula: angle = 360 degrees / number of faces.
2. What is Johnson Solid J2?
Johnson Solid J2 is a type of polyhedron that is made up of triangular and pentagonal faces. It is also known as a deltoidal icositetrahedron.
3. What is the significance of finding a particular angle within Johnson Solid J2?
Finding a particular angle within Johnson Solid J2 is important in understanding the shape and structure of the polyhedron. It can also help in solving mathematical and geometric problems related to
this solid.
4. Can the angle within Johnson Solid J2 be measured in degrees?
Yes, the angle within Johnson Solid J2 can be measured in degrees, as it is a standard unit for measuring angles.
5. Are there any other methods for finding angles within Johnson Solid J2?
Yes, there are other methods such as using trigonometric functions or using geometric constructions to find angles within Johnson Solid J2. However, the formula mentioned in the first question is the
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When to use COUNT vs SUM vs COUNTA vs COUNTBLANK vs COUNTIF
Many people get confused about how and when to use some of the basic Excel functions. For example, when do you use SUM and when do you use COUNT?
In a moment I’ll tackle that specific question.
Then I'll bring 3 other COUNT functions into the mix - COUNTA, COUNTBLANK and COUNTIF and show you where you would use each one.
1. What is the difference between SUM and COUNT?
Very simply, SUM calculates a total for a number of cells or values, so it’s answering the question: HOW MUCH? Or, WHAT IS THE TOTAL?
COUNT tells you HOW MANY cells meet a certain condition.
Consider the following data:
Cell A6 uses a SUM function to add up the values in cells A1 to A5.
Cell C6 uses a COUNT function to find how many cells in the range C1 to C5 contain numbers. The COUNT function ignores blank cells or cells that contain text or symbols.
2. Introducing COUNTA, COUNTBLANK and COUNTIF
There are number of other functions available in Excel. Heres a quick summary of what they do, followed by an example of each.
Consider the following data:
Here’s the results for each formula:
Answer = 3.
There is no single function that tells you the number of text cells but you can work it out with this formula:
=COUNTA(B2:B11) - COUNT(B2:B11)
3. The COUNTIF function
To demonstrate the COUNTIF function, consider the following data:
The COUNTIF function needs 2 bits of information - the range of cells you are looking at and what it is that you’re checking for. The criteria is always encapsulated in double quotation marks (“) and
is not case sensitive.
To find how many tradespeople drive a Toyota:
To find how many plumbers there are:
To find how many tradespeople charge more than $70 per hour:
To find how many of the tradesmen’s names start in the last half of the alphabet:
4. Watch the video (over the shoulder demo)
5. What next?
I hope you found plenty of value in this post. I'd love to hear your biggest takeaway in the comments below together with any questions you may have.
Have a fantastic day.
Jason Morrell is a professional trainer, consultant and course creator who lives on the glorious Gold Coast in Queensland, Australia.
He helps people of all levels unleash and leverage the power contained within Microsoft Office by delivering training, troubleshooting services and taking on client projects. He loves to simplify
tricky concepts and provide helpful, proven, actionable advice that can be implemented for quick results.
Purely for amusement he sometimes talks about himself in the third person.
help me a lot, thank u so much
• You’re welcome Zhou.
It appears double quotation marks are not needed if the criterion is a number in the COUNTIF function.
• Hi Nick. Thank you for taking the time to comment. Yes, it’s true that you don’t need the enclosing quote marks if you are looking to match on an exact number. This is because ‘=’ is implied. For
any other comparison (>, >=, <, <=, <>) the expression must be enclosed in double quote marks.
Jason, you are the best!
• ✌️ Thanks buddy.
wow so simple to understand thank you so much
• You’re welcome Ifechukwu.
Very simple to understand. Thanks for helping with my studies!
• No worries.
Thanks for the help Jason.
• No worries Chukwuefe. I’m glad you found it useful.
Nice and easy to understand. Thanks for the explanation!
• Thank you. You’re welcome.
Thanks, It was helpful
• You’re welcome.
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Swept polygons - Sketch
3.2.6.3 Swept polygons
If swept_object is a polygon, the sweep connects n+1 successive copies of the closed polyline border of the polygon to form body polygons exactly as though the border were a swept polyline as
described in Swept lines. If there are m points in the original polygon, then mn body polygons are formed by this sweep. The body polygons form an extrusion of the boundary of the original polygon
with two holes at the open ends.
Finally, the sweep adds two copies of the original polygon to cover the holes. We call these hole-filling polygons ends. In this manner, sweep forms the boundary of a three-dimensional object from a
two-dimensional polygon.
The order of vertices of end polygons is important for correct culling as described above. An exact copy of the original polygon with vertex order intact forms the first end polygon. The other end
polygon results from transforming and the reversing the order of vertices in the original. The transform places the original polygon at the uncovered hole; it is
T_1^n then T_2^n then ... then T_r^n.
If there are no options on the swept polygon, then the sweep options are copied to each output polygon. If the swept polygon does have options, these are copied to the ends; the sweep options are
copied to the body polygons. In this manner, body and ends may be drawn with different characteristics such as fillcolor. | {"url":"https://ctan.um.ac.ir/graphics/sketch/Doc/sketch/Swept-polygons.html","timestamp":"2024-11-06T04:06:41Z","content_type":"text/html","content_length":"4760","record_id":"<urn:uuid:0947fc15-e9ec-4bb1-a147-3069d12cc20a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00163.warc.gz"} |
How To Think Real Good | Meta-rationality - Aili
Summarize by Aili
How To Think Real Good | Meta-rationality
๐ Abstract
The article discusses the author's long-standing interest in thinking about thinking, and how this led to various projects and collaborations around understanding effective and accurate thinking. The
main focus is on critiquing the LessWrong rationalist community's emphasis on Bayesian probability as the primary tool for rationality, and arguing that it is only a small part of what is needed. The
author proposes that figuring out how to think effectively is a much broader and more complex challenge that requires drawing on a wide range of intellectual tools and methods.
