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C++ to find the factorial of a number using class - CodeVsColor
C++ to find the factorial of a number using a class in 2 ways:
In this post, we will learn how to find the factorial of a number with a class in C++. The factorial of a number is equal to the multiplication of all numbers from 1 to that number.
Using a class, we can put all the factorial-related operations in the class and one method can be exposed to calculate the factorial. We can create one object of the class and call the factorial
method of that class to find the factorial.
It will make the code clean if we move all related functions into one place. With this C++ program, you will learn how to create a different class to calculate the factorial of a number.
Method 1: C++ program to calculate factorial with a class:
The following program uses a separate class Factorial to calculate the factorial of a user-given number:
#include <iostream>
using namespace std;
class Factorial
int num;
unsigned long long factorial = 1;
void calculateFactorial();
void show();
void Factorial::calculateFactorial()
cout << "Enter a number:" << endl;
cin >> num;
if (num == 0 || num == 1)
factorial = 1;
while (num > 1)
factorial = factorial * num;
void Factorial::show()
cout << "Factorial: " << factorial << endl;
int main()
Factorial factorial;
Download the program on Github
• The Factorial class is used to calculate the factorial with user-input value.
• The num is an integer variable to assign the number. The variable factorial is an unsigned long long variable to hold the factorial value. The maximum value of unsigned long long is 2^64-1 or
18446744073709551615. The program will fail if the final factorial result is more than 18446744073709551615.
• The calculateFactorial and show are two public functions. The calculateFactorial function is used to calculate the factorial. The calculateFactorial function reads the user input number and
calculates the factorial. It uses one while loop to calculate the factorial. It multiplies all numbers from num to 1.
• The show function is used to show the factorial value, i.e. the value of the factorial variable.
Enter a number:
Factorial: 24
Enter a number:
Factorial: 120
Method 2: By using a recursive static method:
In the above example, we have used two non-static methods to read the user input and calculate the factorial. We can also use a static method to calculate the factorial. The main method will get the
number from the user and use this method to get the factorial. The advantage of a static method is that we can access this method without creating an instance of the class.
#include <iostream>
using namespace std;
class Factorial
static unsigned long long calculateFactorial(unsigned long long num)
if (num == 0 || num == 1)
return 1;
return num * calculateFactorial(num - 1);
int main()
int num;
cout << "Enter a number :" << endl;
cin >> num;
unsigned long long factorial = Factorial::calculateFactorial(num);
cout << "Factorial: " << factorial << endl;
Download the program on Github
• The calculateFactorial method is a recursive method. It returns 1 if the parameter num is either 0 or 1. Else it multiplies the value of num with the factorial value of num - 1. It uses recursive
calls to find a factorial.
• Since the method is a static method, we can access it without creating an instance of the Factorial class.
It will print similar output:
Enter a number :
Factorial: 2432902008176640000
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MaffsGuru.com - Making maths enjoyableThe general equation of a straight line
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Quantitative Literacy and the Humanities: Probability: Basic calculations
Probability: Basic calculations
Two independent events
Given the probabilities of two independent events, students calculate the probability of both events happening, or of one or the other happening.
• The probability of both Event A and Event B = (probability of Event A) x (probability of Event B)
• The probability of either Event A or Event B = (probability of Event A) + (probability of Event B)
Expected value
Students calculate the expected value of an event in order to assess risk.
• Expected value = sum of all different outcomes
• [(probability of Event A) x (payoff for Event A)] + [(probability of Event B) x (payoff for Event B)]
If you are offered the chance to play a game in which you roll a die, and you receive the dollar value of the roll: $1 for a one, $2 for a two, etc. It costs $3 to play. Is it worth the risk? Answer!
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Forward Secrecy
Forward Secrecy posted December 2014
I was asked during an interview how to build a system where Alice could send encrypted messages to Bob. And I was asked to think outloud.
So I imagined a system where both Alice and Bob would have a set of (public key, private key). I thought of RSA as they all use RSA but ECIES (Elliptic Curve Integrated Encryption Scheme) would be
more secure for smaller keys. Although here ECIES is not a "pure" asymmetric encryption scheme and Elgamal with ECs might be better.
Once one wants to communicate he could send a "hello" request and a handshake protocol could take place (to generate a symmetric encryption key (called a session key in this case)).
I imagined that two session keys would be generated by each end. Different set of keys depending on the direction. One for encrypting the messages and one for making the MAC (that would be then
appended to the encrypted message. So we EtM (Encrypt-then-Mac)).
Then those keys would be encrypted with the public signature of the other one and sent over the wire like this. And Let's add a signature so we can get authentication and they also won't get
tampered. Let's use ECDSA (Elliptic Curve Digital Signature Algorithm) for that.
Although I'm wondering if two symmetric keys for encrypting messages according to the direction is really useful.
I was then asked to think about renewal of keys. Well, the public keys of Alice and Bob are long term keys so I didn't bother with that. About the symmetric keys? What about their TTL (Time To Live)?
My interviewer was nice enough to give me some clues: "It depends on the quantity of messages you encrypt in that time also."
So I thought. Why not using AES in CTR mode. So that after a certain number of iteration we would just regenerate the symmetric keys.
I was then asked to think about Forward Secrecy.
Well I knew the problem it was solving but didn't know it was solving it.
Mark Stamp (a professor of cryptography in California (how many awesome cryptography professors are in California? Seriously?)) kindly uploaded this video of one of his class on "Perfect Forward
So here the trick would be to do an Ephemeral Diffie-Hellman to use a different session key everytime we send something.
This EDH would of course be encrypted as well by our public key system so the attacker would first need to get the private keys to see that EDH.
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Documentation bug in original 15C manual
I couldn't find a list of known bugs in the original manual so sorry if this is not new.
Only minor but page 24 of the 15C Owners Handbook says "Pressing [g][pi] places the first
10 digits of pi into the calculator."
However, strictly speaking, the calculators (both original and LE) do not do this. Instead [g][pi] places a rounded to 10 digits approximation to Pi into the calculator. (And the example on page 62
gets this right.)
05-01-2012, 05:13 PM
I'd also add that the digits are placed in the X-register, not into the calculator as in some unknown place... (I still have my head back in the 80's, when manuals usually cleared issues instead of
bringing them in...)
05-01-2012, 05:14 PM
Quote: I couldn't find a list of known bugs in the original manual so sorry if this is not new.
Only minor but page 24 of the 15C Owners Handbook says "Pressing [g][pi] places the first
10 digits of pi into the calculator."
However, strictly speaking, the calculators (both original and LE) do not do this. Instead [g][pi] places a rounded to 10 digits approximation to Pi into the calculator. (And the example on page
62 gets this right.)
Actually, it doesn't put it into the calculator either. It's already there. Rather it moves it into the X register. Or better yet it makes a copy of the constant rounded to 10 places and puts it into
the X register, which is really a BCD representation of the decimal number system, and not the actual number itself. (assuming of course it doesn't calculate it on the fly)
05-01-2012, 06:51 PM
Quote:(...) BCD representation of the decimal number system, and not the actual number itself. (assuming of course it doesn't calculate it on the fly)
Interesting perspective... If we are talking about memory usage versus program instructions (time to compute being a factor, too), how many digits of a constant (two digits = 1 byte) would be worth
computing instead of storing in system memory? There must be a break even point...
Luiz (Brazil)
Edited: 1 May 2012, 6:55 p.m.
05-01-2012, 07:44 PM
Quote: ... how many digits of a constant (two digits = 1 byte) would be worth computing instead of storing in system memory?
(Sorry, couldn't resist)
05-01-2012, 09:25 PM
Long Live 'The Hitch Hikers' (it will end in tears...)
(neither could I)
GREAT! Loved it!
Edited: 1 May 2012, 9:27 p.m.
05-01-2012, 05:37 PM
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Binomial conditions
When is an experiment described by the binomial distribution? Why do we need both the condition about independence and the one about constant probability?
The binomial distribution is appropriate when we have the following setup:
We perform a fixed number of trials, each of which results in "success" or "failure" (where the meaning of "success" and "failure" is context-dependent). We also require the following two
(i) each trial has an equal probability of success, and
(ii) the trials are independent.
If we let $X$ be the number of successful trials, then $X$ has a binomial distribution.
When we have a situation which looks like it might be binomial, we need to check that all of these conditions hold before we can use the binomial distribution formulae!
We're going to design some scenarios which have a fixed number of trials, each of which results in "success" or "failure", and $X$ is the number of successful trials.
(a) Is it possible that only (ii) holds, but not (i)? That is, can you design a scenario where the probability of success is not equal for all the trials, even though they are independent?
(b) Is it possible that only (i) holds, but not (ii)? That is, can you design a scenario where the trials are not independent, even though each trial has equal probability of success?
Is the probability distribution of $X$ that of a binomial distribution in either of your scenarios? (That is, does it have the form $\mathrm{P}(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ for $r=0$, ..., $n$
for some choice of $n$ and $p$?)
You might find it helpful to work on Binomial or Not? first.This resource is part of the collection Statistics - Maths of Real Life
Copyright information
The icon for this problem is by Matemateca (IME/USP)/Rodrigo Tetsuo Argenton, originally downloaded from Wikimedia Commons (and then adapted for NRICH), licensed under CC BY-SA 4.0.
Getting Started
What does it mean for two trials to be
? Make sure you are absolutely clear on this before you start!
Can you come up with examples where the trials are dependent? What do we mean by the probability of success of the second trial if it is dependent on the first trial?
You might find it helpful to think in terms of tree diagrams or two-way tables (if there are only two trials).
Bear in mind that if each trial is a repeat of the first trial with the same starting conditions, then it is likely to have the same probability of success and be independent of the first trial. So
to find an example where these conditions are
met, the trials cannot possibly all look identical.
There are some example situations that might give you some ideas in
Binomial or Not?
Student Solutions
Some of the situations discussed in this solution also appear in the problem
Binomial or Not?
We focus on the case $n=2$, though these examples can easily be extended.
The probability distribution for $X\sim \mathrm{B}(2,p)$ is:
│$x$ │$0$ │$1$ │$2$ │
│$\mathrm{P}(X=x)$ │$(1-p)^2$│$2p(1-p)$│$p^2$│
(a) Is it possible that only (ii) holds, but not (i)? That is, can you design a scenario where the probability of success is not equal for all the trials, even though they are independent?
We are taking balls from bags of green and red balls. Taking a green ball is considered success. On each trial, we draw a ball at random from a
bag; each bag has a different proportion of green balls. Each trial is independent, but the probability of success is not equal for all the trials. We let $X$ be the number of green balls drawn from
the $n$ bags, where $n$ is fixed.
As an extreme case, let us suppose that we have just two bags ($n=2$), the first bag being all red and the second being all green. Then we will always draw exactly one green ball, so $\mathrm{P}(X=0)
=\mathrm{P}(X=2)=0$. This does not match the binomial distribution in the table above no matter what $p$ is.
A non-example, however, is sampling from the
bag without replacement. We discuss this in (b) below.
There are less extreme examples which have the same non-binomial distribution behaviour.
(b) Is it possible that only (i) holds, but not (ii)? That is, can you design a scenario where the trials are not independent, even though each trial has equal probability of success?
This is quite subtle. Let us imagine that we are counting the number of heads ($X$) appearing on flips of coins, where a head is considered success and a tail is considered failure. We stick two
coins the same way up onto a ruler and toss the ruler. Then the probability of obtaining a head on either coin is equal (and neither 0 nor 1), as they either both land heads or both land tails. But
the results for the two coins are not independent: once we know how the first one landed, we are certain about the result from the second one. So $\mathrm{P}(X=1)=0$, even though $\mathrm{P}(X=0)$
and $\mathrm{P}(X=2)$ are both non-zero. Hence $X$ does not have a binomial distribution.
A more familiar context - though not a perfect example - is asking for the number of sunny days in a certain town during the month of May. The probability of any particular day in May being sunny is
approximately the same. However, if we know that 9th May, say, was sunny, then it is more likely that 10th May will also be sunny. Therefore the probability of any given day being sunny is the same
as the probability of any other given day being sunny, but the sunniness of the days are not independent events.
Note that the probabilities here are only equal (or in the second case, approximately equal) when we are asking for the probabilities before the experiment has started. Once the experiment has
started, we have more information, and so the probabilities of future trials will change.
A more subtle example of the same phenomenon occurs with drawing balls from a bag without replacement. Let us consider the case of a bag with 2 green and 2 red balls initially. We draw two balls, and
count a green ball as a success. $X$ is the total number of green balls drawn. We can calculate probabilities using a tree diagram:
So the probabilities are:
\mathrm{P}(\text{GG}) &= \tfrac{1}{2}\times \tfrac{1}{3} = \tfrac{1}{6} \\
\mathrm{P}(\text{GR}) &= \tfrac{1}{2}\times \tfrac{2}{3} = \tfrac{2}{6} \\
\mathrm{P}(\text{RG}) &= \tfrac{1}{2}\times \tfrac{2}{3} = \tfrac{2}{6} \\
\mathrm{P}(\text{RR}) &= \tfrac{1}{2}\times \tfrac{1}{3} = \tfrac{1}{6} \\
\mathrm{P}(\text{first ball green}) &= \tfrac{1}{6} + \tfrac{2}{6} = \tfrac{3}{6} \\
\mathrm{P}(\text{second ball green}) &= \tfrac{1}{6} + \tfrac{2}{6} = \tfrac{3}{6}
Therefore the first ball and second ball each have a probability of $\frac{1}{2}$ of being green. However, the probability of the second ball being green given that the first ball is green is only $\
frac{1}{3}$, so the trials are not independent. The probability distribution of $X$ is also not binomial. (Why the trials have equal probabilities of green is an interesting question and worth
A final note
Another way of describing the two conditions (i) and (ii) is with the single condition: "the probability of any trial being successful is the same, regardless of what happens on any other trial". The
phrase "regardless of what happens on any other trial" is equivalent to saying that the trials are independent.
Teachers' Resources
Why do this problem?
It is common for students who have studied the binomial distribution to be quite unfamiliar with the conditions necessary for this to be an appropriate distribution to model an experiment. The
constant probability one is fairly straightforward, but the independence condition is quite poorly understood. Even some textbooks fail to describe the conditions correctly.
In this problem, students are asked to construct scenarios in which one of the two conditions holds but the other does not. The first is easier, but the second requires a clear understanding of the
term "independent". Students are likely to deepen their understanding of this concept by working on this problem, as well as gain a deeper appreciation for the need for these conditions. It will
help students to be able to distinguish between situations which are described by a binomial distribution and those which are not.
Note that this problem does not address the need for the number of trials to be fixed from the start, and for the random variable to be the total number of successes.
Possible approach
This problem will be most helpful after students have had some exposure to the binomial distribution and have developed a familiarity with it in "regular" situations. The teacher could first ask
their students if they can recall the conditions for a binomial distribution to be appropriate, and remind them if they have forgotten. The teacher should then check that the students understand the
terms in the conditions, and in particular the term "independent". Once this is clear, students could work on their own initially to construct such examples, and then share their ideas with a
partner before feeding back to the whole class.
There might be significant confusion around the idea of the probability of a dependent event. If the second trial is not independent of the first one, then the probability of success on the second
trial will change after the first trial has been performed. So when we say that "each trial has an equal probability of success", what we mean is "before the experiment begins, each trial has equal
probability of success". If instead we meant "regardless of what happens on earlier trials, each trial has an equal probability of success", we would be saying that the trials are independent and
they each have an equal probability of success, which is just (i) and (ii) together. This point may well come up through discussion.
Key questions
What does "independent" mean?
How could the separate trials be best represented?
What are some key features of the binomial distribution probabilities?
Possible extension
How many different scenarios can you construct where the binomial distribution fails to be the correct distribution, even though it has some of the features required?
Possible support
If students have not yet worked on
Binomial or Not?
, this might be a useful starting point.
For the question of independence, it may be easier to come up with reasons why two trials are
. If you can't do that, then they are likely to be independent.
Limit yourself to just two trials. How could these be represented? (At least two different ways!)