๐ Q&A
[01] Problem Formulation
1. What are the key points made about problem formulation?
• Before applying any technical method, you need to already have a pretty good idea of what the form of the answer will be.
• A successful problem formulation has to make the distinctions that are used in the problem solution, and make the problem small enough that it's easy to solve.
• All problem formulations are "false" because they abstract away details of reality, but the goal is to find an idealization that is useful for the particular problem at hand.
• Heuristics for good problem formulation include: working through specific examples, iterating between formulation and solution, and solving a simplified version of the problem first.
2. Why is problem formulation so important according to the author? The author argues that finding a good formulation for a problem is often most of the work of solving it. Formal methods like
Bayesian probability require a precise specification of the problem, but figuring out that specification is a major challenge in itself that the Bayesian framework does not address.
3. What is the author's view on the relationship between problem formulation and problem solving? The author states that problem formulation and problem solution are mutually-recursive processes -
you need to go back and forth between trying to formulate the problem and trying to solve it, as progress in one area informs the other.
[02] Rationality Without Probability
1. What examples does the author provide of solving problems without using probability theory? The author discusses his work on classical planning problems in robotics, where he was able to solve the
problem using modal logic and model theory, without any use of probability. He also describes work with Phil Agre on developing a theory of effective action that dealt with uncertainty without
representing it probabilistically.
2. What are the author's key points about the limitations of probability theory for rationality? The author argues that probability theory is not the only, or even the best, way to deal with
uncertainty. It can collapse together many different sources of uncertainty in unhelpful ways. The author also suggests that Bayesians sometimes inappropriately attribute unconscious probabilistic
reasoning as an explanation when the actual reasoning process is unclear.
3. What is the author's view on learning from diverse fields outside one's own? The author emphasizes the importance of learning from fields very different from your own, as each has ways of thinking
that can be useful in surprising ways. Applying tools and perspectives from anthropology, psychology, and philosophy was crucial to the work he did with Phil Agre.
[03] Heuristics and Cognitive Styles
1. What heuristics or rules of thumb does the author suggest for effective thinking? Some key heuristics mentioned include:
• If a problem seems too hard, the formulation is probably wrong - go back and observe the real-world situation.
• Learn as many different kinds of math as possible, as it's difficult to predict what will be relevant.
• Get a superficial understanding of many mathematical fields, so you can recognize when they might apply.
• Collect a "bag of tricks" - a set of obscure technical methods that you've mastered and can apply opportunistically.
• Find teachers who are willing to explain how a field works, not just the subject matter.
2. How does the author characterize the cognitive styles of highly effective problem solvers? The author notes that very smart people often have unique cognitive styles that give them an edge. He
contrasts his own tendency towards "interesting" approaches with his collaborator Ajay's preference for the simplest, most obvious solutions that work. The author suggests trying to figure out and
develop your own cognitive style as a strength.
3. What is the author's view on the role of formal methods versus informal reasoning in effective thinking? The author suggests that insights from informal reasoning and observation are probably more
important than understanding technical rationality methods. He argues that the Bayesian framework and other formal tools only address a small part of what is needed for effective thinking.
[04] Conclusions
1. What are the key takeaways from the author's overall perspective on thinking about thinking? The main points are:
• Figuring out how to think effectively is an extremely difficult, open-ended challenge without a single general method.
• Problem formulation and selection is as important as problem solving itself, and requires careful observation of the real world.
• A good problem formulation exposes the relevant distinctions and abstracts away irrelevant details, making the problem easier to solve.
• Progress often requires applying a diverse range of intellectual tools and methods, not just focusing on one formal framework like Bayesian probability.
• Meta-level knowledge about how different fields and methods work is critical, but very hard to come by.
2. Why does the author see this as a challenge that requires a collaborative community effort? The author acknowledges that his own attempt to write a comprehensive book on "How to Think Real Good"
was too ambitious for a single person. He suggests that the LessWrong community's collaborative approach is well-suited to tackling this broad and difficult challenge of understanding effective
3. What does the author hope to achieve by sharing these ideas and inviting further discussion? The author seems to hope that by outlining some of his key insights and heuristics around thinking
about thinking, and inviting further discussion and collaboration, the community can make progress on this important but elusive problem. He recognizes the limitations of his own "brain dump"
approach, and sees the value in a more systematic, collective effort.
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Converting Decimal Numbers To Binary Worksheet
Converting Decimal Numbers To Binary Worksheet serve as foundational devices in the realm of mathematics, offering an organized yet functional system for students to check out and master numerical
concepts. These worksheets use an organized approach to understanding numbers, nurturing a strong structure upon which mathematical proficiency grows. From the simplest checking exercises to the
complexities of sophisticated calculations, Converting Decimal Numbers To Binary Worksheet deal with students of varied ages and ability degrees.
Unveiling the Essence of Converting Decimal Numbers To Binary Worksheet
Converting Decimal Numbers To Binary Worksheet
Converting Decimal Numbers To Binary Worksheet -
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Sub-Total in Footer for each Page | Microsoft Community Hub
Forum Discussion
Sub-Total in Footer for each Page
My Problem: -
Excel Version 2016
Firstly, apologies if I have not adhered to any of the Forum rules, it is my first post.