What would the probabilities have to be if the number of successes is described by the binomial distribution? How can you break this? | {"url":"https://nrich.maths.org/problems/binomial-conditions","timestamp":"2024-11-02T14:04:54Z","content_type":"text/html","content_length":"49403","record_id":"<urn:uuid:f0520ac3-5d54-47f8-a508-31ed768c5b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00897.warc.gz"} |
EMS | Jobs | University Professor of Mathematical General Relativity
University Professor of Mathematical General Relativity
Faculty of Mathematics, University of Vienna | University of Vienna
The position holder should conduct excellent research in the field of low regularity spacetime geometry, singularity theorems and synthetic extensions of Lorentzian geometry. He*she should be
internationally visible in both Lorentzian geometry and mathematical general relativity. The position holder is expected to collaborate with the FWF Emerging Fields Programme "A new geometry for
Einstein’s theory of relativity and beyond". In addition, the position holder is expected to make a significant contribution to the teaching profile of the faculty, especially in teacher training.
He*she is also expected to collaborate with the Mathematics Education group at the faculty and to work at the interface between mathematics and mathematics didactics research.
Last updated: 30 October 2024
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Karen Smilowitz
Sequencing activities over time is a fundamental optimization problem. The problem can be modeled using a directed network in which activities are represented by nodes and pairs of activities that
can be performed consecutively are represented by arcs. A sequence of activities then corresponds to a path in the directed network, and an optimal sequence … Read more | {"url":"https://optimization-online.org/author/ksmilowitz/","timestamp":"2024-11-08T08:43:42Z","content_type":"text/html","content_length":"82989","record_id":"<urn:uuid:643f2966-79cc-4571-b379-d5724eb92d9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00758.warc.gz"} |
Interesting Information.....
Re: Interesting Information.....
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
The Answer II 165 is correct. Excellent, bobbym!
II 166. What are the composition of Alpha brasses and Beta brasses?
II 167. What is the composition of Constantan, a copper-nickel alloy also known as Eureka, Advance, and Ferry?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
Although not within the range of 5%, it is very close, a good attempt! Neat work, bobbym! (II 163)
II 164. What is the unit franklin (Fr)?
II 165. What is the unit maxwell, abbreviated as Mx?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
II 162. What is the area in square kilometers or square miles of the nation Democratic Republic of the Congo? (±5%)
II 163. What is the area in square kilometers or square miles of the nation Republic of the Sudan? (±5%)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
Good attempt, bobbym! (II 158)
II 160. What is the density of Bromine in kilograms per meter cube? (room temperature)
II 161. What is the density of Diethyl ether in grams per cubic centimeters or kilograms per meter cube?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
II 158. What is the density of concrete in kilograms per meter cube?
II 159. What is the density of cotton in kilograms per meter cube?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
The Answer II 167 is correct. Neat work, bobbym!
II 168. Give the value of
up to three decimals. (rounded)
II 169. Give the value of
up to three decimals. (rounded)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
Both the Answers, II 168 and II 169, are perfect. Brilliant, bobbym!
II 170. What is the Surface Gravity in meters per second square of Jupiter? (±10%)
II 171. What is the Surface Gravity in meters per second square of Saturn? (±10%)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
The Answer II 170 is perfect (well within the range). Excellent, bobbym!
II 172. What is the Escape velocity in kilometers per second of Jupiter?
II 173. What is the Escape velocity in kilometers per second of Saturn?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Interesting Information.....
Both the Answers, II 172 and II 173, are correct. Perfect, bobbym!
II 174. What is the refractive index of Silver Bromide?
II 175. What is the refractive index of Aluminium oxide (British English) or aluminum oxide (American English)?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 176. What is the value : 1° of latitude at equator (Kilometers or miles)?
II 177. What is the value : 1° of latitude at poles (Kilometers or miles)?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 178. What is the value of a Nautical mile?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 179. Blood pressure is one of the vital signs—together with respiratory rate, heart rate, oxygen saturation, and body temperature—that healthcare professionals use in evaluating a patient's
health. What is the normal resting blood pressure, in an adult is approximately? (with units)
II 180. The blood sugar level, blood sugar concentration, or blood glucose level is the measure of concentration of glucose present in the blood of humans or other animals. What is the international
standard way of measuring blood glucose levels? (units)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 181. What is the mean radius of planet Uranus in kilometers approximately?
II 182. What is the mean radius of planet Neptune in kilometers approximately?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 183. What are the Alkali Metals?
II 184. What are the Halogens?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 185. What is the density of Helium?
II 186. What is the density of Sodium?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 187. What is the density of Uranium?
II 188. What is the density of Platinum?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Re: Interesting Information.....
II 189. What is special about the number 14,641?
II 190. What is special about the number 360,360?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking. | {"url":"https://mathisfunforum.com/viewtopic.php?pid=399353","timestamp":"2024-11-13T01:38:32Z","content_type":"application/xhtml+xml","content_length":"72323","record_id":"<urn:uuid:85caa1fa-5fcd-4a50-85a0-224fc5e53bd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00019.warc.gz"} |
Authors: F. Cazals and T. Dreyfus and T. O'Donnell
This package discusses the representations used to represent molecular conformations. Such coordinates are cornerstones of the following tasks [157] , [174] , [97] :
• Energy calculations: computing the potential energy of a (macro-)molecular system, using a so-called force field, see the package Molecular_potential_energy.
• Structure minimization: minimizing the potential energy of a system by following its negative gradient.
• Structure generation: generating a novel structure given a known structure – aka a move set.
Pre-requisites: representing molecular conformations
Internal coordinates
We consider a molecular graph, as introduced in the package Molecular_covalent_structure.
Internal coordinates represent the geometry of a molecule in terms of bond lengths, valence angles, and dihedral angles [141] .
Bond lengths.
Bonds are defined by two points connected in the molecular covalent graph~(Fig. fig-ic-bl-va (A)).
Valence angles.
Valence angles are defined by around a particle participating in two such bonds, thereby defining an angle (Fig. fig-ic-bl-va (B)).
Dihedral angles.
Dihedral angles come into two guises: proper and improper.
For proper angles, consider four consecutive atoms on a path: the dihedral angle in the angle between the two planes defined by the first three and the last three atoms (Fig. fig-ic-dh (A)).
For improper angles, consider a central atom fig-ic-dh (B)). Pick a second atom the define a hinge, e.g. fig-ic-dh (B) ). Note that an improper angle can be thought as the off planarity angle of atom
│Bond length and valence angle │
│The bond length is the distance between two atoms. The valence angle is the angle at the apex of the triangle formed by two covalent bonds sharing a central atom.│
│Dihedral angle: proper and improper │
│A proper dihedral angle is defined by three consecutive atoms. An improper dihedral angle is defined by a central atom connected to three others: the angle is that defined by two planes sharing an │
│edge of the tetrahedron involving the four atoms. │
Coordinates and degrees of freedom
Primitive internal coordinates refer to the set of all internal coordinates (bond lengths, valence angles, dihedral angles) defined from a covalent structure.
Internal coordinates are classically represented using a so-called Z-matrix (Z-matrix).
To understand subtleties between the various types of coordinates, the following notations will be useful:
We now discuss several examples illustrating these definitions.
Consider the fluoroethylene from Fig.
(D). It is easily seen that there are 5 bond lengths, 6 valence angles, 4 dihedral angles. For the latter, observe that once one has fixed the
│Covalent structure and coordinates: Cartesian coordinates, internal coordinates, and degrees of freedom. │
│See examples exple-ic-one , exple-ic-cyclobutane and exple-ic-fluoroethylene for details. (A) A molecular graph with 4 covalent bond lengths, 2 valence angles, and 1 dihedral angle. (B) A molecular│
│graph with 3 covalent bond lengths and 3 valence angles (C) Cyclobutane: 4 bond lengths, 4 valence angles, 4 dihedral angles. (D) Fluoroethylene: 5 bond lengths, 6 valence angles, 4 dihedral │
│angles. Inset: indices of the atoms for the z-matrix representation of Fig. fig-z-matrix-fluoroethylene . │
APtclcactv11091612353D 0 0.00000 0.00000
6 5 0 0 0 0 0 0 0 0999 V2000
-1.0606 0.1723 0.0001 F 0 0 0 0 0 0 0 0 0 0 0 0
0.1319 -0.4627 -0.0005 C 0 0 0 0 0 0 0 0 0 0 0 0
1.2458 0.2325 0.0001 C 0 0 0 0 0 0 0 0 0 0 0 0
0.1690 -1.5420 0.0030 H 0 0 0 0 0 0 0 0 0 0 0 0
2.1991 -0.2751 -0.0004 H 0 0 0 0 0 0 0 0 0 0 0 0
1.2087 1.3119 0.0010 H 0 0 0 0 0 0 0 0 0 0 0 0
M END
Fluoroethylene in MOL/SDF format.Generated with Jmol (right mouse click / File / Save.
Internal coordinates: primitive, natural, delocalized
As illustrated by example exple-ic-fluoroethylene, the choice of a coordinate system to represent a molecule may be non trivial. It is even more so in the presence of cycles. The choice of the
representation depends on the problem tackled, and is of paramount importance when geometric optimization (minimization of the potential energy) is carried out. The reader is referred to the
excellent overview provided in the Q-Chem user manual (Q-Chem and more specifically here).
In short:
• Primitive internal coordinates are encoded in the graph topology. As illustrated by example exple-ic-fluoroethylene, such coordinates are usually redundant, so that deciding of a non redundant
set of coordinates does not admit a unique solution. See also [156] .
• To reduce the coupling, both harmonic and anharmonic, between internal coordinates, natural internal coordinates were designed [147] , and algorithms to derive them proposed [84] . These
algorithms remain complex, though.
• Following the spirit of natural internals, yet with much simpler generation algorithms, delocalized internal coordinates were finally proposed [15] . In a nutshell, these coordinates, which stem
for a linear transformation between redundant internal coordinates and Cartesian coordinates, are characterized as follows:
□ they form a complete and non redundant set of
□ each such coordinate is a linear combination of a number of all internal coordinates.
In the sequel, we focus on primitive internal coordinates (PIC) and delocalized internal coordinates (DIC).
Primitive internal coordinates (PIC)
Given a molecular graph, see package Molecular_covalent_structure, all primitive internal coordinates are generated as follows:
• bond lengths: one iterates over the edges of the graph.
• valence angles: one iterates over the pairs of consecutive edges of the graph.
• proper dihedral angles: one iterates over the edges exple-ic-fluoroethylene .
The primitive internal coordinates are computed using the class SBL::CSB::T_Molecular_primitive_internal_coordinates < ConformationType , CovalentStructure >, where ConformationType represents a
conformation with Cartesian coordinates as defined in the package Molecular_conformation, and CovalentStructure represents a covalent structure as defined in the package Molecular_covalent_structure.
The class SBL::CSB::T_Molecular_primitive_internal_coordinates provides methods for computing the primitive internal coordinates of :
• a given bond : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_distance ,
• a valence angle : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_bond_angle ,
• a proper dihedral angle : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_dihedral_angle .
The class SBL::CSB::T_Molecular_primitive_internal_coordinates provides also methods to fill arrays (or other compliant data structures) with all primitive internal coordinates for :
• bonds : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_bond_distances ,
• valence angles : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_bond_angles ,
• dihedral angles : SBL::CSB::T_Molecular_primitive_internal_coordinates::get_dihedral_angles .
From Cartesian coordinates to internal coordinates
Internal coordinates are defined as follows:
Bond length. The bond length between two particles
Valence angle. The valence angle at particle
Dihedral angle (proper). Denoting
Note that the orientations of
The two possible orientations of cis-trans isomerism: in the trans configuration, the dihedral angle is expected to be cis configuration, it is expected to be
From internal coordinates to Cartesian coordinates.
There exist several strategies to convert back internal coordinates to Cartesian coordinates, and these differ in several respects, namely the number and the type of floating point operations. As
shown in [141] , the most efficient one in the so-called SN-NeFR method.
The Wilson B matrix
Displacements in internal and Cartesian coordinates are related by the so-called B matrix. To define it, consider:
Following [157] , [97] and the references therein, we define:
The Wilson
Practically, a row of matrix eq-bond-length) or Eq. (eq-valence-angle) or Eq. (eq-torsion-angle)) with respect to the Cartesian coordinates defining the variable associated with the row.
From internal coordinates to Cartesian coordinates
We cover several embedding operations:
• Embedding a 4-rth atom using a dihedral angle
• Embedding the
Embedding for a dihedral angles
The operation which consists of computing the Cartesian coordinates of one atom is called the embedding step. This operation requires a context, that is 3 atoms already embedded, with respect to
which the new atom is positioned.
This operation is then repeated to embed all atoms.
The Nerf and SN-NeRF algorithms
In the following, we present the [141].
Given a set of points fig-nerf-embedding), the aim is to embed
The first operation plainly consists of using spherical coordinates in a suitable coordinate system centered at specialized reference frame in [7] .) This yields the following coordinates for
The second operation consists performing a rotation + translation to transform the previous coordinate system into that of the world/lab. The final position of point
we obtain
│Cartesian embedding of a point given (i) a context defined by three points/atoms, and (ii) a dihedral angle.│
Iterative embedding
We use the previous operation to iteratively embed a molecule, given all internal coordinates.
Initialization. The initialization consists of embedding three particles connected in a path, using two distances and an angle in an arbitrary Cartesian reference frame. In our case (Fig.
Iterative embedding of the remaining particles. The embedding of the remaining particles in the coordinate system defined by the first three is computed while performing a traversal of the molecular
covalent graph. To describe the algorithm, we use the notion of context, namely three connected atoms which have already been embedded, and with respect to which the new particle is positioned.
The traversal is performed using two stacks: one corresponding to the points to be embedded next; the second one refers to the contexts (one for each atom to be embedded).
Note that the initialization makes it possible to stack all the neighbors of the first three atoms – their context is defined by these three atoms.
Then, the algorithm proceeds iteratively as follows:
• The particle on the top of the stack is popped, and is embedded using its context. The particles linked to it by an edge in the covalent graph and not already visited are stacked, together with
their context.
• Each time an embedded particle is stacked it is tagged to avoid processing twice.
The process terminates when the stacks are empty.
│Conversion from internal to Cartesian coordinates. In this illustration the graph is traversed from left to right and each colored particle is embedded using the previous three as context and the │
│three internal coordinates with the same color. │
Implementation: main class. The conversion is performed using the operator of class SBL::CSB::T_Molecular_cartesian_coordinates. The template arguments are the Conformation type and the Covalent
structure. As parameter for the operator the covalent structure and the internal coordinates are required (bond lengths, bond angles and dihedral angles).
Implementation: utilities. Static functions are defined in SBL::CSB::Molecular_coordinates_utilities. These can be used to compute bond and dihedral angles given vectors and to embed a particle given
three others and internal coordinates as done in
Embedding the Cbeta carbon atom
To graft the
Analytically, define the following –see [odonnell2022modeling] :
One obtains the two possible embeddings for
│Placing the Using three valence angles and one distance yields two solutions, the correct one being selected by a chirality argument. ││
The correct solutions is selected by chirality, noting that the volume of the tetrahedron defined by
Delocalized internal coordinates (DIC)
To derive the DIC, we follow [15] :
And their derivatives.
For the gradient in internal coordinates, see [157] , [97] and the references therein.
subsection mol-coords-dic-rep Representation
matrices from eigen
Delocalized internal coordinates are not yet implemented.
Later: rep is using matrices
And their derivatives.
This package offers also offers sbl-coordinates-converter.exe and sbl-coordinates-file-converter.exe to perform various conversions of coordinates.
Converting IC to CC : back and forth
The application sbl-coordinates-converter.exe provides the following conversions:
• Converting Cartesian coordinates into internal coordinates, and
• Converting internal coordinates into Cartesian coordinates,
• converting XTC files s (if Gromacs is installed) into the Point-d file format used.
We note in passing that our encoding of internal coordinates is based on atom ids, as provided in PDB files.
Converting Cartesian coordinates to internal coordinates. Given a PDB file, this executable generates all IC (bond lengths, valence angles, dihedral angles):
sbl-cartesian-internal-converter.exe --pdb-file data/ala5.pdb
This call simply generates one txt file for each type of coordinate:
==> ala5_bonds.txt <==
#Chain X
#Atom ids bond length(Angstrom)
1 5 1.45994
5 11 1.51006
==> ala5_bond_angles.txt <==
#Chain X
#Atom ids bond angle(radii)
5 1 2 1.90268
5 1 3 1.90217
==> ala5_dihedral_angles.txt <==
#Chain X
#Atom ids dihedral angle(radii)
2 1 5 7 0.979966
Converting internal coordinates into Cartesian coordinates: all coordinates provided. A simple case is that were all IC are provided. In that case, one needs to provide
• A pdb file which is used to learn the topology of the molecule. (Nb: the Cartesian coordinates are of no interest.)
• Three files respectively providing the bond length, valence angles, and dihedral angles:
sbl-cartesian-internal-converter.exe --pdb-file ala5.pdb --bond-lengths-file ala5_bonds.txt --bond-angles-file ala5_bond_angles.txt --dihedral-angles-file ala5_dihedral_angles.txt
This call simply generates a txt file containing the Cartesian coordinates – ala5_cartesian.txt in the previous example.
Converting internal coordinates into Cartesian coordinates: selected coordinates missing. In case selected IC are missing, a force field can be passed so as to use the equilibrium values of the
corresponding models. For example, using
sbl-cartesian-internal-converter.exe --pdb-file ala5.pdb --bond-angles-file ala5_bond_angles.txt --dihedral-angles-file ala5_dihedral_angles.txt --force-field-file data/amber-ff14sb.xml
Conversion xtc – point_d format
The executable sbl-coordinates-file-converter.exe convertes xtc files to the Point_d format. This conversion is useful by several executables from the SBL – see e.g. the package Landscape_explorer . | {"url":"https://sbl.inria.fr/doc/Molecular_coordinates-user-manual.html","timestamp":"2024-11-13T17:34:58Z","content_type":"application/xhtml+xml","content_length":"50337","record_id":"<urn:uuid:bc0c91a6-9e15-4c86-a2a5-f35e7417f321>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00870.warc.gz"} |
Problem statement:
The alternative subgraphs assembly line balancing problem (ASALBP) is an extension of SALBP. It additionally considers that there might be different technological modes for mounting parts. This leads
to a balancing problem problem where it is additionally required to select one of several mounting modes, i.e., one of alternative subgraphs (of the precedence graph), for certain parts or group of
parts. The respective subgraphs differ by processing times and/or precedence relations. Therefore, solving this problem implies simultaneously selecting an assembly subgraph for each part of the
assembly which has alternatives and balancing the line (i.e., assigning the task to the workstations).
The following figure (taken from Scholl et al. 2009) visualizes the mode selection problem. The precedence graph contains two alternative parts. Between the pair of „or-nodes“ 2 und 10, one of the
three alternative subgraphs has to be chosen. Between the „or-nodes“ 12 and 17, one of both alternative subgraphs is to be realized. Thus, in total, 3*2=6 ways of assembling the product are
Data sets for ASALBP-1: For the problem version 1 (minimize the number of stations given the cycle time), there are two different data sets available. | {"url":"https://assembly-line-balancing.de/asalbp/","timestamp":"2024-11-03T07:25:42Z","content_type":"text/html","content_length":"41408","record_id":"<urn:uuid:bfb165d2-23bc-478e-b8f0-e7c146127c99>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00517.warc.gz"} |
Where are the vertical asymptotes of f(x) = tan x? | Socratic
Where are the vertical asymptotes of #f(x) = tan x#?
1 Answer
The vertical asymptotes of a function are usually located in points, where the function is undefined. In this case since $\tan x = \sin \frac{x}{\cos} x$, the asymptotes are located where $\cos x =
0$ (denominator of a fraction cannot be zero) which leads to the answer: $x = \frac{\pi}{2} + k \pi , x \in \mathbb{Z}$
Impact of this question
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Worst scoring flaw?
Also, creepy disco dancers aside, the 1978 revival is still the best NTT iteration.
...joined by any "both players write down titles simultaneously" game (like Sing-a-Tune) so the contestants got a chance to use different skills during the show.
I liked the version with the $100,000 Mystery Tune way better. Well orchestrated, suspenseful, and it was all season long.
Everyone may disagree, but I've always been bugged that in Melody Roulette, the wheel didn't affect the score. I'd rather they tallied the money at the end as the score so the wheel means something
for the game than best of five tunes with a shiny object. I liked Money Trees, too, even if it was tacky.
Writing the answers in Sing-a-Tune was awkward. Art Fleming showed you what his players wrote, but with Tom, you're taking his word for it. Maybe they could have built a wall a la Dotto, where we
could see what they're writing without them seeing each other. | {"url":"https://www.gameshowforum.org/index.php?topic=35839.45","timestamp":"2024-11-07T22:53:43Z","content_type":"application/xhtml+xml","content_length":"59844","record_id":"<urn:uuid:aaf84f7c-012b-4612-a140-3e0da439e356>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00262.warc.gz"} |
PPT - Double PowerPoint Presentation, free download - ID:9337305
Télécharger la présentation
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MainLine vs Crackatoa
I have already built the Mainline which sounds fantastic but a little not enough power.
Along with what PJ says, if you have a source with weak output voltage like a phone (usually around 300mV) and you turn the Crack or the Crack-a-two-a all the way up, you're nowhere near approaching
the maximum output level. A 2V DAC should be more than enough to get the job done though, just be sure if you're using it with a computer that all of the level controls are set to maximum.
That DAC has a ton of output. What happens if you turn the Mainline all the way up?
The Susvara is one of the more demanding headphones on the market and doesn't sound its best from the Mainline IME. You'll also struggle pairing it with OTL amps like the Crackatwoa (and other brand
OTL amps) because of it's sub-150 ohm impedance. You're best off running the Susvara off a decent solid state amp or a tube amp designed for speakers (like the Kaiju) where you have more power
available at 60 ohms. The biggest issue with the Susvara is its low sensitivity meaning that it does require a lot of power relative to most headphones on the market. It will still do well with
anything producing around 500mW of clean power at 60 ohms, but I felt like it was stretching the limits on the Mainline. | {"url":"https://forum.bottlehead.com/index.php?topic=14324.0;prev_next=next","timestamp":"2024-11-10T04:35:38Z","content_type":"application/xhtml+xml","content_length":"49134","record_id":"<urn:uuid:ced58c1e-8e57-49e1-9737-30b20090dee5>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00158.warc.gz"} |
The area of the trapezoid with the middle line 12 and height 15 equal to 180
A trapezoid is a quadrilateral whose two sides are parallel (bases) and the other two are not parallel (sides).
Trapezoid area formula
S = h·m
Answer: the area of the trapezoid with the middle line 12 and height 15 equal to 180 | {"url":"https://binary2hex.com/trapezoid-area.html?id=45","timestamp":"2024-11-06T22:05:40Z","content_type":"text/html","content_length":"45356","record_id":"<urn:uuid:c7915356-76de-4594-b84f-0a41513a4bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00765.warc.gz"} |
TSys class¶
Deterministic Transition System¶
Represents a transition system [Principles of Model Checking, Def. 2.1]. $$ TSys = (S, A, T, AP, L) $$ In the TSys class, each component is represented as a function.
• The set of states $S$ is represented by TSys.states function,
• The set of actions $A$ is represented by TSys.actions function,
• The transition function $T$ is represented by TSys.delta function,
• The set of atomic propositions is represented by TSys.atoms function,
• The labeling function $L$ is represented by TSys.label function.
All of the above functions are marked abstract. The recommended way to use TSys class is by subclassing it and implementing its component functions.
The example shows an example of defining a deterministic transition system.
Car Parking Transition System¶
Consider a car parking with $n$ slots where cars can enter or exit. There are two properties of interest, namely whether the parking lot is empty or full.
The CarParking class demonstrates how a parameterized transition system can be defined.
In [36]:
# This code block is necessary only when using `ggsolver:v0.1` docker image.
import sys
from examples.jupyter_patch import *
In [37]:
import logging
logger = logging.getLogger()
A deterministic transition system is defined as follows:
• First we define a class that inherits from ggsolver.models.TSys class. We will call it CarParking.
• Define its __init__ method to define instance variables that store input parameters. In our case, the only input parameter is the number of slots in parking lot: num_slots.
• Implement the component functions states, actions, delta, atoms, label that define the transition system.
Note: If `__init__` method is defined for your derived class, then a call to `super(DerivedTSys, self).__init__()` must be made.
In [38]:
from ggsolver.models import TSys
class CarParking(TSys):
def __init__(self, num_slots):
super(CarParking, self).__init__()
self.num_slots = num_slots
def states(self):
To determine whether parking is full or empty, we can maintain a count of the number of cars
in the parking lot. Hence, the states can be represented by integers from 0 to num_slots.
return list(range(self.num_slots + 1))
def actions(self):
There are two actions: enter and exit, that represent a car has entered or exited.
return ["enter", "exit"]
def delta(self, state, act):
The transition function should determine the next state, i.e. how many cars will be there in the parking lot, based on how many cars are currently in the parking lot and the action: whether a car is entering or exiting.
if state == 0 and act == "exit":
return 0
elif state == self.num_slots and act == "enter":
return self.num_slots
elif act == "enter":
return state + 1
else: # if act == "exit":
return state - 1
def atoms(self):
We care about two properties: whether the parking lot is "empty" or it is "full".
Hence, we will define two atomic propositions.
return ["empty", "full"]
def label(self, state):
The parking lot is empty if count is 0, and it is full when count is equal to num_slots.
if state == 0:
return ["empty"]
elif state == self.num_slots:
return ["full"]
return []
Note that our class definition closely follows the mathematical definition of a transition system. This provides an easy interface to implement various real-life or research paper based transition
sysetsm models directly.
Moreover, the ability to define parameters such as num_slots allows us to define a family of parameterized transition systems. This is particularly useful when running batch simulations.
However, the above style of defining transition system is inefficient. Especially, when running planning algorithms on large transition system, a large number of calls are made to states and delta
function. Since there is non-trivial computation occurring in these functions, it slows the algorithms.
In such cases, it is efficient to use discrete graph representation of the transition system. We provide an easy interface to construct and visualize the equivalent graph representation of the
defined transition system. This involves three steps:
• Instantiate your derived transition system class.
• Graphify the instance.
• Save to PNG the graph to a PNG for visualization.
Note: Graphs with more than 500 nodes cannot be saved to PNG. At present the best way to visualize such graphs is to serialize them and manually check them.
We demonstrate these steps next.
Instantiate CarParking¶
This is same as instantiating any class in Python.
In [39]:
tsys = CarParking(num_slots=5)
In case the initial state of the transition system is known, it can be set by calling initialize function as shown below. In our case, assume that parking lot starts empty.
The initial state of the transition system can be checked using init_state property.
In [40]:
print("init_state:", tsys.init_state())
The graphify function returns a ggsolver.graph.Graph object that represents a multi-digraph. See Graph Class API and Graph Class Example for more information about Graph class.
Note: If any changes are made to the transition system after a call to graphify, the changes will not reflect in the graph. The graphify must be called again.
[SUCCESS] <Graph with |V|=6, |E|=12> generated.
Visualize using to_png function¶
A graph with less than 500 nodes can be visualized using to_png function. The to_png function requires one positional argument:
• fpath: Path to where the generated PNG should be stored.
In [42]:
# Define path where to save the generated PNG.
fpath = "out/car_parking_nolabels.png"
# Generate a PNG
# Show PNG in Jupyter Notebook
html = img2html(fpath)
As you may notice, the generated PNG has the structure as expected. However, it would be nice to be visualize what atomic propositions hold in which state and which edges correspond to which actions.
For this purpose, we pass two optional arguments to to_png function.
In [43]:
# Define path where to save the generated PNG.
fpath = "out/car_parking_labeled.png"
# Generate a PNG
graph.to_png(fpath, nlabel=["state", "label"], elabel=["input"])
# Show PNG in Jupyter Notebook
html = img2html(fpath)
The graph of a transition system can be serialized into a dictionary. This allows us to save it or share it easily over a communication channel.
See Graph.Serialize() API Documentation to understand the components of generated dictionary.
{'graph': {'nodes': 6,
'edges': {0: {1: 1, 0: 1},
1: {2: 1, 0: 1},
2: {3: 1, 1: 1},
3: {4: 1, 2: 1},
4: {5: 1, 3: 1},
5: {5: 1, 4: 1}},
'node_properties': {'state': {'default': None,
'dict': {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}},
'label': {'default': None,
'dict': {0: ['empty'], 1: [], 2: [], 3: [], 4: [], 5: ['full']}}},
'edge_properties': {'prob': {'default': None, 'dict': []},
'input': {'default': None,
'dict': [{'edge': (0, 1, 0), 'pvalue': 'enter'},
{'edge': (0, 0, 0), 'pvalue': 'exit'},
{'edge': (1, 2, 0), 'pvalue': 'enter'},
{'edge': (1, 0, 0), 'pvalue': 'exit'},
{'edge': (2, 3, 0), 'pvalue': 'enter'},
{'edge': (2, 1, 0), 'pvalue': 'exit'},
{'edge': (3, 4, 0), 'pvalue': 'enter'},
{'edge': (3, 2, 0), 'pvalue': 'exit'},
{'edge': (4, 5, 0), 'pvalue': 'enter'},
{'edge': (4, 3, 0), 'pvalue': 'exit'},
{'edge': (5, 5, 0), 'pvalue': 'enter'},
{'edge': (5, 4, 0), 'pvalue': 'exit'}]}},
'graph_properties': {'inp_domain': ['enter', 'exit'],
'is_probabilistic': False,
'actions': ['enter', 'exit'],
'is_deterministic': True,
'init_state': 0,
'atoms': ['empty', 'full']}}} | {"url":"https://akulkarni.me/docs/ggsolver/examples/models_tsys.html","timestamp":"2024-11-08T00:52:21Z","content_type":"text/html","content_length":"690300","record_id":"<urn:uuid:4695aadf-b7ce-408d-aca9-e6de566f20a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00650.warc.gz"} |
Science:Math Exam Resources/Courses/MATH103/April 2013/Question 02 (d)
MATH103 April 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) i • Q3 (b) ii • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7
(b) • Q7 (c) • Q7 (d) • Q7 (e) • Q8 (a) • Q8 (b) • Q8 (c) • Q9 • Q10 •
Question 02 (d)
Integration: Short Answer Problems - Evaluate the following integrals: state if a definite integral does not exist; use limits for improper integrals. (full marks for correct answer; work must be
shown for partial marks)
${\displaystyle \displaystyle I_{d}=\int \sin(\ln x)\,dx}$
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
Start with the substitution u = ln x.
Hint 2
Next, do integration by parts twice to solve for the integral you wound up with after the substitution.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
The integral will become less complicated if we make a substitution ${\displaystyle u=\ln x}$.
First substitution
If u = ln x then ${\displaystyle du=(1/x)\ dx}$ so ${\displaystyle dx=x\ du=e^{u}\ du}$. For the last equality we solved ${\displaystyle u=\ln x}$ for ${\displaystyle x}$. Therefore,
${\displaystyle I=\int \sin(\ln x)\ dx=\int e^{u}\sin u\ du}$
First integration by parts
To find this integral, we will need to integrate by parts twice.
Let ${\displaystyle f=e^{u}}$ with ${\displaystyle df=e^{u}\ du}$ and ${\displaystyle dg=\sin u\ du}$ with ${\displaystyle g=-\cos u}$. Then
{\displaystyle {\begin{aligned}I&=fg-\int gdf=-e^{u}\cos u-\int e^{u}(-\cos u)\ du\\&=-e^{u}\cos u+\int e^{u}\cos u\ du\end{aligned}}}
Second integration by parts
We repeat the process on the second integral, this time with ${\displaystyle f=e^{u}}$ and ${\displaystyle df=e^{u}\ du}$, and ${\displaystyle dg=\cos u\ du}$ with ${\displaystyle g=\sin u}$. Now we
${\displaystyle I=-e^{u}\cos u+(e^{u}\sin u-\int e^{u}\sin u\ du)}$
Inspect the result
We recognize the integral above as I! We can now solve for I:
${\displaystyle \displaystyle I=-e^{u}\cos u+e^{u}\sin u-I}$
${\displaystyle \displaystyle 2I=e^{u}\sin u-e^{u}\cos u}$
${\displaystyle \displaystyle I={\frac {1}{2}}e^{u}(\sin u-\cos u)+C}$
We added the arbitrary constant C because we are computing an indefinite integral.