I want a sum total (in a footer) for each page. Pretty simple I thought until I discovered excel do not provide the function to do this i.e. do sum totals in footers Grrrrrr!!!!
We produce ‘Schedules of Works’ with the furthest most right-hand column (this being ‘F’) used as a pricing column for contractors to enter their prices alongside our work descriptions. These prices
/ costs should then total at the bottom for each page.
Each blank page roughly starts off with 50 rows, so, I could just put a sum total in row 51. However, as we write the schedule, this could change to as little as 10 rows per page dependent upon the
size of the descriptions we write in each row. As such the sum-total would be pushed off the bottom and onto the next page.
This wouldn’t be too bad for our Admin to re-configure if it were only a few pages, however each schedule can be up to 80 pages long, times that by 20 or so people in the office producing many
schedules. This now becomes a lot of editing work.
Not one to let this defeat me, I have spent the last week Googling a hopeful answer from many excel forums / pod cast videos etc as follows: -
1st Try: - (using the formula bar)
This works fine to a degree however with the following problems for me: -
1) – If I need to insert 5 or more rows it pushes the sub-total onto the next page
2) – If I the row height increase through needing to enter a large description / amount of txt it pushes the sub-total onto the next page
3) The sub-total isn’t in a footer anyway, just at the bottom of the page. Although maybe it could be referenced in the footer via a macro or something.
2nd Try: - (using a macro)
Sub PrintWithTotals()
Fill in this information
FirstDataRow = 6
RowsPerPage = 50
ColToTotal = 6
‘ Find how many rows today
HeadRow = FirstDataRow - 1
FinalRow = Cells(Rows.Count, 1).End(xlUp).Row
PageCount = (FinalRow - HeadRow) / RowsPerPage
PageCount = Application.WorksheetFunction.RoundUp(PageCount, 0)
For i = 1 To PageCount
ThisPageFirstRow = (i - 1) * RowsPerPage + HeadRow + 1
ThisPageLastRow = ThisPageFirstRow + RowsPerPage - 1
TotalThisPage = Application.WorksheetFunction. _
Sum(Cells(ThisPageFirstRow, ColToTotal).Resize(RowsPerPage, 1))
‘ Change the Footer for this page
Application.PrintCommunication = False
With ActiveSheet.PageSetup
.RightFooter = "Total This Page $" & Format(TotalThisPage, "#,##0.00")
End With
Application.PrintCommunication = True
' Print this page
ActiveWindow.SelectedSheets.PrintOut From:=i, To:=i, Copies:=1, Collate _
:=True, IgnorePrintAreas:=False[/B][/U]
Next i
‘ Clear the footer in case someone prints without the macro
Application.PrintCommunication = False
With ActiveSheet.PageSetup
.LeftFooter = "Use PrintWithTotals Macro "
.RightFooter = " "
End With
Application.PrintCommunication = True
End Sub
This dosn't produc any errors, nevertheless it doesn’t work. When I step through the code either using F8 or the debug step-in procedure it misses out the section I have noted above. So I'm guessing
it isn't running this part properly.
Watching the video from MrExcel, he steps through it fine and it works fine.
To re-cap, I am just after getting a sub-total (for each page) in a footer regardless of how many rows I have for that page, delete, add or increase in size.
I am at a loss what else to try. If anyone could please help, the winner gets a Big Hug!
Cheers Steve
• I'd be glad to help you with the Excel issue of calculating subtotals for each page in a footer, even with varying row counts and descriptions. Here are some approaches that address the
challenges you mentioned:
1. Dynamic Formula with Conditional Formatting:
This method combines a formula and conditional formatting to automatically adjust the subtotal's position and visibility based on the number of rows.
□ MOD(ROW()-ROW(B1),45)=0: This checks if the current row number is divisible by 45 (adjustable based on your typical page size).
□ SUM(G2:G&ROW()-ROW(B1)): If the condition is true, this calculates the sum of column G from row 2 to the current row.
□ ROW(B1) is a reference to the first row with data (adjust if needed).
Conditional Formatting:
□ Apply conditional formatting to the cell containing the formula.
□ Set the rule to format the cell as hidden (e.g., white font on white background) when the formula result is empty.
2. Macro with Footer Update:
This approach uses a macro to calculate the subtotal for each page and update the footer dynamically.
Sub UpdateFooters()
Dim pageCount As Integer, currentPage As Integer, firstRow As Integer, lastRow As Integer
Dim total As Double
pageCount = Application.WorksheetFunction.RoundUp(Rows.Count / 45, 0) ' Adjust divisor if needed
For currentPage = 1 To pageCount
firstRow = (currentPage - 1) * 45 + 1 ' Adjust base row if needed
lastRow = firstRow + 44 ' Adjust offset if needed
total = Application.WorksheetFunction.Sum(Range("G" & firstRow & ":G" & lastRow))
' Update footer on current page
ActiveSheet.PageSetup.Footer.RightFooter = "Total Page " & currentPage & ": $" & Format(total, "#,##0.00")
Next currentPage
End Sub
□ The macro loops through each page, calculates the subtotal using the specified range, and updates the right footer with the total and page number.
3. Advanced Formula with Table and Named Ranges:
This method uses a more advanced formula with a table and named ranges for flexibility and easier customization.
1. Create a table encompassing your data (Ctrl+T).
2. Define named ranges:
☆ DataRange: The entire table (e.g., Table1).
☆ RowsPerPage: The typical number of rows per page (e.g., 45).
3. Use this formula in the desired footer cell:
□ The formula sums values in the "Price" column (adjust if needed) based on the condition of being on the last row of a page (adjustable).
Additional Tips:
□ Consider using a combination of methods for more robust or customized solutions.
□ Thoroughly test any formulas or macros before implementing them in your actual spreadsheets.
□ If you encounter specific errors or need further assistance, provide more details about your spreadsheet setup and the issues you're facing.
I hope these improved approaches help you achieve the desired subtotals and footers in your Excel workbook! | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/sub-total-in-footer-for-each-page/166806","timestamp":"2024-11-11T20:24:05Z","content_type":"text/html","content_length":"324666","record_id":"<urn:uuid:9507c9d2-c694-4513-afa8-5eaf19da1a9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00246.warc.gz"} |
Binary Relation
A binary relation is a finitary relation [math]\displaystyle{ R(D,B) }[/math] with two relation arguments (it pairs each Set Member of Relation Domain Set [math]\displaystyle{ D }[/math] with at
least one Set Member of relation range set B).
• AKA: R, One Set Binary Relation.
• Context:
• Example:
□ The Subsumption Relation (IsA) is a binary relation. E.g. The tree of life relation always compares two classes of animals.
□ The Parthood Relation (PartOf) is a binary relation.
□ The LessThan relation on integers.
• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/binary_relation Retrieved:2016-1-18.
□ In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A^2 = . More generally, a binary
relation between two sets A and B is a subset of . The terms correspondence', dyadic relation and 2-place relation are synonyms for binary relation.
An example is the “divides” relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with
no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with
numbers that include 0, 6, and 9, but not 4 or 13.
Binary relations are used in many branches of mathematics to model concepts like “is greater than", “is equal to", and "divides" in arithmetic, “is congruent to” in geometry, "is adjacent to"
in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in
computer science.
A binary relation is the special case of a n-ary relation R ⊆ A[1] × … × A[n], that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A[j] of the
relation. An example for a ternary relation on Z×Z×Z is "lies between … and ...", containing e.g. the triples , , and .
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is
an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.
• http://planetmath.org/encyclopedia/NullaryRelation.html
□ Basically, a binary relation $ R$ involves objects coming from two collections $ A,B$, where the objects are paired up so that each pair consists of an object from $ A$, and an object from $
□ More formally, a binary relation is a subset $ R$ of the Cartesian product of two sets $ A$ and $ B$. One may write
☆ $\displaystyle a\: R\: b$
□ to indicate that the ordered pair $ (a, b)$ is an element of $ R$. A subset of $ A\times A$ is simply called a binary relation on $ A$. If $ R$ is a binary relation on $ A$, then we write
☆ $\displaystyle a_1 \: R \: a_2 \: R \: a_3 \: … \: a_{n-1} \: R \: a_n $
to mean $ a_1 \: R\: a_2, a_2\} R\: a_3, \ldots,$ and $ a_{n-1}\: R \: a_n$.
□ Given a binary relation $ R\subseteq A\times B$, the domain $ \operatorname{dom}(R)$ of $ R$ is the set of elements in $ A$ forming parts of the pairs in $ R$. In other words,
☆ $\displaystyle \operatorname{dom}(R):=\lbrace x\in A\mid (x,y)\in R$ for some $\displaystyle y \in B \rbrace$
□ and the range $ \operatorname{ran}(R)$ of $ R$ is the set of parts of pairs of $ R$ coming from $ B$:
☆ $\displaystyle \operatorname{ran}(R):=\lbrace y\in B\mid (x,y)\in R$ for some $\displaystyle x\in A \rbrace.$
• (Roth & Yih, 2002) ⇒ Dan Roth, and Wen-tau Yih. (2002). “Probabilistic Reasoning for Entity & Relation Recognition.” In: Proceedings of the 20th International Conference on Computational
Linguistics (COLING 2002).
□ Definition 2.2 (Relation) A (binary) relation R[i,j] = (E[i];E[j]) represents the relation between E[i] and E[j], where E[i] is the first argument and E[j] is the second. In addition, R[ij]
can range over a set of entity types CR.
□ Figure2 : Dole ’s wife, Elizabeth, is a native of Salisbury, N.C.
□ Example 2.2 In the sentence given in Figure 2, there are six relations between the entities: R[1,2] = (“Dole”, “Elizabeth”), R[2,1] = (“Elizabeth”, “Dole”), R[1,3] = (“Dole”, “Salisbury,
N.C.”), R[3,1] = (“Salisbury, N.C.”, “Dole”), R[2,3] = (“Elizabeth”, “Salisbury, N.C.”), and R[3,2] = (“Salisbury, N.C.”, “Elizabeth”) | {"url":"https://www.gabormelli.com/RKB/Binary_Relation","timestamp":"2024-11-02T11:00:20Z","content_type":"text/html","content_length":"53145","record_id":"<urn:uuid:04228a96-4fc6-4a9a-ab5e-78f86c3fd5fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00709.warc.gz"} |
Time Value of Money Calculator
TVM Calculator
TVM Calculator is a tool to calculate the time value of money. Based on the interest rate and principal, you can calculate the TVM after a number of years.
Present Amount: $
Interest Rate: %
Years: years
Compound Period:
Additional Contributions: $
The time value of money calculator is useful for you to estimate how much your money will be worth in the future. If you make monthly or annual deposit, the TVM calculator will show you how much your
money will be compounded. | {"url":"https://tvmcalculator.com/","timestamp":"2024-11-06T10:20:42Z","content_type":"application/xhtml+xml","content_length":"7675","record_id":"<urn:uuid:41d7c071-434a-4641-9b0b-15d0c5d6cce8>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00075.warc.gz"} |
Newton's method to find square root, inverse - DSP LOG
Newton’s method to find square root, inverse
Some of us would have used Newton’s method (also known as Newton-Raphson method) in some form or other. The method has quite a bit of history, starting with the Babylonian way of finding the square
root and later over centuries reaching the present recursive way of finding the solution. In this post, we will describe Newton’s method and apply it to find the square root and the inverse of a
Geometrical interpretation
We know that the derivative of a function $f(x)$ at $x_0$ is the slope of the tangent (red line) at $x_0$ i.e.,
$f^'(x_0) = \frac{f(x_0)}{x_0-x_1}$.