Bring x back
Finally, the integral we desire can be found by replacing ${\displaystyle u}$ by ${\displaystyle \ln x}$ so that ${\displaystyle e^{u}=x}$:
${\displaystyle \displaystyle I={\frac {1}{2}}x(\sin(\ln x)-\cos(\ln x))+C}$
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Integration by parts, MER Tag Substitution, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag | {"url":"https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH103/April_2013/Question_02_(d)","timestamp":"2024-11-12T04:01:54Z","content_type":"text/html","content_length":"68554","record_id":"<urn:uuid:4b9d214a-ae1b-47f6-b72b-9e2fce66d6ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00337.warc.gz"} |
Euclidean Calculation Crossword - Temz Calculators
Euclidean Calculation Crossword
The Euclidean calculation is a fundamental tool in mathematics and various fields of science. It is used to determine the shortest distance between two points in Euclidean space. The Euclidean
distance is calculated using the Pythagorean theorem and provides the straight-line distance between two points.
Euclidean Calculation Formula
The Euclidean distance between two points (x1, y1) and (x2, y2) in a 2D plane is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
• Distance is the Euclidean distance between the two points.
• (x1, y1) and (x2, y2) are the coordinates of the two points.
To calculate the Euclidean distance, subtract the x-coordinates and y-coordinates of the two points, square the differences, sum them, and then take the square root of the result.
What is Euclidean Calculation?
Euclidean calculation refers to the process of determining the Euclidean distance between two points in Euclidean space. This calculation is essential in various applications such as geometry,
physics, and computer science. Understanding Euclidean calculation is crucial for solving problems involving distances in a plane or in higher-dimensional spaces.
How to Calculate Euclidean Distance?
The following steps outline how to calculate the Euclidean distance using the given formula.
1. First, determine the coordinates of the two points (x1, y1) and (x2, y2).
2. Next, subtract the x-coordinates (x2 - x1) and y-coordinates (y2 - y1) of the two points.
3. Square the differences of the coordinates.
4. Sum the squared differences.
5. Take the square root of the sum to find the distance.
6. After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Point 1 = (3, 4)
Point 2 = (7, 1)
Calculate the Euclidean distance using the formula provided above.
1. What is Euclidean distance?
Euclidean distance is the straight-line distance between two points in Euclidean space, calculated using the Pythagorean theorem.
2. How is Euclidean distance used?
Euclidean distance is used in various fields such as geometry, physics, computer science, and data analysis to measure the distance between points.
3. Can the calculator handle multiple points?
Yes, the advanced calculator can handle multiple points to calculate the total Euclidean distance between a series of points.
4. How accurate is the Euclidean distance calculator?
The calculator provides accurate results based on the inputs provided. For precise measurements, ensure correct and accurate data entry.
5. Is the Euclidean distance formula applicable in 3D space?
Yes, the formula can be extended to 3D space by including the z-coordinates of the points: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2). | {"url":"https://temz.net/euclidean-calculation-crossword","timestamp":"2024-11-07T10:04:02Z","content_type":"text/html","content_length":"73698","record_id":"<urn:uuid:07039ac1-0283-4f43-a939-35f1f4af6f72>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00621.warc.gz"} |
Uniform Electric Field (8 of 9) Voltage Needed Across the Plates | Video Summary and Q&A | Glasp
Uniform Electric Field (8 of 9) Voltage Needed Across the Plates | Summary and Q&A
Uniform Electric Field (8 of 9) Voltage Needed Across the Plates
Calculate the voltage across parallel plates using electric field strength and plate separation.
Key Insights
• 😥 Parallel plates have a uniform electric field, unlike point charges.
• 🍽️ The equation for calculating electric field strength in parallel plates involves potential difference and plate separation.
• 🍽️ To determine the potential difference, multiply the electric field strength by the plate separation distance.
• 🍽️ Understanding the concept and calculation of voltage across parallel plates is crucial in physics.
• 🍽️ The relationship between electric field strength, potential difference, and plate separation is essential in solving problems.
• 🍽️ Accomplishing the specified electric field strength between parallel plates requires calculating the appropriate voltage.
• 🚠 Being able to manipulate the formula correctly helps in finding the desired potential difference value.
okay in today's video we are going to go over number four problem number four for our potential difference problems that we are using to help you get a better understanding about potential difference
in electric potential energy all right and today's problem we are going to try to figure out what voltage we want to put across our parallel plates if... Read More
Questions & Answers
Q: How does the electric field between parallel plates differ from point charges?
The electric field between parallel plates is uniform, unlike the varying electric field around point charges. It remains consistent between the plates as long as you are near the middle.
Q: What is the equation used to calculate electric field strength for parallel plates?
The equation relates electric field strength (E), potential difference (ΔV), and plate separation (d). It can only be applied to parallel plates due to the uniform field.
Q: How is the formula rearranged to determine the potential difference across the plates?
By rearranging the equation, ΔV = E * d, where E is the electric field strength in volts per meter and d is the distance between the plates converted to meters.
Q: In the given example, what voltage is needed across plates 15mm apart for an electric field of 750 volts per meter?
By substituting the values into the formula, the potential difference required is calculated as 11.25 volts to achieve the specified electric field strength.
Summary & Key Takeaways
• Explaining potential difference in parallel plates with uniform electric field.
• Using the equation relating electric field strength, potential difference, and plate separation.
• Calculation example to determine the voltage required across the plates.
Explore More Summaries from Step by Step Science 📚 | {"url":"https://glasp.co/youtube/p/uniform-electric-field-8-of-9-voltage-needed-across-the-plates","timestamp":"2024-11-04T02:30:10Z","content_type":"text/html","content_length":"357386","record_id":"<urn:uuid:2f19207f-7b5e-4d14-83c0-f3e55970f04c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00698.warc.gz"} |
Asymptotic Behavior
L to R: horizontal, vertical and oblique asymptotes.
Asymptotic behavior is a mathematical concept that describes how a function behaves as the input (or independent variable) approaches infinity. In other words, asymptotic behavior is concerned with
the long-term behavior of a function as the input values get larger and larger.
Asymptotic behavior can be used to predict the behavior of a function in the limit, as well as to simplify complex functions so that they are easier to work with. Asymptotic behavior is a useful tool
for mathematicians, engineers, and scientists who need to understand how functions behave at the limit.
Types of Asymptotic Behavior
There are two main types of asymptotic behavior: vertical asymptotes and horizontal asymptotes. Vertical asymptotes occur when the function approaches infinity or negative infinity, while horizontal
asymptotes occur when the function approaches a finite number. In both cases, the asymptotic behavior of the function can be determined by analyzing its graph.
Asymptotic Expansion
An asymptotic expansion is a representation of a function as a sum of terms that becomes exact in the limit as some variable goes to infinity. Typically, the variable is a quantity called the
asymptotic parameter, which measures the distance from some starting point. For example, consider the function f(x) = 1/x. If we take x to be very large, then f(x) will be very small. We can
therefore write f(x) as an asymptotic expansion in powers of 1/x:
f(x) ≈ 1 + 1/2x + 1/3x + …
In this case, each term in the expansion provides a better approximation to f(x) as x gets larger and larger. In general, asymptotic expansions are useful in situations where it is difficult to
compute a function exactly, but we can still get some idea of its behavior by looking at its leading terms.
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Substituting variable/function
Substituting variable/function
asked 2017-03-19 16:07:00 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
I got a problem with substituting a solution into another equation. I could break down my problem to a simple example:
r = var('r');
p = function('p')(r);
v = var('v');
v = p + p.derivative(r);
p1(r)=desolve(p.derivative(r)==0,p,ivar=r); p1(r)
_C + diff(p(r), r)
What's happening here? Why is p substituted, but not diff(p,r)=0? What's the correct way to do this substitution?
2 Answers
Sort by ยป oldest newest most voted
You should use the method substitute_function instead of subs and use a symbol different from p to denote p(r) (otherwise there is a confusion between the function p and its value at r). Besides the
semicolons at the end of each line are not necessary in Python and the line v = var('v') is useless, since the Python variable v is redeclared in the next line. So basically, your code should be:
sage: r = var('r')
sage: P = function('p')(r)
sage: v = P + P.derivative(r)
sage: p1(r) = desolve(P.derivative(r)==0, P, ivar=r); p1(r)
sage: v.substitute_function(p, p1)
edit flag offensive delete link more
Thank you! My lack of understanding was the difference between P and 'p'. This helped a lot
ceee ( 2017-03-20 12:17:13 +0100 )edit
Thank you for that explanation. I will try by myself your substiture_fonction method.
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Research Seminar “Modern Problems of Mathematical Physics” (Term 1-8) / DG120268 / 23-25
Instructor: Andrei Marshakov
Course “Modern problems of mathematical physics” is a student seminar, so participants are expected to give talks based on the modern research papers. Current topic of the seminar can vary from time
to time. Topics that were already covered, or can be covered in the future, are:
classical integrable equations, complex curves and theta-functions,
quantum integrable models (quantum-mechanical and field-theoretical),
models of statistical physics, stochastic integrability,
quantum/classical duality, supersymmetric gauge theories, cluster algebras etc.
Representations of Finite Groups (Term 1-2) / MA060595 / 23-25
Instructor: Grigori Olshanski
Representation theory is used in many areas of mathematics (algebra, topology, algebraic groups, Lie groups and Lie algebras, quantum groups, algebraic number theory, combinatorics, probability
theory, …), as well as in mathematical physics. Therefore, mastering the basic technique of representation theory is necessary for mathematicians of various specialties. The aim of the course is to
give an introduction to representation theory on the material of finite groups. Particular attention will be paid to representations of the symmetric groups.
Tentative program:
1. Reminder of the basics from the algebra course: group algebra of a finite group, irreducible representations, Schur’s lemma, characters, orthogonality relations, Maschke’s theorem, Burnside’s
2. Representations of finite Abelian groups, duality for finite Abelian groups, Fourier transform, biregular representation
3. Intertwining operators, induced representations, Frobenius duality
4. Mackey machine, projective representations, coverings over symmetric groups
5. Functional equation for characters, Gelfand pairs, spherical functions, connection with orthogonal polynomials
6. Representations of the symmetric group: various approaches to the classification and construction of irreducible representations
7. If time permits: principal series representations for the group GL(N) over a finite field, Hecke algebra, Harish-Chandra theory
Cluster Varieties and Integrable Systems (Term 1-2) / MA060597 / 23-25
Instructor: Andrei Marshakov
Geometric Representation Theory (Term 1-2) / DA060271 / 23-25
Instructors: Michael Finkelberg
Geometric representation theory applies algebraic geometry to the problems of representation theory. Some of the most famous problems of representation theory were solved on this way during the last
40 years. The list includes the Langlands reciprocity for the general linear groups over the functional fields, the Langlands-Shelstad fundamental Lemma, the proof of the Kazhdan-Lusztig conjectures;
the computation of the characters of the finite groups of Lie type. We will study representations of the affine Hecke algebras using the geometry of affine Grassmannians (Satake isomorphism) and
Steinberg varieties of triples (Deligne-Langlands conjecture). This is a course for master students knowing the basics of algebraic geometry, sheaf theory, homology and K-theory.
Full Syllabus
Differential Topology (Term 1-2) / MA060599 / 23-25
Instructor: Alexander Gaifullin
The course will cover two topics, which are central in topology of smooth manifolds, the de Rham cohomology theory and Morse theory. The course will culminate in two famous results of differential
topology: Smale’s h-cobordism theorem and Milnor’s discovery of exotic smooth structures on the 7-dimensional sphere. The h-cobordism theorem proved by S. Smale in 1962 is the main (and almost the
only) tool for proving that two smooth manifolds (of dimension greater than or equal to 5) are diffeomorphic. In particular, this theorem implies the high-dimensional Poincare conjecture (for
manifolds of dimensions 5 and higher). Milnor’s discovery of exotic smooth structures on the 7-dimensional sphere and further results of Kervaire and Milnor were the first steps towards surgery
theory, which is the most powerful tool for classifying smooth manifolds.
= = Homology and cohomology //
De Rham cohomology. Singular homology. Pairing between homology and cohomology. Multiplication in cohomology and intersection of cycles. Poincare duality
= = Morse theory //
Morse functions. Cobordisms corresponding to critical points. Morse inequalities. Lefschetz theorem on hyperplane sections. Smale’s h-cobordism theorem. High-dimensional Poincare conjecture
= = Characteristic classes //
Principal bundles and their characteristic classes. Chern-Weil theory. Chern classes and Pontryagin classes. Integral Chern classes and Pontryagin classes. Smooth structures on the 7-dimensional
Phase Transitions, Rigorous (Term 3-4) / MA060600 / 23-25
Instructor: Semen Shlosman
This is a course on rigorous results in statistical mechanics, random fields and percolation theory. Some of it will be dedicated to the theory of phase transitions, uniqueness or non-uniqueness of
the lattice Gibbs fields. We will also study the models at the criticality, where one hopes to find (in dimension 2) the onset of conformal invariance. We will see that it is indeed the case for the
percolation (and the Ising model, if time permits).
The topics will include:
= Crossing probabilities as a characteristic of sub-, super- and at- criticality.
= Critical percolation and its power-law behavior.
= The Russo-Seymour-Welsh theory of crossing probabilities – a cornerstone of critical percolation
= Cardy’s formula for crossing probabilities
= Parafermionic observables and S. Smirnov theory
= Conformal invariance of two-dimensional percolation a la Khristoforov.
= Ising model in 1D, 2D and 3D.
= Ising model on Cayley trees.
= Conformal invariance of two-dimensional Ising model
= O(N)-symmetric models
= Continuous symmetry in 2D systems: The Mermin–Wagner Theorem and the absence of Goldstone bosons.
= The Berezinskii–Kosterlitz–Thouless transition
= Reflection Positivity and the chessboard estimates in statistical mechanics
= Infrared bounds and breaking of continuous symmetry in 3D
Introduction to quantum field theory (Term 3-4) / MA060505 / 23-25
Instructors: Vladimir Losyakov, Petr Dunin-Barkowski
As you know, the modern theory of fundamental physics (the “standard model of elementary particle physics”) is a quantum field theory (QFT). In addition to this central role in modern physics,
quantum field theory also has many applications in pure mathematics (for example, from it came the so-called quantum knot invariants and Gromov-Witten invariants of symplectic manifolds).
“Ordinary” quantum mechanics deals with systems with a fixed number of particles. In QFP, the objects of study are fields (not in the sense of a “field of complex numbers”, but in the sense of an
“electromagnetic field”), whose elementary perturbations are analogs of quantum mechanical particles, but can appear and disappear (“born” and “die”); at the same time, the number of degrees of
freedom turns out to be infinite.
Within the framework of this course, the basic concepts of QFT will be introduced “from scratch”. The Fock space and the formalism of operators on it, as well as the formalism of the “continuum
integral” will be defined. The main example under consideration will be the quantum scalar field theory. A scalar field in physical terminology is a field that, at the classical level, is defined by
one number at each point (i.e., in fact, its state at a given time is just a numerical function on space), unlike a vector field (an example of which, in particular, is an electromagnetic field).
However, considering the quantum theory of a scalar field (even separately, and simpler than for the Higgs field) is in any case very useful, since it allows you to get acquainted with the apparatus
and phenomena of QFT on a simpler example than vector and spinor fields. The course will consider the “perturbation theory” (that is, in fact, a method for calculating the first orders of smallness
in a small parameter expansion) for a scalar field and describe ways to calculate various probabilities of events with particles.