Rearranging, the intercept of the tangent at x-axis is,
$x_1 = x_0 - \frac{f(x_0)}{f^'(x_0)}$.
From the figure above, can see that the tangent (red line) intercepts the x-axis at $x_1$ which is closer to the $f(x)=0$ where compared to $x_0$. Keep on doing this operation recursively, and it
converges to the zero of the function OR in another words the root of the function.
In general for $n^{th}$iteration, the equation is :
$\Huge x_{n+1} = x_{n} - \frac{f(x_n)}{f^'(x_n)}$.
Finding square root
Let us, for example try to use this method for finding the square root of D=100. The function to zero out in the Newton’s method frame work is,
$f(x) = x^2 - D$, where $D=100$.
The first derivative is
$f^'(x) = 2x$.
The recursive equation is,
$x_{n+1} = x_{n} - \frac{x_n^2-D}{2x_n}$.
Matlab code snippet
clear ; close all;
D = 100; % number to find the square root
x = 1; % initial value
for ii = 1:10
fx = x.^2 - D;
f1x = 2*x;
x = x - fx/f1x;
x_v(ii) = x;
x_v' =
We can see that the it converges within around 8 iterations. Further, playing around with the initial value,
a) if we start with initial value of x = -1, then we will converge to -10.
b) if we start with initial value of x = 0, then we will not converge
and so on…
Finding inverse (division)
Newton’s method can be used to find the inverse of a variable D. One way to write the function to zero out is$f(x) = xD - 1$, but we soon realize that this does not work as we need know $\frac{1}{D}$
in the first place.
Alternatively the function to zero out can be written as,
$f(x) = \frac{1}{x} - D$.
The first derivative is,
$f^'(x) = -\frac{1}{x^2}$.
The equation in the recursive form is,
$\begin{array}{lll}x_{n+1} &= & x_{n} - \(\frac{\frac{1}{x}-D}{-\frac{1}{x^2}}\)\\ & = & x_n\(2-x_n)\end{array}$.
Matlab code snippet
clear ; close all;
D = .1; % number to find the square root
x = [.1:.2:1]; % initial value
for ii = 1:10
fx = (1./x) - D;
f1x = -1./x.^2;
x = x - fx./f1x;
x_v(:,ii) = x;
legend('0.1', '0.3', '0.5', '0.7', '0.9');
grid on; xlabel('number of iterations'); ylabel('inverse');
title('finding inverse newton''s method');
The following plot shows the convergence of inverse computation to the right value for different values of$x_0$ for this example matlab code snippet.
Figure : convergence of inverse computation
Finding the minima of a function
To find the minima of a function, we to find where the derivative of the function becomes zero i.e. $f^'(x) = 0$.
Using Newton’s method, the recursive equation becomes :
$x_{n+1} = x_n - \frac{f^'(x)}{f^{''}(x)} = 0$.
We have briefly gone through the Newton’s method and its applications to find the roots of a function, inverse, minima etc. However, there are quite a few aspects which we did not go over, like :
a) Impact of the initial value on the convergence of the function
b) Rate of the convergence
c) Error bounds of the converged result
d) Conditions where the convergence does not happen
and so on…
Hoping to discuss those in another post… 🙂
One thought on “Newton’s method to find square root, inverse”
1. very thankse | {"url":"https://dsplog.com/2011/12/25/newtons-method-square-root-inverse/","timestamp":"2024-11-01T20:25:15Z","content_type":"text/html","content_length":"97533","record_id":"<urn:uuid:ea6e6c13-12b8-4c4c-b574-7819020462b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00023.warc.gz"} |
Characterize | Disruptive Planets
top of page
discovering and studying new worlds to consolidate our understanding of planets and habitats.
The next step after discovering an object of astronomical interest is to initiate its study. In this quest, we work towards developing robust/reliable characterization pipelines for exoplanetary
science. Theoretical works aim at supporting our community’s best use of the new James Webb Space Telescope. In parallel, applications with the Spitzer and Hubble Space Telescopes provide us with the
first glimpses in the atmospheres of other worlds.
Exploring the Atmosphere of Terrestrial Planets​
Terrestrial planets around Dwarf stars are uniquely suited for atmospheric exploration owing to the small size of their host. In this subquest, we leverage such optimal configuration to initiate the
atmospheric characterization of new worlds.”
Showcased Papers:
"Atmospheric reconnaissance of the habitable-zone Earth-sized planets orbiting TRAPPIST-1"
de Wit*, Wakeford*, Lewis*, et al. 2018
"Temporal Evolution of the High-energy Irradiation and Water Content of TRAPPIST-1 Exoplanets"
Bourrier, de Wit, et al. 2017
Transmission spectra of TRAPPIST-1 d, e, f and g compared with synthetic atmospheres dominated by hydrogen (H2), water (H2O), carbon dioxide (CO2) and nitrogen (N2). Trace gases are given in
parentheses following the dominant gas. HST/WFC3 measurements are shown as black circles with 1σ error bars. Each spectrum is shown shifted by its average over the WFC3 band. The measurements are
inconsistent with the presence of a cloud-free H2-dominated atmosphere at greater than 3σ confidence for planets d, e and f (only the values larger than 3σ are reported in the legends). Figure from
de Wit, Wakeford, Lewis, et al. 2018.
Searching for signs of exospheres around the TRAPPIST-1 planets. Ly- line proles of TRAPPIST-1. Solid-line profiles correspond to our best estimates for the theoretical intrinsic Ly-line in Visit 1-3
(blue) and in Visit 4 (black). They yield the dashed-line profiles after ISM absorption and convolution by STIS LSF. ISM absorption profile in the range 0-1 has been scaled to the vertical axis range
and plotted as a dotted black line. The dashed-line profile in Visit 4 was adjusted to the observations (red histogram, equal to the average of all spectra in Visit 4) outside of the hatched regions,
and excluding the variable range between -187 and -55kms^-1 (highlighted in orange).
Enabling the Reliable Decryption of Exoplanet Transmission Spectra​
As we enter a new era of Space Exploration with the JWST and the ELTs, the tools we use to decrypt astronomical data need to evolve to ensure that we reliably decode new signals and do not
misinterpret their groundbreaking content.
Showcased Papers:
"The Impending Opacity Challenge in Exoplanet Atmospheric Characterization"
Niraula*, de Wit*, et al. 2022
"Constraining Exoplanet Mass from Transmission Spectroscopy"
de Wit & Seager 2013
Propagation of the ensemble of opacity-model perturbations to the level of retrieved atmospheric properties. PPDs of the retrieved atmospheric parameters (that is, final data product) for the
super-Earth scenario highlighting the biases induced by perturbations to our opacity model. Each cross-section is identified by its colour and label on the right. The dotted black vertical lines
represent the true values used in generating the synthetic spectrum. Deviations with a statistical significance of up to ~20σ and physical significance of over 1 dex are reported. This suggests the
significant sensitivity of retrieved atmospheric properties to opacity models in upcoming atmospheric retrieval efforts. Figure from Niraula, de Wit, et al. 2022.
Probing the information content of exoplanet transmission spectra. A line profile (fν) depends on the pressure (p) as a rational function at fixed temperature (T) and frequency (ν). (A) Dependency of
fν on T and p, at fixed ν, that shows the four domains of different dependency regimes of fν on p whose boundaries are T-dependent (gray dot-dash lines). The black dot-dash lines represents the
position of the slices in the {T − p − f} space used to highlight that fν behaves as a rational function of p, at {T, ν} fixed. Planels B and C present the slices at T = 1200K and T = 200K,
respectively. These slices show that fν behaves as a rational function of p with a zero and a pair of conjugated zeros (Fig. S.20). In particular, the absolute value of the zero is less than the
poles’, as underscored by the sequential transition from the following dependency regimes ∝ p0, ∝ p1, ∝ p∼0, and ∝ p−1 with increasing p—the dotted and the dashed lines represent a slope of 1 and 2,
respectively. (D) Dependency of the absorption coefficient (αν) on p, at T = 200K. The exponent of the αν dependency on p increases by one compared to fν dependency on p. The exponent increase by one
because of αν’s additional zero at p = 0 that originates from the number density. Figure from de Wit & Seager 2013.
Towards 3D Maps of Other Worlds​
Although humanity does not yet have the capabilities to spatially resolve stars other than our Sun, there exist ways to start mapping exoplanets. Using such approaches, it will be possible to study
tri-dimensional structures in the atmospheres of other worlds and their evolution. Stay tune for some exo weather forecast!
Showcased Papers:
"Inference of Inhomogneeous Clouds in an Exoplanet Atmosphere"
Demory, de Wit, et al. 2013
"Towards consistent mapping of distant worlds: secondary-eclipse scanning of the exoplanet HD 189733b"
de Wit et al. 2012
The inhomogeneous cloud cover of Kepler-7b. Left Panel: Longitudinal brightness maps of Kepler-7b. Kepler-7b’s longitudinal brightness distributions Ip/I as retrieved in Kepler’s bandpass. Right
Panel: Artistic rendering of Kepler-7b’s dayside showcasing a permanent cloud coverage on its western side. Figure from Demory, de Wit, et al. 2013.
Schematic description of the anomalous occultation ingress/egress induced by the shape or the brightness distribution of an exoplanet. The red curve indicates the occultation photometry for a
non-uniformly-bright disk (hot spot in red). The yellow curve indicates the occultation photometry for an oblate exoplanet (yellow ellipse). Both synthetic scenarios show specific deviations from the
occultation photometry of uniformly-bright disk (black curve) in the occultation ingress/egress. Figure from de Wit, et al. 2012.
Know the Star, Know the Planet (In Construction)​
Probing the Interior of Other Astronomical Bodies​
Constraining the density of astronomical bodies is key to discussing their interior properties, and even their formation history. In this subquests, we consider the reliability of existing strategies
and explore new ones.
Showcased Papers:
"On the Effects of Planetary Oblateness on Exoplanet Studies"
Berardo & de Wit 2022
"Constraining the Interiors of Asteroids Through Close Encounters"
Dinsmore & de Wit 2022 (link to follow)
Absence of observational constraints on oblateness limits the access to tight constraints on exoplanets’ density. The cumulative number of planets with a fractional density error below a certain
value (blue line). The black lines show the percentage by which the density would change if a previously assumed spherical planet were to be oblate to a certain degree, calculated using equation 10.
The red line indicating a value of f = 0.25 is in reference to the limits of current observations (pre-JWST). Figure from Berardo & de Wit 2022.
Data, best-fitting results, and residuals for a fit to synthetic data simulated for a reference asteroid. Uncertainty bands are also shown. The best fit results are consistent with the data. Right
Panel: Cross-sectional slices of the extracted uncertainty distribution on density, which is nearly identical to and statistically consistent with the true distribution for this synthetic case with a
dense asymmetric core. Figure from Dinsmore & de Wit 2022.