Full Syllabus
Symplectic Geometry (Term 1-2) / MA060596 / 23-25
Instructor: Maxim Kazarian
Introduction to Quantum Groups (Term 1-2) / MA060426 / 23-25
Modern Dynamical Systems (Term 3-4) / MA060257 / 23-25
Instructors: Aleksandra Skripchenko, Sergei Lando
Dynamical systems in our course will be presented mainly not as an independent branch of mathematics but as a very powerful tool that can be applied in geometry, topology, probability, analysis,
number theory and physics. We consciously decided to sacrifice some classical chapters of ergodic theory and to introduce the most important dynamical notions and ideas in the geometric and
topological context already intuitively familiar to our audience. As a compensation, we will show applications of dynamics to important problems in other mathematical disciplines. We hope to arrive
at the end of the course to the most recent advances in dynamics and geometry and to present (at least informally) some of results of A. Avila, A. Eskin, M. Kontsevich, M. Mirzakhani, G. Margulis.
In accordance with this strategy, the course comprises several blocks closely related to each other. The first three of them (including very short introduction) are mainly mandatory. The decision,
which of the topics listed below these three blocks would depend on the background and interests of the audience.
Full Syllabus
Some Uses of Twistors in Field Theory (Term 3-4) / MA060601 / 23-25
Instructor: Alexei Rosly
The subject of this course will be mainly the complex geometry. The choice of topics, however, is determined by their uses in Field Theory and Theory of Integrable Systems
Integrable Many-Body Systems and Nonlinear Equations (Term 3-4) / MA060602 / 23-25
Instructor: Anton Zabrodin
This course is devoted to many-body integrable systems of classical mechanics such as Calogero-Moser, Ruijsenaars-Schneider and their spin generalizations. These systems play a significant
role in modern mathematical physics. They are interesting and meaningful from both mathematical and physical points of view and have important applications and deep connections with different
problems in mathematics and physics. The history of integrable many-body systems starts in 1971 from the famous Calogero-Moser model which exists in rational, trigonometric or hyperbolic and (most
general) elliptic versions. Later it was discovered that there exists a one-parametric deformation of the Calogero-Moser system preserving integrability, often referred to as relativistic extension.
This model is now called the Ruijsenaars-Schneider system. In its most general version the interaction
between particles is described by elliptic functions.
The integrable many-body systems of Calogero-Moser type have intimate connection with nonlinear integrable equations such as
Korteveg-de Vries and Kadomtsev-Petviashvili (KP) equations. Namely, they describe dynamics of poles of singular solutions (in general, elliptic solutions) to the nonlinear integrable partial
differential equations. The Ruijsenaars-Schneider system plays the same role for singular solutions to the Toda lattice equation.
In this course the algebraic structure of the integrable many-body systems will be presented. The Lax representation and interrals of motion will be obtained using the corerspondence with the
equations of the KP type. The necessary material about the latter will be given in the course. The construction of the spectral curves will be discussed.
Path integral: stochastic processes and basics of quantum mechanics (Term 1-2) / MA060542 / 23-25
Instructor: Andrei Semenov
One of the most powerful methods of modern theoretical physics is the method of functional integration or path integration. The foundations of this approach were developed by N. Wiener at the
beginning of the 20th century, but it spread widely after R. Feynman, who applied this approach in quantum mechanics. At present, the functional integral has found its application in the theory of
random processes, polymer physics, quantum and statistical mechanics, and even in financial mathematics. Despite the fact that in some cases its applicability has not yet been mathematically rigorous
proven, this method makes it possible to obtain exact and approximate solutions of various interesting problems with surprising elegance. The course is devoted to the basics of this approach and its
applications to the theory of random processes and quantum mechanics. In the first part of the course, using the example of stochastic differential equations, the main ideas of this approach will be
described, as well as various methods for exact and approximate calculation of functional integrals. Further, within the framework of the course, the main ideas of quantum mechanics will be
considered, and both the operator approach and the approach using functional integration will be considered. It will be demonstrated that, from the point of view of formalism, the description of
random processes and the description of quantum mechanical systems are very similar. This will make it possible to make a number of interesting observations, such as, for example, the analogy between
supersymmetric quantum mechanics and the diffusion of a particle in an external potential. In the final part of the course, depending on the interests of the audience, various applications of the
functional integration method will be discussed, such as polymer physics, financial mathematics, etc.
Full Syllabus | {"url":"https://crei.skoltech.ru/cas/education/mscprog/curriqulum-2023-24/","timestamp":"2024-11-05T19:18:55Z","content_type":"text/html","content_length":"83843","record_id":"<urn:uuid:9bc8ab8b-b965-46e8-a796-c672649658ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00758.warc.gz"} |
z test formula
In the realm of statistics, the z test stands as a fundamental tool for making inferences about population parameters. It’s a statistical method used to determine whether there’s a significant
difference between a sample mean and a population mean when the population standard deviation is known. In this article, we’ll delve into the intricacies of the z test formula, its applications, and
how it’s calculated.
Introduction to z test
The z test is a hypothesis test that helps researchers assess whether the mean of a sample differs significantly from the mean of a population. It’s especially helpful while managing huge example
sizes and when the populace standard deviation is known.
Understanding the z test formula
The z test formula is relatively straightforward and is based on the standard normal distribution. It can be expressed as:
• �ˉ
• x
• ˉ
• is the sample mean,
• �
• μ is the population mean,
• �
• σ is the population standard deviation, and
• �
• n is the sample size.
Applications of the z test
The z test finds applications in various fields, including:
• Quality control in manufacturing
• Market research
• Clinical trials
• Educational assessment
• Finance and economics
Steps to calculate z score
To calculate the z score using the z test formula, follow these steps:
• Determine the sample mean (
• �ˉ
• x
• ˉ
• ).
• Identify the population mean (
• �
• μ).
• Determine the population standard deviation (
• �
• σ).
• Determine the sample size (
• �
• n).
• Plug the values into the z test formula and calculate the z score.
Interpretation of z score
The z score indicates the number of standard deviations a data point is from the mean. A positive z score indicates that the data point is above the mean, while a negative z score indicates that it’s
below the mean. The farther the z score is from zero, the more significant the deviation from the mean.
Importance of z test in statistics
The z test plays a crucial role in inferential statistics by providing a systematic way to test hypotheses and make inferences about population parameters based on sample data.
Advantages of using z test
Some advantages of the z test include:
• Suitable for large sample sizes
• Utilizes known population parameters
• Provides a standardized measure of deviation
Limitations of the z test
Despite its usefulness, the z test has limitations, including:
• Requires knowledge of population standard deviation
• Less robust with small sample sizes
• Assumes normality of data distribution
Practical examples of z test application
Let’s consider a practical example where the z test can be applied:
Suppose a pharmaceutical company wants to test the effectiveness of a new drug in lowering cholesterol levels. They conduct a clinical trial with a large sample size and compare the mean cholesterol
levels of the treatment group with the population mean.
Comparing z test with other hypothesis tests
The z test is just one of many hypothesis tests available in statistics. Comparing it with tests like the t-test or chi-square test can provide insights into when each test is most appropriate based
on the data characteristics and assumptions.
Tips for conducting a successful z test
To ensure the validity and reliability of the results obtained from a z test, consider the following tips:
• Ensure the sample is representative of the population.
• Confirm the suspicions of ordinariness and homogeneity of fluctuation.
• Use appropriate significance levels and confidence intervals.
Real-world scenarios where z test is used
The z test finds application in various real-world scenarios, including:
• A/B testing in marketing
• Quality control in manufacturing
• Educational research
• Financial analysis
Considerations when applying the z test formula
When applying the z test formula, it’s essential to consider factors such as the size of the sample, the variability of the population, and the assumptions underlying the test to draw accurate
Common misconceptions about the z test
One common misconception about the z test is that it requires a large sample size to be valid. While the z test is robust with large sample sizes, it can still be used effectively with smaller
samples under certain conditions.
In conclusion, the z test formula is a powerful statistical tool that enables researchers to make informed decisions based on sample data. By understanding its applications, advantages, limitations,
and practical considerations, statisticians and researchers can leverage the z test to draw meaningful conclusions and drive evidence-based decision-making.
1. Can the z test be used with small sample sizes?
While the z test is more robust with large sample sizes, it can still be used with small samples if certain conditions are met, such as the data being approximately normally distributed.
2. What is the significance level in a z test?
The significance level (alpha) in a z test formula represents the probability of rejecting the null hypothesis when it is true. Normal importance levels incorporate 0.05 and 0.01.
3. Is the z test the same as the t-test?
No, the z test and t-test are different statistical tests. The z test formula is utilized when the populace standard deviation is known, while the t-test is utilized when it is obscure and should be
assessed from the example.
4. What does a negative z score indicate?
A negative z score indicates that the data point is below the mean of the distribution.
5. How do you interpret the p-value in a z test?
The p-value in a z test represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. | {"url":"https://nexgennsphere.com/2024/04/06/z-test-formula/","timestamp":"2024-11-09T06:45:00Z","content_type":"text/html","content_length":"106205","record_id":"<urn:uuid:59dfec92-0b0a-4312-8141-6ed067c92173>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00532.warc.gz"} |
When is midnight? Nomenclature question (was Re: Guatemala DST?)
Markus Kuhn Markus.Kuhn at cl.cam.ac.uk
Sun Apr 23 09:04:03 UTC 2006
"Dave Cantor" wrote on 2006-04-22 22:34 UTC:
> On 22 Apr 2006 at 16:06, Oscar van Vlijmen wrote:
> > DST [for Guatemala] starts on Sunday, April 30, 2006 at 12:00 Midnight local
> > standard time DST ends on Sunday, October 1, 2006 at 12:00
> > Midnight local daylight time
> I am not commenting on the veracity of the cited material at all,
> and I quote Oscar only because the quotation serves as an
> example.
> I'm not even sure that people on this list are the right ones to
> complain to, but surely most everyone on this list, must have
> noticed the ambiguity.
> I am troubled by the specification of midnight on a certain date.
> In the old days (and by that I mean roughly before computers were
> commonly used by non-computer-geeks like us) to keep time,
> "midnight Tuesday" meant the minute after 11:59 p.m. Tuesday
> night. I think (but am not sure) that most people still mean
> that when they say "midnight Tuesday". Midnight _used_to_be_ a
> synonym for 2400 hrs., the end of the day.
> But we pretty much don't use 2400 hrs. any more, and "midnight"
> has become a synonym, in some contexts, for 0000 hrs., the start
> of the day. So "midnight Tuesday" might refer to the minute
> before 12:01 a.m. Tuesday morning (Monday night!).
> Is anyone else concerned? What should be done? Who should do it?
Government officials who use obsolete and ambiguous terms such as "12:00
midnight" should be gently pointed to the relevant official standards
for time notation, which have solved this problem adequately and
ISO 8601: 00:00 is midnight at the start of the given date
24:00 is midnight at the end of the given date
(= 00:00 of the next day)
All this is discussed in great detail, for example, at
Markus Kuhn, Computer Laboratory, University of Cambridge
http://www.cl.cam.ac.uk/~mgk25/ || CB3 0FD, Great Britain
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ands to Microns
Hands to Microns Converter
⇅ Switch toMicrons to Hands Converter
How to use this Hands to Microns Converter 🤔
Follow these steps to convert given length from the units of Hands to the units of Microns.
1. Enter the input Hands value in the text field.
2. The calculator converts the given Hands into Microns in realtime ⌚ using the conversion formula, and displays under the Microns label. You do not need to click any button. If the input changes,
Microns value is re-calculated, just like that.
3. You may copy the resulting Microns 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 Hands to Microns?
The formula to convert given length from Hands to Microns is:
Length[(Microns)] = Length[(Hands)] / 0.000009842519685
Substitute the given value of length in hands, i.e., Length[(Hands)] in the above formula and simplify the right-hand side value. The resulting value is the length in microns, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a horse is measured to be 16 hands tall.
Convert this height from hands to Microns.
The length in hands is:
Length[(Hands)] = 16
The formula to convert length from hands to microns is:
Length[(Microns)] = Length[(Hands)] / 0.000009842519685
Substitute given weight Length[(Hands)] = 16 in the above formula.
Length[(Microns)] = 16 / 0.000009842519685
Length[(Microns)] = 1625600
Final Answer:
Therefore, 16 hand is equal to 1625600 µ.
The length is 1625600 µ, in microns.
Consider that a racehorse stands at 15.5 hands.
Convert this measurement from hands to Microns.
The length in hands is:
Length[(Hands)] = 15.5
The formula to convert length from hands to microns is:
Length[(Microns)] = Length[(Hands)] / 0.000009842519685
Substitute given weight Length[(Hands)] = 15.5 in the above formula.
Length[(Microns)] = 15.5 / 0.000009842519685
Length[(Microns)] = 1574800
Final Answer:
Therefore, 15.5 hand is equal to 1574800 µ.
The length is 1574800 µ, in microns.
Hands to Microns Conversion Table
The following table gives some of the most used conversions from Hands to Microns.
Hands (hand) Microns (µ)
0 hand 0 µ
1 hand 101600 µ
2 hand 203200 µ
3 hand 304800 µ
4 hand 406400 µ
5 hand 508000 µ
6 hand 609600 µ
7 hand 711200 µ
8 hand 812800 µ
9 hand 914400 µ
10 hand 1016000 µ
20 hand 2032000 µ
50 hand 5080000 µ
100 hand 10160000 µ
1000 hand 101600000.0004 µ
10000 hand 1016000000.0041 µ
100000 hand 10160000000.0406 µ
A hand is a unit of length used primarily to measure the height of horses. One hand is equivalent to 4 inches or approximately 0.1016 meters.
The hand is defined as 4 inches, providing a standardized measurement for assessing horse height, ensuring consistency across various contexts and practices.
Hands are used in the equestrian industry to measure the height of horses, from the ground to the highest point of the withers. The unit offers a convenient and traditional method for expressing
horse height and remains in use in equestrian competitions and breed standards.
A micron, also known as a micrometer (µm), is a unit of length in the International System of Units (SI). One micron is equivalent to 0.000001 meters or approximately 0.00003937 inches.
The micron is defined as one-millionth of a meter, making it an extremely precise measurement for very small distances.
Microns are used worldwide to measure length and distance in various fields, including science, engineering, and manufacturing. They are especially important in fields that require precise
measurements, such as semiconductor fabrication, microscopy, and material science.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Hands to Microns in Length?
The formula to convert Hands to Microns in Length is:
Hands / 0.000009842519685
2. Is this tool free or paid?
This Length conversion tool, which converts Hands to Microns, is completely free to use.
3. How do I convert Length from Hands to Microns?
To convert Length from Hands to Microns, you can use the following formula:
Hands / 0.000009842519685
For example, if you have a value in Hands, you substitute that value in place of Hands in the above formula, and solve the mathematical expression to get the equivalent value in Microns. | {"url":"https://convertonline.org/unit/?convert=hands-microns","timestamp":"2024-11-14T02:01:24Z","content_type":"text/html","content_length":"90139","record_id":"<urn:uuid:41c76d8e-57e6-41e8-98b9-00d0580d47e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00871.warc.gz"} |
The Order of the Spiritual Master is Our Dharma
A devotee washes Srila Prabhupada's feet with milk, San Diego 1975. Photo by Višākhā dd. Image courtesy of © The Bhaktivedanta Book Trust International, Inc. www.Krishna.com
[Discussing Śrīmad Bhāgavatam 6.1.7]
HH Bhaktividyā Pūrṇa Svāmī Mahārāja: So this subject matter is a very interesting subject because the living entity, the conditioned living entity is always going to do something wrong. And when one
does something wrong then that is not proper. The proper... Human life is meant to elevate oneself, is actually to become God conscious - that is the purpose of this human form. So now, if the human
form can be engaged in the Lord's service, that is a proper perfection. But when we perform our activities, unless we have that inclination to serve the Lord, then according to conditioned nature we
are going to work for our own pleasure, the own pleasure of the mind and senses. Therefore then if one has that mentality, one still has to do so according to God's direction. If one wants to do
things, one has to follow God's direction, and that is the Dharma-śāstras. So they give what the Lord says is how a materialist should act, so that there will not be difficulty. And if they don't act
according to that, then as described in the previous Canto, then they go to the hellish regions where they have so many difficulties. So then the idea is, is the human form is not meant for hellish
life, it is not even meant for pious life, but it is meant for transcendental elevation. But the living entity, because of his conditioning, he does not want to engage himself properly in these
transcendental activities. He wants to engage in sense gratification because of the false identity with the body and mind. So then the point comes up here, is how does one save oneself from all these
hellish conditions?
So now what is happening, is Prabhupāda points out, is that Śukadeva Gosvāmī wants to test Parīkṣit Mahārāja to see how serious he is about actually understanding this point. Because it can be taken
in two levels, it could be taken: 1) How do I save myself from hellish conditions? But I establish myself in very nice pious conditions and I enjoy it because of this. Or, 2) how to get out of the
whole cycle of birth and death because whether pious or impious, it is still material, it still binds us to the material world. So which does he mean? When he is talking about becoming freed from
these things and liberated, what liberation is he talking about? Simply being freed from material disturbances, or does he mean being freed from the actual material condition?
So he first will explain, he is starting to explain about the Dharma-śāstras, that if you follow them, they give, 'Okay, if you have done this material thing wrong, you do this material activity, it
is pious, means you have done something wrong, it means it is sinful, impious. So the impious material activity is countered by a pious material activity, something that is beneficial.' Like that,
some kind of something that corrects. Now, why is it done in this way? Because the materialist only knows the body and mind, so therefore, if they do something wrong with the body and mind, you can
correct that by bringing in another element of the body and mind to deal with that. Then one can relate to this, and that's like the... One can understand this. Just like someone may not behave
properly, so you will find, he can understand someone else who doesn't behave properly because they can relate to that, they can understand the mood and motivations and all that. So the
Dharma-śāstras are meant for this purpose. We are engaged in material activities, but we don't know how to... We don't know anything beyond that. So we talk about something transcendental, there is
no interest in it, so one won't be able to apply, one won't be able to relate to it, what has it got to do with this? You know, it's just like I have a gross object and there is a problem with it, I
expect that there is a gross dealing with it to correct it. I may not understand subtle dealings with it, right? I cannot understand that someone is behaving in a particular way and therefore I will
deal with them also, they have done something rough, we will deal with them rough back. I can relate to that. But if I tell them, 'No, the person is behaving roughly because of, you know, some
emotional disturbance and this and that, so if you deal with them in this subtle way, dealing with the emotions and the mind, you can change that rough behavior.' But you'll find, most people will
not appreciate that. You know, it's rough dealings deserves rough feelings, you know, an eye for an eye, a tooth for a tooth, and all this kind-of stuff. You know, it works, it is very practical,
right? And even better is two eyes for one eye and two teeth for one teeth, that's even better. Like that, you know, teach them a lesson, set an example, right? Like that. But the subtle things are
not even appreciated, even on the material platform, let alone when you say the solution is actually spiritual. So, therefore the Dharma-śāstras are there. They are there for the materialistic
So now, Kṛṣṇa doesn't say, 'Give up these performances of prescribed duties' in the beginning stages of the Gītā, but He says ultimately that's what has to be done. And he will protect one, there
will be no sinful reaction from giving up those duties, as long as you replace that with surrendering to Him. In other words, how much devotional service you establish, to that much the, of all the
Dharma-śāstras don't need to be taken into consideration. Because Prabhupāda mentions at the end of the purport, is if you don't follow this then... He says, 'Actually it is a fact that one who does
not take to devotional service must follow the decision of these scriptures by performing pious acts to counteract his impious...' They have to, it is not a question of 'I don't believe in it' or 'I
don't want.' It's just the way it is. You know, fire will burn whether we believe it or not. So those who are not devotees, if they do not act according to the Dharma-śāstras, then they will go to
hell, it is that easy, that clear. There is not another alternative, it's not that the alternative cultures, alternative societies can overcome that. You know, it just doesn't work, it can't. We may
think what we like, but it doesn't.
So as we take up devotional service and get fixed in that, then that means these things start dropping off, and then we don't worry about this. Therefore he starts off with the point of atonement,
because atonement is one of the first things we drop off. That's one of the first things we drop off. Because we understand that the reason someone has done something wrong is because of the material
disease. So the solution is, you may have a system of atonement that may correct their mentality from sinful to pious, but it still is material. And according to the Gītā the modes of nature are
always fighting with each other, so, therefore, you know, you are piously situated with the mode of goodness, but with time passion and ignorance will prevail. So it is no solution. It may be a
temporary solution, it may be a good solution for today, but it is not a long-term solution. So, therefore, only devotional service can change that.
So if someone has a material problem, it can only be solved by spiritual understanding. That is the thing. We see, otherwise, if that wasn't the fact, then how is it that we all joined this movement
so easily? We just walked in the door clean, you know, started following the regulative principles, chanting Hare Kṛṣṇa, and that's it. What atonement did we do for all the different sinful
activities that we performed? You know, you kill one cow, and for every hair on that cow's body you have to take birth and be killed, right? And generally we find that hamburgers by the nature of it
is not made out of one cow, but many cows, right? So one can understand, you know, especially the modern Western culture, how sinful someone is! Everyone walked in the door clean. All Prabhupāda did
was ask, 'You chant Hare Kṛṣṇa.' He didn't say, 'Well, now you have to become, go through some atonement.' So such sinful things can be counteracted by chanting, that means anything can be
counteracted by chanting. So this is the first principle, is that atonement actually has no meaning, because all it does is correct the material, the conditioned nature, but it still leaves it
conditioned. So unless that is connected to Kṛṣṇa, that there is a purification, in other words, you correct by devotional service, so unless it is a service, it doesn't really correct.
So that is the point that will be made as it progresses, that the materialists, they become very much impressed by these things, but it doesn't actually free one from hellish life. It may
immediately, but it doesn't in the long run. So only by devotional service can actually one be freed because that frees us from the conditioned life. By instruction, that is what frees us: oṁ
ajñāna-timirāndhasya jñānāñjana-śalākayā. Śalākayā is cutting, it is removing. So the material life is being removed by instruction. So by association of devotees and by devotional service, by
hearing instruction and performing service for Krishna, in other words, jñāna and vairāgya, then one's devotion increases, one's realization of Kṛṣṇa increases. So this is, this is what was the
actual platform. Therefore Kṛṣṇa makes this point: devotional service is so strong, if one fully surrenders to Kṛṣṇa and takes fully to His devotional service, the reactions of the sinful life
immediately stop. Therefore Prabhupāda was establishing and pressing this full surrender, that you take up this surrender to Kṛṣṇa, because then if we do that, then all these reactions stop, and
one's obligations to the Manu-saṁhitā are very small, very small. Means, you have a conditioned nature, you have to engage it in some activities, so then some of the aspects of just social
interaction are maintained, and that's it. But more than that we don't bother. Means, you have to have some way to interact with others, to establish relationship, so then that is done by this
following of the prescribed duties here. But more than that not. But if we don't take up fully, to that degree that we are not, to that degree then these things we are liable to.
So if we are following the order of the Spiritual Master very strictly, then that is our dharma, that is our Dharma-śāstra, you know, what was it? In that... That is our Dharma-śāstra, just like
Narottama Dāsa Ṭhākura saying that the instructions of Rūpa Mañjarī, these are his dharma, you know, and following these, these are his life. So one is following dharma, but now the dharma, the
material dharma becomes replaced with the instruction of the Spiritual Master, the instruction of Kṛṣṇa.
But if we don't, then as Prabhupāda pointed out, then we must follow these other things because we are humans, so we are liable, so we must follow. Therefore Kṛṣṇa recommends this, this full
surrender, or at least endeavor for full surrender because then one has less to follow. But if not then one has to follow, that is the difficulty. So Kṛṣṇa Himself personally recommends, and the
Ācāryas all recommend that endeavor for full surrender and coming to that platform. Because otherwise, getting involved so much in the material, all the Dharma-śāstra, is very difficult, very
difficult. Like that. And they will just give you a material result. Of course, if one wants a material result, then that's the place to look. If one is interested in dharma, artha, kāma, then that's
the place to look. But if one is interested in pure devotional service, then he will follow the path of devotional service for Kṛṣṇa. Then that will bring one to the transcendental platform, the
liberated platform, and then from there one is freed from all these sinful reactions, nothing is left.
Excerpt from a lecture on Śrīmad Bhāgavatam 6.1.7, given on the Day of Disappearance of Śrīla Prabhupāda, Śrī Māyāpura. This passage starts at 7 min 40 s.
All comments. | {"url":"https://bhaktividyapurnaswami.com/2023/11/18/gurus-order-dharma/","timestamp":"2024-11-04T15:18:35Z","content_type":"text/html","content_length":"193758","record_id":"<urn:uuid:0ff9c4dc-1a62-4ae6-8f81-be607e7f5b04>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00050.warc.gz"} |
The Limits of Computing - Weizmann Wonder Wander - News, Features and Discoveries
Computers can solve complex problems but they are far from omnipotent. Prof. Ran Raz of the Weizmann Institute's Computer Science and Applied Mathematics Department is trying to portray the limits of
a computer's abilities and to estimate the resources required to solve certain problems.
For example, one problem a computer will never be able to solve is this: Will a computer running a given program stop at the end of the process? Surprisingly enough, it has been shown that this
problem, which scientists call the "halting problem," cannot be solved by any machine.
In addition, there are problems that in principle can be solved by computers, but arriving at the solution would take centuries. Raz is trying to analyze the resources (such as time or memory space)
needed to perform certain computational tasks. In most cases, such analyses are extremely difficult. To make progress, scientists utilize simpler, more basic, and sometimes weaker models of
"Boolean circuits" are an example of a basic model in which the computation is performed by applying the logical operations AND, OR, and NOT to Boolean variables (bits). A weaker model allows only
the operations AND and OR. Investigating the computational limitations of such basic models is imperative if the limits of a real computer's ability are to be determined. Understanding the basic
limitations of man's electronic best friend could also prove critical in devising future computerized systems.
Prof. Ran Raz holds the Elaine Blond Career Development Chair.Born - Jerusalem, Israel Ph.D. - Hebrew University of Jerusalem Postdoctoral research - Princeton University, New Jersey Weizmann
Institute of Science - Since 1994 | {"url":"https://wis-wander.weizmann.ac.il/math-computer-science/limits-computing","timestamp":"2024-11-07T16:19:19Z","content_type":"text/html","content_length":"59630","record_id":"<urn:uuid:5787df2e-1181-43c2-91ff-823cabfd96b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00457.warc.gz"} |
(Homotopy) Type Theory: Chapter One
In what is old news by now, the folks at the Institute for Advanced Study have released Homotopy Type Theory: Univalent Foundations of Mathematics. There has been some (meta)commentary (Dan Piponi,
Bob Harper, Andrej Bauer, François G. Dorais, Steve Awodey, Carlo Angiuli, Mike Shulman, John Baez) on the Internet, though, of course, it takes time to read a math textbook, so don’t expect detailed
technical commentary from non-authors for a while.
Of course, being a puny grad student, I was, of course, most interested in the book’s contribution of yet another Martin-Löf intuitionistic type theory introduction, e.g. chapter one. The classic
introduction is, of course, the papers that Martin Löf wrote (nota bene: there were many iterations of this paper, so it’s a little hard to find the right one, though it seems Giovanni Sambin’s notes
are the easiest to find), but an introduction of type theory for homotopy type theory has to make certain adjustments, and this makes for some novel presentation. In particular, the chapter’s
discussion of identity types is considerably more detailed than I have seen elsewhere (this is not surprising, since identity is of central importance to homotopy type theory). There is also a
considerable bit of pedantry/structure in the discussion of the types that make up the theory, reminiscent of the PFPL (though I believe that this particular chapter was mostly written by others).
And, of course, there are many little variations in how the theory is actually put together, expounded upon in some detail in the chapter notes.
In more detail:
Definitional and propositional equality. The chapter spends a little bit of time carefully distinguishing between definitional equality (a purely syntactic notion up to computation) and propositional
equality (which involves evidence), which I appreciated. The difference between connectives which show up inside and outside the deductive system was a major point of confusion for me when I was
originally learning logic.
The general pattern of the introduction of a new kind of type. The modern style for introducing logical connectives is to classify the rules into various kinds, such as introduction rules and
elimination rules, and then hew to this regularity in the presentation. Often, readers are expected to “see it”, but this book makes a helpful remark laying out the style. I found a useful exercise
was to take the rules and reorganize them so that, for example, all of the elimination rules are together and compare them.
Recursion and induction. I’ve written about this subject before, arguing that recursion and induction aren’t the same thing, since induction needs to work over indexed types. This is true, but there
is an important point I did not make: induction is generalized recursion. This is because when you specify your type family P to be the constant type family which ignores its index, the dependence is
erased and you have an ordinary recursor. In fact, this is a CPDT exercise; I think it clarifies things to see this in both Coq and informal mathematics, as the informal presentation makes the
dimension of generalization clearer.
Identity types. I won’t lie: I had a difficult time with this section, and I don’t think I fully understand why path induction works, even after a very long remark at the end of the section.
(Additionally, while the notes point to some prior literature about the subject, I took a look at the papers and I did not see anything that resembled their presentation of path induction.) By
default, Coq thinks the inductive principle for equality types should be what is referred to in this book as the indiscernability of identicals:
> Check eq_rect.
: forall (A : Type) (x : A) (P : A -> Type),
P x -> forall y : A, x = y -> P y
(As a tangent, the use of family C is confusingly overloaded; when discussing the generalization of the previous principlem the reader is required to imagine C(x) -> C(y) === C(x, y)—the C’s of
course being distinct.) Path induction asks for more:
: forall (A : Type), forall (C : forall (x y : A), x = y -> Type),
(forall (x : A), C x x (eq_refl x)) -> forall (x y : A), forall (p : x = y), C x y p
This is perhaps not too surprising, since this machinery is principally motivated by homotopy type theory. Additionally, the inductive principle follows the same pattern as the other inductive
principles defined for the other types. The trouble is a frustrating discussion of why this inductive principle valid, even when you might expect, in a HoTT setting, that not all equality was proven
using reflexivity. My understanding of the matter is that is has to do with the placement of the forall (x : A) quantifier. It is permissible to move one of the x's to the top level (based path
induction), but not both. (This is somewhat obscured by the reuse of variable names.) There is also a geometric intuition, which is that when both or one endpoints of the path are free
(inner-quantification), then I can contract the path into nothingness. But I have a difficult time mapping this onto any sort of rigorous argument. Perhaps you can help me out.
As an aside, I have some general remarks about learning type theory from a functional programming background. I have noticed that it is not too hard to use Coq without knowing much type theory,
and even easier to miss the point of why the type theory might be helpful. But in the end, it is really useful to understand what is going on, and so it’s well worth studying why dependent
products and sums generalize the way they do. It also seems that people find the pi and sigma notation confusing: it helps if you realize that they are algebraic puns. Don’t skip the definition
of the inductive principles.
I apologize if any of this post has been inaccurate or misleadingly skewed. My overall impression is that this first chapter is a very crisp introduction to type theory, but that the segments on
identity types may be a little difficult to understand. Now, onwards to chapter two! | {"url":"http://blog.ezyang.com/2013/06/homotopy-type-theory-chapter-one/","timestamp":"2024-11-07T01:16:34Z","content_type":"text/html","content_length":"32559","record_id":"<urn:uuid:81cfe640-67f4-427c-99ec-98ae55bfbfa8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00541.warc.gz"} |
Programmatic Boolean Simplification and ClamAV Signatures
ClamAV uses boolean logic in its
LDB signatures
. Each LDB signature has a set of subsignatures that, when present, evaluate to True in its logical statement. When the logical statement evaluates to True, the signature alerts. My goal was to
create something to clean up the logical statement in these signatures.
Logical & (and) and logical | (or) are the primary operators in ClamAV LDB signatures. An example representative of most LDB logical blocks would look something like 0&(1|2). This would indicate that
for this virus, subsignature 0 must be present, and one of subsignature 1 or subsignature 2 must also be present for an alert to be triggered. ClamAV does allow you to specify how many times a
subsignature can alert with the operators =, <, and >. For example, 0=0&1 would indicate that subsignature 0 should occur zero times in a file and subsignature 1 must be present. These conditional
statements (example, 0>3) are treated in the same manner as a single subsignature (example, 0) in that they evaluate to True or not. This simplifies our problem because negations (N=0) are treated as
positive matches and can simply be substituted with a dummy variable. For a more detailed description of ClamAV LDB signatures, please refer to
My goal starting out was to take a boolean equation and represent it in a form with minimal bytes. This means that unnecessary logic would need to be eliminated, as well, any boolean identity to
reduce the number of bytes used should be applied. Some examples:
Original Reduced
0|(0&1) 0
(0&1)|(0&2) 0&(1|2)
Over some months, I kept seeing cases where this capability would be useful. For example, if you had some set of subsignatures and some set of files, automatically construct an LDB signature that
alerts on all those files. Another situation would be when LDB signatures are submitted. With an algorithm like this, a tool to could be written to automatically verify that the logical block is in a
minimal representation. Signature verification and improvement is what I will focus on in this post.