Background Photo Credit: NASA/JPL-Caltech/R. Hurt, T. Pyle (IPAC)
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Dispaly Multiple Charts In Excel 2024 - Multiplication Chart Printable
Dispaly Multiple Charts In Excel
Dispaly Multiple Charts In Excel – You could make a multiplication graph in Excel simply by using a web template. You can find numerous types of templates and discover ways to file format your
multiplication graph utilizing them. Here are some tips and tricks to create a multiplication graph or chart. When you have a web template, all you need to do is version the formulation and mixture
it in the new cell. After that you can make use of this method to flourish several phone numbers by another established. Dispaly Multiple Charts In Excel.
Multiplication kitchen table web template
If you are in the need to create a multiplication table, you may want to learn how to write a simple formula. Initial, you need to fasten row one of the header line, then increase the number on row A
by mobile phone B. An additional way to produce a multiplication desk is to apply mixed referrals. In this case, you would get into $A2 into column A and B$1 into row B. The outcome is really a
multiplication desk with a formula that really works for columns and rows.
You can use the multiplication table template to create your table if you are using an Excel program. Just open the spreadsheet with your multiplication dinner table template and change the title to
the student’s title. You may also modify the page to fit your personal requires. There is an option to modify the color of the cellular material to improve the look of the multiplication dinner
table, too. Then, you may change the plethora of multiples to suit your needs.
Developing a multiplication graph in Shine
When you’re using multiplication table software program, you can actually develop a simple multiplication table in Stand out. Merely create a page with columns and rows numbered from a to thirty.
Where rows and columns intersect is the solution. If a row has a digit of three, and a column has a digit of five, then the answer is three times five, for example. The same goes for the opposite.
First, you may enter in the amounts that you should grow. For example, if you need to multiply two digits by three, you can type a formula for each number in cell A1. To help make the figures larger,
choose the cellular material at A1 and A8, and then click the correct arrow to decide on a selection of tissues. After that you can sort the multiplication formulation from the cellular material in
the other columns and rows.
Gallery of Dispaly Multiple Charts In Excel
How to Graph Three Sets Of Data Criteria In An Excel Clustered Column
How To Display Multiple Charts From Excel Worksheet On Userform Times
How To Display Multiple Charts In One Chart Sheet
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Publication : t90/178
Non-perturbative effects in 2D gravity and matrix models
David F.
Two dimensional Euclidean quantum gravity may be formulated as a functional integral over 2--dimensional Riemannian manifolds. This infinite dimensional integral may be discretized in such a way
that the topological expansion in terms of the genus of the manifold is mapped onto the $1/N$ expansion of some zero--dimensional matrix model \refs{\Planar}. The $N=\infty$ limit exhibits
critical points which can be shown to describe the continuum limit of 2--dimensional gravity on a genus zero manifold, eventually coupled to some matter fields. Recently it was shown that a
scaling limit can be constructed \ref\Scaling{E. Br\'ezin and V. A. Kazakov, Phys. Lett. 236B (1990) 2125. M. R. Douglas and S. H. Shenker, Nucl. Phys. B335 (1990) 635. D. J. Gross and A. A.
Migdal, Phys. rev. Lett. 64 (1990) 27.} . In this limit all the terms of the topological expansion survive and thus one obtains a fully non--perturbative solution for two dimensional gravity.
However in the most interesting cases, in particular for pure gravity, the solution is defined as a solution of a non--linear differential equation of the Painlev\'e type and presents some
non--perturbative ambiguities, related to the delicate issue of boundary conditions, which are usually attributed to some ``non--perturbative effects" of the theory. In this talk I shall review
some attempts to get a better understanding of these effects. For simplicity and shortness I shall mainly deal with the case of pure gravity, which seems to embody the main problems. The approach
that I have followed consists in trying to relate those non--perturbative issues to the non--perturbative effects which are present in the original matrix models.
Année de publication
: 1990
Chapitre de livre
: Random Surfaces and Quantum Gravity, volume 262 in NATO ASI series B
Maison d'édition
: Plenum Press, 1990
: 21-34
Ecole - Communication invitée
: Random Surfaces and Quantum Gravity ;
Cargèse, France
; 1990-05-28 / 1990-06-01
: Anglais
: pp. 21-34
: Alvarez O., Marinari E., Windey P.
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Click here to download this file as a Word document.
DESIGN, DEVELOPMENT, AND USE OF WEB-BASED
INTERACTIVE INSTRUCTIONAL MATERIALS
Michael R. Colvin David A. Smith
Department of Mathematics Duke University
California Polytechnic State University Box 90320
San Luis Obispo, CA 93407 Durham, NC 27708-0320
mcolvin@calpoly.edu das@math.duke.edu
Lawrence C. Moore Franklin A. Wattenberg
Duke University Department of Mathematics
Box 90320 Montana State University
Durham, NC 27708-0320 Bozeman, MT 59717
lang@math.duke.edu frankw@math.montana.edu
(on leave at
William J. Mueller National Science Foundation,
Department of Mathematics DUE, Suite 835
University of Arizona 4201 Wilson Blvd.
Tucson, AZ 85721 Arlington, VA 22230
mueller@math.arizona.edu fwattenb@nsf.gov)
The authors are participants in the Connected Curriculum Project, a World Wide Web project with home page at:
and materials available on servers at Cal Poly San Luis Obispo, Duke University, and Montana State University. The project is supported by National Science Foundation grant DUE-9555407.