With the utility for this growing, I began. The biggest difficulty I had starting out was that I had two assumptions. First, that there would be a single algorithm able to achieve this. That was
incorrect. As well, I thought that the method for doing this would be available online. That was also incorrect.
The first step was to parse the logical block into something that is more easily manipulated programmatically. For this I implemented the
Shunting Yard Algorithm
to parse the equation into polish notation. Instead of rebuilding a prefix string, the logical block is translated into an expression tree. This program was implemented in Python, so the expression
tree consisted of dicts and lists. For example:
{'&':[0, {'&': [1, {'|': [2, 3]}]}]}
Once the expression tree is built, it is converted to disjunctive normal form. This makes the statement an "or (|) of ands (&)". If that description is hard to digest, it can be thought of as the
boolean equivalent of a sum of products. This form is reached by recursively applying the boolean identity for distribution in order to expand the equation. The algorithm for this is to identify a
the situation where an and (&) sub-tree has an or (|) child. You then expand and distribute the children. The difficult thing to figure out with this part was that once a branch was changed, the
algorithm must be run again on that branch until it stops changing. An example of the input and output:
Any duplicate and (&) sub-statement is eliminated. This prepares the logical block for the
Quine-McCluskey algorithm
. This algorithm is the programmatic equivalent of a
Karnaugh Map
. That is, it will minimize a boolean expression. However, it will only do this minimization in a logical sense. In terms of representation, the equation will likely still not be optimized. In the
Quine-McCluskey algorithm, a truth table is constructed. This was represented by a list of lists in Python. Using the above example, we would get:
Truth table:
| 0 1 2 |
| ===== |
| 1 1 1 | // 0&1&2
| 1 0 1 | // 0&2
[[1,1,1], [1,0,1]]
Already, the redundant 0&0 has been reduced to a single 0 in the second statement. The algorithm then sorts the truth table entries based on their count of 1s. Any two terms that differ only by one
digit can have that digit marked with a dash (-) to indicate that the term does not matter. For instance:
Truth table:
| 0 1 2 |
| ===== |
| 1 - 1 |
| 1 - 1 |
I used 2 in place of a dash, since this was a list of integers it would keep processing simple.
[[1,2,1], [1,2,1]]
This shows that we can actually eliminate subsignature 1. Since the full signature will fire with or without that subsignature, it is not necessary to keep. Since this simple example does not
demonstrate it, I will skip over the selection portion of Quine-McCluskey. The full algorithm is detailed on Wikipedia (linked above). We now have deleted subsignature 1 and have changed subsignature
2 to 1, leaving us with:
Now that we have a logically efficient equation we must minimize it. For this, the script recursively applies boolean identity for distribution. However, this time, it applies it in a way that
extracts common terms from statements. If the current key is or (|), the function finds the most common value in this branch at a depth of 1. This means that it will return if it reaches another or
(|) after passing an and (&).
The previous case is completely reduced, so I will use a different example here.
/ \
& &
/ \ / \
& 0 0 &
/ \ / \
It would start at the root node (|), then find the most common value in its subtree. Both 0 and 1 occur twice in the subtree, so it selects whichever it saw first. It then extracts that value and
rebuilds the tree.
/ \
1 |
/ \
& &
/ \ / \
It continues to do this until no more values can be extracted.
This algorithm was not producing the results I wanted on larger equations until I added the capability for it to select subtrees as the most common value in addition to integers (leaf nodes).
The program then rebuilds the logical statement in prefix notation from the expression tree. It checks the difference in length between the original and simplified signature. If there is an
improvement, the original and simplified equations are converted to a form that Z3 Python can read and equivalence is verified. Z3 has a slightly different syntax, but the biggest consideration is
that ClamAV's method of representing variables as number is incompatible, so each number must be converted to a letter.
slvr.add(Not(simp_z3 == orig_z3))
if slvr.check() == unsat:
if opt_demo:
print 'Equivalence proven!'
return True
if opt_demo:
print 'Not equivalent!'
print 'Counterexample:'
print s.model()
return False
You ask Z3 to prove that the simplified equation does not equal the original equation. If this request is unsatisfiable, then you have proven the two equations are equivalent. Once equivalence is
verified, the LDB signature is reconstructed with the improved logic and printed out.
This first example demonstrates the scripts ability to extract common values and reduce a signature via the distribution identity.
ORIG: (0&2&3&4)|(1&2&3&4)
SIMP: (0|1)&2&3&4
Equivalence proven!
Reduced size by 8 bytes.
In this example the script combines terms to simplify the final signature.
ORIG: 0&(1|2)&((3&(5|6))|(4&(5|6)))
SIMP: 0&(1|2)&(3|4)&(5|6)
Equivalence proven!
Reduced size by 10 bytes.
A simple situation would be eliminating redundant logic.
ORIG: ((0&1)|(1&0))
SIMP: 0&1
Equivalence proven!
Reduced size by 10 bytes.
As well, there are situations where the logic can make a subsignature unnecessary. When a subsignature is deleted, there is a little bookkeeping that needs to be done to make sure Z3 still verifies
it. When building the Z3 equations it is important to note that subsignature 1 in the simplified equation is subsignature 2 in the original.
Removing subsig 1
ORIG: 0&(1|0)&2
SIMP: 0&1
Equivalence proven!
Reduced size by 15 bytes.
ClamAV Signature Set
The next question is what happens if we run this against the ClamAV LDB signature set. As of writing this, there are currently 615 LDB signatures in daily.cld. If this tool were to have been used on
all the LDB signatures prior to publishing, it would have saved 712 bytes. A large portion of these savings comes from parentheses around the outside of the equation that are dropped. The tool would
take something like (0&1) and produce 0&1.
If we were to update all the signatures with this tool there is an additional factor to consider. The first time a signature is updated a number is appended to the end of the signature name. For
example, if you were to update the signature Win.Trojan.Agent it would be renamed Win.Trojan.Agent-1. That is a cost of 2 bytes. Any additional update increments the number on the end of the
signature name. If the signature ended in -9, -99, etc., then there would be a cost of one byte when it is updated. Considering this, an update to the signature database would save 446 bytes. The
fact that this is a relatively low number is reassuring to me - it means that we are writing pretty good signatures.
This problem was a lot of fun to solve since the solution was not available anywhere online. In future research, having this functionality will enable me to generate robust LDBs to cover clusters of
malware with a single signature. On a functional level, this tool will save space and improve the overall quality of signatures published in the future. Thanks for reading. | {"url":"https://blog.clamav.net/2014/03/programmatic-boolean-simplification-and.html","timestamp":"2024-11-13T04:14:25Z","content_type":"application/xhtml+xml","content_length":"123754","record_id":"<urn:uuid:f5c75be9-b161-4603-a816-d5a024121e72>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00336.warc.gz"} |
Pandas pivot_table Silently Drops Indices with NaNs
In this post, we will discuss when pivot_table silently drops indices with NaN-s. We will give an example, expected behavior and many resources.
Let's have a DataFrame like:
import pandas as pd
import numpy as np
df = pd.DataFrame({'foo': ['one', 'one', 'one', 'two', 'two', 'two'],
'bar': ['A', 'B', np.nan, 'A', 'B', 'C'],
'baz': [1, 2, 3, 4, 5, 6],
'zoo': ['x', 'y', 'z', 'q', 'w', 't']})
with data:
│ │foo│bar│baz│zoo│
│0│one│A │1 │x │
│1│one│B │2 │y │
│2│one│NaN│3 │z │
│3│two│A │4 │q │
│4│two│B │5 │w │
silent drop of NaN-s indexes
Now let's run two different examples:
df.pivot_table(index='foo', columns='bar', values='zoo', aggfunc=sum)
result is:
│bar │A│B│ C │
│foo │ │ │ │
│one │x│y│NaN │
│two │q│w│t │
Even trying with dropna=False still results in the same behavior in pandas 2.0.1:
df.pivot_table(index='foo', columns='bar', values='zoo', aggfunc=sum, dropna=False)
pivot_table and dropna
Below you can read what is doing parameter dropna:
dropna bool, default True
Do not include columns whose entries are all NaN. If True, rows with a NaN value in any column will be omitted before computing margins.
while pivot will give us different result:
df.pivot(index='foo', columns='bar', values='zoo')
which returns NaN-s from the bar column:
│bar │nan │A│B│ C │
│foo │ │ │ │ │
│one │z │x│y│NaN │
│two │NaN │q│w│t │
Stop silent drop
Once you analyze the error and data a potential solution might be to fill NaN values with default value (which differs from the rest):
df['bar'] = df['bar'].fillna(0)
df.pivot_table(index='foo', columns='bar', values='zoo', aggfunc=sum)
After this change pivot_table will not drop the NaN indexes:
│ │foo│bar│baz│zoo│
│0│one│A │1 │x │
│1│one│B │2 │y │
│2│one│0 │3 │z │
│3│two│A │4 │q │
│4│two│B │5 │w │
The image below show the behaviour before and after the silent drop of NaN-s:
You can always refer to the official Pandas documentation for examples and what is expected: Reshaping and pivot tables
Pandas offers a variety of methods and functions to wrangle data. Sometimes the results might be unexpected. In this case test the results against another method or sequence of steps.
If you notice a Pandas bug or unexpected behavior you can open ticket or check Pandas issues like: ENH: pivot/groupby index with nan #3729 | {"url":"https://datascientyst.com/pandas-pivot_table-silently-drops-indices-with-nans/","timestamp":"2024-11-09T05:51:54Z","content_type":"text/html","content_length":"74514","record_id":"<urn:uuid:6cfd5240-7223-4d79-b2fc-d89df4365083>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00280.warc.gz"} |
Documentation > Explore
Profile parameters ๐
The profile (or calibration profile) method is designed to test the sensitivity of the input parameters in a calibration context. Calibration profile algorithm differs from traditional sensitivity
analysis: it captures the full effect of a parameter variation on the model fitness,
every other input being calibrated
to optimize the calibration criterion.
In the following we will use the term of
to denote the calibration error function, or calibration criterion. Several types of evaluation functions can be used as fitness. Most of the time, they are either some kind of distance between model
dynamics and data, or statistical measures performed on the output dynamics.
Method's score ๐
The calibration profile method is suited to reveal a model's sensitivity regarding its parameter, hence the highest score possible in sensitivity. However, it does not retrieve information about the
input space nor the output space structures, as it focus on
parameter/input, every other input being let free (to be calibrated). As the studied parameter varies, the other parameter are calibrated, so this method scores very well regarding calibration. For
the same reason, it can handle stochasticity. Finally, the profile method realizes calibrations on the other inputs for each interval of the input under study, so the more inputs, the more
sensitivity to dimensionality of input space.
Given a fitness function, the profile of a selected parameter
is constructed by dividing its interval into subintervals of equal size. For each subinterval,
is fixed, and the model is calibrated to minimise the fitness, similarly to
. The optimisation is performed over the other parameters of the model.
As an example, let's consider a model with 3 parameters
, each taking real values between 1 and 10. The profile of the parameter
is made by splitting the [1,10] interval into 9 intervals of size 1. Then, calibration is performed in parallel within each subinterval of
At the end of the calibration, we obtain sets of
values minimising the fitness, with
taking values in each subinterval. By plotting the fitness against the values of
, one can visually determine if the model, within each of
subintervals, is able to produce acceptable dynamics.
An important nuance is that for each point of this fitness vs
values plot, fitness values has been obtained by "adjusting"
to counterbalance the effect of
's values on the model dynamics. This means that
values vary for each point of the
parameter profile plot.
Profile within OpenMOLE ๐
Specific constructor ๐
constructor takes the following arguments:
• evaluation: the model task, that has to be previously declared in your script,
• objective: the fitness, a sequence of output variable defined in the OpenMOLE script that is used to evaluate the model dynamics, to be minimized (for this method it is advised to use a single
• profile: the sequence of parameter to profile, use in key word to set the number of intervals (i.e.g Seq(myParam) or Seq(myParam in 100) or Seq(myParam in (1.0 to 5.0 by 0.5))),
• genome: a list of the model input parameters with their values interval declared with the in operator,
• termination: a \"quantity\" of model evaluation allocated to the profile task. Can be given in number of evaluations (e.g. 20000) or in computation time units (e.g. 230 minutes, 2 days...),
• parallelism: optional number of parallel calibrations performed within i subinterval, defaults to 1,
• stochastic: optional the seed provider, mandatory if your model contains randomness,
• distribution: optional computation distribution strategy, default is \"SteadyState\".
• reject: optional a predicate which is true for genomes that must be rejected by the genome sampler (for instance \"i1 > 50\").
The output of the genetic algorithm must be captured with a hook which saves the current optimal population. The generic way to use it is to write either
hook(workDirectory / "path/of/a/directory"
to save the results as CSV files in a specific directory, or
hook display
to display the results in the standard output.
The hook arguments for the
• output: the file in which to store the result,
• keepHistory: optional, Boolean, keep the history of the results for future analysis,
• frequency: optional, Long, the frequency in generations where the result should be saved, it is generally set to avoid using too much disk space,
• keepAll: optional, Boolean, save all the individuals of the population not only the optimal ones.
For more details about hooks, check the corresponding
Use example ๐
To build a profile exploration, use the
constructor. Here is an example :
val param1 = Val[Double]
val param2 = Val[Double]
val fitness = Val[Double]
val mySeed = Val[Long]
evaluation = modelTask,
objective = fitness,
profile = param1,
genome = Seq(
param1 in (0.0, 99.0),
param2 in (0.0, 99.0)
termination = 2000000,
parallelism = 500,
stochastic = Stochastic(seed = mySeed, sample = 100)
) by Island(10 minutes) hook(workDirectory / "path/to/a/directory")
Interpretation guide ๐
A calibration profile is a 2D plot with the value of the profiled parameter represented on the X-axis and the best possible calibration error (
fitness value) on the Y-axis. To ease the interpretation of the profiles, we define an acceptance threshold on the calibration error (
fitness). Under this acceptance threshold, the calibration error is considered sufficiently satisfying and the dynamics exposed by the model sufficiently acceptable. Over this acceptance threshold
the calibration error is considered too high and the dynamics exposed by the model are considered unacceptable, or at least non-pertinent.
The computed calibration profiles may take very diverse shapes depending on the effect of the parameter of the model dynamics, however some of this shapes are recurrent. The most typical shapes are
shown on the figure bellow.
• Shape 1 occurs when a parameter is constraining the dynamic of the model (with respect to the fitness calibration criterion). The model is able to produce acceptable dynamics only for a single
specific range of the parameter. In this case a connected (i.e. \"one piece\" domain) validity interval can be established for the parameter.
• Shape 2 occurs when a parameter is constraining the dynamic of the model (with respect to the fitness calibration criterion), but the validity domain of the parameter is not connected. It might
mean that several qualitatively different dynamics of the model meet the fitness requirement. In this case, model dynamics should be observed directly to determine if every kind of dynamics is
suitable or if the fitness should be revised.
• Shape 3 occurs when the model \"resists\" to calibration. The profile does not expose any acceptable dynamic according to the calibration criterion. In this case, the model should be improved or
the calibration criterion should be revised.
• Shape 4 occurs when a parameter does not constrain sufficiently the model dynamics (with respect to the fitness calibration criterion). The model can always be calibrated whatever the value of
the profiled parameter. In this case this parameter constitute a superfluous degree of liberty for the model : its effect is compensated by a variation of the other parameters. In general it
means that either this parameter should be fixed, or that a mechanism of the model should be removed, or that the model should be reduced by expressing the value of this parameter as a function
of the other parameters.