The Connected Curriculum Project
CCP is the result of a happy meeting of a new medium the World Wide Web with some older ideas. Those actively involved in the calculus reform movement have generally shared the following three goals:
To break down the barriers between disciplines to study mathematics in the context of real problems. To engage our students in active learning. To help our students and ourselves develop a sense of
ownership of the concepts and techniques discussed in our classes. Almost all of lower division mathematics was developed to study problems from the real world the same problems that most of our
students hope eventually to be able to solve. By studying mathematics in context, students learn more about doing mathematics, learn more about the world around them, and are better prepared to use
mathematics effectively outside the mathematics classroom. Mathematics is an essential language for studying other subjects, and it should be routinely used across the curriculum. Our first goal is
Mathematics Across the Curriculum.
The architecture of the World Wide Web is ideally suited for accomplishing all three of these goals. Web browsers are designed to encourage very active and interactive engagement with material. We
use a browser as a conductor, orchestrating a mix of text, graphics, movies, and sound with helper applications, such as Mathematica, Maple, Mathcad, and TI Graph Link. In a typical session, students
go back and forth between their browser window, their CAS window, and "hands-on" laboratory equipment, exploring real phenomena, mathematical models, and the mathematics underlying the models at the
same time.
The hypertext structure of web material emphasizes links and connections rather than compartmentalization. Students starting with material in one subject will find themselves following links to other
subjects. Hypertext also emphasizes choice. By choosing applications and examples that interest them, and by selecting background material they need, students construct their own individualized
courses, tailored to their own interests and background.
The malleability of web-based material encourages multiple authorship and, most importantly, multiple ownership. Web-based courses are living courses, evolving as our world evolves. Our goal is to
create and maintain a large volume of highly interconnected material that will support learning across the curriculum with bridges rather than barriers.
Design, Implementation, and Learning Issues
Over the past few years we have benefitted from a study of the ways materials can and should be presented on a computer screen. Issues such as use of color, placement, and font face and size, and
arrangement of text, interactive mathematics, and graphics are important factors in the pedagogical success of interactive modules. The CCP materials are designed to translate sound design principles
to a web-based environment in which the author has only limited control over the final rendering of the display.
Our materials guide exploration by a variety of tools, e.g., computer algebra systems such as Maple, Mathematica, or Mathcad, Java-based interactions, and CBL data gathering. Our live presentation
shows how to tie all these activities together through the web and how to use related web-based materials as part of the exploration.
How does use of these materials in a course affect the development of the course and the student approach to learning? As we outline in the next section, the materials may be used in a variety of
ways. What effects do these different approaches and combinations of them have on learning? Our live examples illustrate the lessons we have learned.
Uses of CCP Materials
Our materials are being and will be used in various ways: As supplements to existing courses that use currently available textbooks. To support new courses that draw material from several different
fields. By students on their own to supplement and complement material in their courses, and to help them see connections between courses and fields that are often overlooked in a departmentalized
These materials can also be used in different settings: In regularly scheduled laboratory components for traditionally structured classes. In self-scheduled laboratory components for traditionally
structured classes, with work at various sites on campus or at home or in dorms wherever students have web access, plus a helper application and (if needed) lab equipment. In courses where some
lecturing is replaced by group work, with students at workstations that include computers and lab equipment, and with an instructor circulating to work with students in their small groups. By
students who choose to use CCP as an additional resource working alone or in groups at home, in dorms, or in open labs.
Contributing Authors and Participating Sites
In addition to the authors of this paper, participants in and contributors to CCP include:
William Barker (Bowdoin College, ME)
Lewis Blake (Duke University, NC)
Robert Cole (Evergreen State College, WA)
Lester Coyle (Loyola College, MD)
Donald Hartig (Cal Poly San Luis Obispo, CA)
Leonard Lipkin (University of North Florida, FL)
Charles Patton (MathTech Services, Eugene, OR)
James Peters (Weber State University, UT)
Richard Schori (Oregon State University, OR)
In the future we expect to accept contributions to a Connected Curriculum Library that will comprise peer-reviewed and professionally-edited materials residing on a single site, with several mirror
sites. For the time being, our materials reside on three separate (but linked) sites, each with a special focus.
At the Cal Poly Site the focus is on interdisciplinary projects, student-directed investigations of extensions and applications of classroom material. These projects cover mathematical topics from
agriculture, architecture and enviromental design, business, engineering, liberal arts, various sciences, and mathematics itself. The mathematical level ranges from precalculus to differential
equations. Each project contains links to support modules in the relevant areas of mathematics.
The Duke Site focuses on interactive modules, self-contained lessons for use as laboratory activities, classroom demonstrations, or self-study. Mathematical topics come from precalculus, single- and
multivariable calculus, differential equations, linear algebra, engineering math, and probability and statistics. The modules also cover a wide range of applications of interest to students and
teachers across disciplines. This site is a descendant of the NSF-sponsored Project CALC reform calculus project. In particular, it will eventually contain web-based versions of all the Project CALC
The focus of the Montana State Site is on interactive texts, detailed, full-length expositions of mathematical subjects with frequent opportunities for reader participation and investigation.
Subjects include precalculus, calculus, modeling, linear algebra, differential equations, and probability and statistics. The texts are cross-referenced, linked to a library of supplementary help
material, and tied together by a number of recurring "case studies." Through links to many applications, the texts seek to break down barriers between disciplines and to encourage student and teacher
"ownership" of individual courses.
The Connected Curriculum Project is producing innovative interactive materials in areas of mathematics from precalculus to linear algebra and differential equations, as well as materials that
illustrate the use of mathematics in other disciplines across the curriculum. Some of the units are designed to be used as integral parts of courses others are free-standing reference modules for
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