The calibration profile algorithm has been published in the following paper: Romain Reuillon, Clara Schmitt, Ricardo De Aldama, and Jean-Baptiste Mouret, ยซA New Method to Evaluate Simulation Models:
The Calibration Profile (CP) Algorithm ยป published in
Journal of Artificial Societies and Social Simulation
(JASSS) , Vol 18, Issue 1, 2015.
[online version] [bibteX]
Stochastic models ๐
You can check additional options to profile stochastic models on this page. | {"url":"https://next.openmole.org/Profile.html","timestamp":"2024-11-08T04:15:19Z","content_type":"text/html","content_length":"114398","record_id":"<urn:uuid:35d93e35-5959-41cf-aeca-fb94174c4cc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00421.warc.gz"} |
Page ID: 3239
M | Motion in a circle | Angular speed | «Roundabout»
v = r \omega
where \omega is the angular speed in rad \, s^{-1} ,
r is the radius of the circle in metres,
v is the velocity in ms^{-1} .
Software/Applets used on this page
This page uses
You can get a better display of the maths by downloading special TeX fonts from
. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.
This question appears in the following syllabi:
Syllabus Module Section Topic Exam Year
AQA A-Level (UK - Pre-2017) M2 Motion in a circle Angular speed -
AQA AS Further Maths 2017 Mechanics Circular Motion Angular Speed -
AQA AS/A2 Further Maths 2017 Mechanics Circular Motion Angular Speed -
CCEA A-Level (NI) M2 Motion in a circle Angular speed -
CIE A-Level (UK) M2 Motion in a circle Angular speed -
Edexcel A-Level (UK - Pre-2017) M3 Motion in a circle Angular speed -
Edexcel AS Further Maths 2017 Further Mechanics 2 Motion in a Circle Angular Speed -
Edexcel AS/A2 Further Maths 2017 Further Mechanics 2 Motion in a Circle Angular Speed -
OCR A-Level (UK - Pre-2017) M2 Motion in a circle Angular speed -
OCR AS Further Maths 2017 Mechanics Motion in a Circle Angular Speed -
OCR MEI A2 Further Maths 2017 Mechanics B Circular Motion Angular Speed -
OCR-MEI A-Level (UK - Pre-2017) M3 Motion in a circle Angular speed -
Universal (all site questions) M Motion in a circle Angular speed - | {"url":"https://mathsnet.com/3239","timestamp":"2024-11-05T22:19:03Z","content_type":"text/html","content_length":"36386","record_id":"<urn:uuid:59e69606-83a0-467b-ad2a-772044a337df>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00367.warc.gz"} |
I throw a die 1 time. I will pay you $1 for each dot that turns up on the die. How much would you pay to play this game? You may assume that you are risk-neutral.
Betting games are pretty popular for interviews at trading firms. These are typically disguised expected value questions. In this case, we simply need to take the expected value of rolling a die.
Interviewers, may ask follow up questions such as: "Now assume that you are risk averse. How does that change your answer?"
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What is the sum of integers from 1 to 100?
We need to remember the Gauss sum equation:
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Which is one of these exponents is bigger:
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There are 25 horses, each of which runs at a constant speed that is different from the other horses. Since the track only has 5 lanes, each race can have at most 5 horses. If you need to find the 3
fastest horses, what is the minimum number of races needed to identify them?
Let's label each horse 1 - 25. You first need to race each horse: (1, 2, 3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20) and (21, 22, 23, 24, 25). That is 5 races. Then we take
the winner of each heat. Let's say that is the lowest horse number in each heat and race them (1, 6, 11, 16, 21). This guarantees that #1 is the fastest horse. This is the 6th race. Are #6 and #11
necessarily the 2nd and 3rd fastest? No, because it is possible that #2 and #3 just lost in the first heat to the fastest horse (#1). We need a final 7th race between (6, 11, 2, 3, x) where x in any
What are the smallest and largest values that Delta can take for a vanilla put option?
The delta of an option is the 1st derivative of the value of the option with respect to the price of the stock. As stock price increases, the value of a put option decreases thus we know the delta
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probability the option ends up in-the-money, thus a value less than -1 would not make sense.
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I toss a coin 400 times. What is the probability of getting < 220 heads?
Use the binomial distribution. Since p = 0.50 and N = 100, the mean is 50, the variance is 25 and the standard deviation is 5. So 40 corresponds to a one sided 2 std move. A two sided 2 std move
contains 95% of the area, so a 1 sided 2 std move contains 95% + 2.5% = 97.5%.
You have a chessboard. From any single diagonal, you remove two square. There are just 62 squares left on the board. You also have 31 domino tiles, each of which is conveniently the same size as two
of the chessboard squares. Is it possible to perfectly cover the board with these dominoes, without any domino edges hanging over?
The opposite corners of a chess board have the same color. Placing a domino on the board requires covering both a white and black square. So, we will not be able to cover the chessboard.
What are you waiting for? Registration takes under 1 minute and you can then start practicing for your quant interviews.
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Stock X has an expected return of 5% and stock Y has an expected return of 15%. Stock X has a volatility of 10% and stock Y has volatility of 25%. The correlation between their returns is 25%. What
is the allocation to stock X that creates the minimum variance portfolio?
We have that
To find a local minimum, we need to differentiate, set equal to zero and solve for our parameter
We evaluate the expression with the given inputs. The weight of stock X is 93.75%.
Which of the following is NOT an assumptions of OLS?
Spherical errors
Normal errors
No linear dependence
Linear relationship
The normal errors assumption is only required when doing OLS on small samples. In most financial applications of OLS, you should have sufficient data so won't need worry about having normal errors.
You got -y-/9 questions correct. You've got some work to do, but we can help you! You can train on tons of questions once you join. This will help you get a job in -x-. That's a good start, but we
know you can do better. You can train on tons of questions once you join. This will help you get a job in -x-. Great work, you really knocked it out of the park! You can train on tons of other
difficult problems on our site. This will ensure that you ace even the hardest interviews. | {"url":"https://www.quantquestions.com","timestamp":"2024-11-14T17:58:34Z","content_type":"text/html","content_length":"61847","record_id":"<urn:uuid:f0a93a0d-8914-4753-b567-e929a0f67b08>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00876.warc.gz"} |
Year 10 Autumn 2 2021-22
Circles and Equations (7.5 school weeks)
Pre-Covid lessons from Y10 Autumn 2 can be found here
Note: the Quadratics lessons are from Y9, check understanding carefully with these key concepts here these lessons were affected by lockdown.
Lesson 1: Solving simultaneous equations by elimination (subtraction)
Lesson 2: Solving simultaneous equations by elimination (addition)
Lesson 3: Difference or sum
Lesson 4: Balancing coefficients
Lesson 5: Simultaneous Scenarios
Lesson 6: Olaf’s boat hire
Lesson 7: Points of Intersection
Lesson 8: Drawing quadratic graphs
Lesson 9: Finding the roots of quadratic equations by factorising
Lesson 10: Forming and solving quadratic equations using factorising
Lesson 11: Sector Area
Lesson 12: Arc Length
Half Term Check Up
Lesson 1: Solving simultaneous equations by elimination (subtraction)
Lesson 2: Solving simultaneous equations by elimination (addition)
Lesson 3: Difference or sum
Lesson 4: Balancing coefficients
Lesson 5: Simultaneous Scenarios
Lesson 6: Olaf’s boat hire
Lesson 7: Points of intersection
Lesson 8 : Drawing quadratic graphs (from Y9)
Lesson 9 : Finding the roots of quadratic equations by factorising (from Y9)
Lesson 10: Forming and solving quadratic equations using factorising (from Y9)
Lesson 11: Forming and solving quadratic equations (a>1) (from Y9)
Lesson 12: The Quadratic Formula (from Y9)
Lesson 13: The Equation of a Circle
Lesson 14: Quadratic meets Linear
Lesson 15: Equation of the tangent
Lessons 16, 17, 18 : Circle Theorems
Lesson 19 Sector Area
Lesson 20: Arc Length
Half Term Check Up | {"url":"https://archwaymaths.com/year-10-autumn-2-2020-21/","timestamp":"2024-11-08T12:46:01Z","content_type":"text/html","content_length":"39614","record_id":"<urn:uuid:c1812f3f-e4b4-4404-b40c-d80f215e3fad>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00725.warc.gz"} |
Smoothing and Worst Case Complexity for Direct-Search Methods in Non-Smooth Optimization
For smooth objective functions it has been shown that the worst case cost of direct-search methods is of the same order as the one of steepest descent, when measured in number of iterations to
achieve a certain threshold of stationarity. Motivated by the lack of such a result in the non-smooth case, we propose, analyze, and test a class of smoothing direct-search methods for the
optimization of non-smooth functions. Given a parameterized family of smoothing functions for the non-smooth objective function, this class of methods consists of applying a direct-search algorithm
for a fixed value of the smoothing parameter until the step size is relatively small, after which the smoothing parameter is reduced and the process is repeated. One can show that the worst case
complexity (or cost) of this procedure is roughly one order of magnitude worse than the one for direct search or steepest descent on smooth functions. The class of smoothing direct-search methods is
also showed to enjoy asymptotic global convergence properties. Some preliminary numerical experience indicates that this approach leads to better values of the objective function, pushing in some
cases the optimization further, apparently without an additional cost in the number of function evaluations.
Preprint 12-02, Dept. Mathematics, Univ. Coimbra.
View Smoothing and Worst Case Complexity for Direct-Search Methods in Non-Smooth Optimization | {"url":"https://optimization-online.org/2012/01/3331/","timestamp":"2024-11-12T09:35:22Z","content_type":"text/html","content_length":"85200","record_id":"<urn:uuid:e6edbcf8-ad7b-4645-ac75-134d34e2c670>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00173.warc.gz"} |
What is Context Transition in DAX, and how does it affect ca...
What is Context Transition in DAX, and how does it affect calculations? Can you provide an example?
In DAX, context transition refers to how row context can transition to filter context when a calculated column or measure is evaluated. Understanding this concept is essential for working with DAX in
Power BI because context transition affects how data is filtered and evaluated in different scenarios, particularly when using functions like CALCULATE.
Breaking Down Context Types
1. Row Context: Applies when DAX evaluates a row in a calculated column or when iterating through each row in a table (e.g., using functions like SUMX or FILTER).
2. Filter Context: Refers to the filters that are applied to a calculation, such as report-level filters or filters applied in a visual.
Context Transition occurs when a row context is converted into a filter context—a change in how DAX interprets the data and calculations. This transition primarily happens when you use functions like
CALCULATE, which allows row-by-row evaluations to act as filters, applying these filters to the entire calculation.
Example: Demonstrating Context Transition with CALCULATE
Let’s say you have a Sales table with columns for ProductID, SalesAmount, and Quantity. You also have a Products table with columns ProductID and ProductCategory.
Now, imagine you want to create a calculated column in the Sales table that sums up all sales in the same product category as the current row.
Without Context Transition (Incorrect Approach):
This formula won’t work correctly because the row context of each ProductCategory is not transitioned to filter context across the entire Sales table.
With Context Transition (Correct Approach):
To get the correct result, you need to use CALCULATE to apply context transition:
In this formula:
1. CALCULATE transitions the row context of the current row’s ProductCategory to filter context.
2. The ALLEXCEPT function removes any existing filters except for the ProductCategory, so the calculation sums up all rows in the same category, ignoring other filters in the Sales table.
Why is Context Transition Important?
• Essential for Complex Measures: Many complex DAX measures rely on context transition, especially those that require evaluating data dynamically based on row-by-row conditions.
• Enables Row Context to Act as Filter Context: Context transition allows DAX functions that rely on filter context, like CALCULATE, to apply row-level logic more broadly.
Key Takeaway: Always remember that CALCULATE is the primary function that enables context transition. Whenever you’re dealing with row-by-row calculations that should also filter across the table,
consider if context transition is necessary to get accurate results.
Interview Tip: Explaining context transition with an example is a great way to demonstrate understanding. Highlighting how CALCULATE and ALLEXCEPT work together to manage filter context can show your
grasp of advanced DAX principles. | {"url":"https://sqlqueries.in/community/qa/what-is-context-transition-in-dax-and-how-does-it-affect-calculations-can-you-provide-an-example/","timestamp":"2024-11-03T07:35:05Z","content_type":"text/html","content_length":"154763","record_id":"<urn:uuid:d2637f62-ea93-4c57-b26d-fe7e5b51ffc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00259.warc.gz"} |
Net Present Value Pension Calculator - Certified Calculator
Net Present Value Pension Calculator
Managing pension investments involves evaluating their present value, considering factors like initial investment, annual cashflow, and discount rates. The Net Present Value (NPV) provides a
comprehensive measure of the investment's profitability, accounting for the time value of money.
Formula: The NPV formula is calculated as the sum of the initial investment and the present value of annual cashflows, discounted at the specified rate over the investment period.
How to Use:
1. Enter the initial investment amount.
2. Input the annual cashflow generated by the pension investment.
3. Specify the discount rate as a percentage.
4. Define the investment period in years.
5. Click the "Calculate" button to obtain the Net Present Value.
Example: Suppose you invest $50,000 initially, receive an annual cashflow of $10,000, use a discount rate of 5%, and plan to invest for 10 years. The NPV will be calculated based on these parameters.
1. What is Net Present Value (NPV)?
□ NPV is a financial metric that assesses the profitability of an investment by discounting future cashflows to their present value.
2. Why is NPV important for pension calculations?
□ NPV accounts for the time value of money, providing a realistic measure of the pension's value.
3. Can NPV be negative?
□ Yes, a negative NPV indicates that the investment may not meet the required rate of return.
4. How is the discount rate determined?
□ The discount rate reflects the investor's required rate of return and the perceived risk of the investment.
5. Is the NPV always accurate in predicting profitability?
□ While NPV is a valuable metric, external factors can impact actual investment performance.
6. What happens if the discount rate is zero?
□ With a zero discount rate, the NPV calculation simplifies to the sum of all cashflows, ignoring the time value of money.
Conclusion: The Net Present Value Pension Calculator simplifies the assessment of pension investments, providing a reliable indicator of their financial viability. By considering factors such as
initial investment, annual cashflows, and the discount rate, investors can make informed decisions regarding their pension portfolios. Use this tool to streamline your pension investment evaluation
process and plan for a secure financial future.
Leave a Comment | {"url":"https://certifiedcalculator.com/net-present-value-pension-calculator/","timestamp":"2024-11-10T00:24:06Z","content_type":"text/html","content_length":"55023","record_id":"<urn:uuid:e321c13b-649c-4eaa-9e94-3cd215806b35>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00857.warc.gz"} |
NCERT Solutions Maths Class 10 Exercise 7.1 Chapter 7 - Coordinate Geometry
NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry Exercise 7.1 - FREE PDF Download
The NCERT Solutions for Class 10 Maths Chapter 7 Exercise 7.1 on Coordinate Geometry offers detailed answers to the exercises given. These solutions are designed to help students prepare for their
CBSE Class 10 board exams. It's important for students to go through these solutions carefully as they cover various types of questions related to Coordinate Geometry. By practicing with these
solutions, students can enhance their understanding and be better equipped to tackle similar questions in their Class 10 board exams.
1. NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry Exercise 7.1 - FREE PDF Download
2. Glance on NCERT Solutions Maths Chapter 7 Exercise 7.1 Class 10 | Vedantu
3. Access PDF for Maths NCERT Chapter 7 Coordinate Geometry Exercise 7.1 Class 10
4. Class 10 Maths Chapter 7: Exercises Breakdown
5. Other Study Materials of CBSE Class 10 Maths Chapter 7
6. A Chapter-Specific NCERT Solutions for Class 10 Maths
7. Study Resources for Class 10 Maths
Glance on NCERT Solutions Maths Chapter 7 Exercise 7.1 Class 10 | Vedantu
• In this article, students can enhance their problem-solving skills and develop a deeper insight into coordinate geometry concepts.
• Understanding the Cartesian plane with X and Y axes, and representing points using coordinates (x, y).
• Applying the distance formula to find the distance between two points on the coordinate plane.
• Plotting points on the graph based on their coordinates.
• Finding the distance between points with given coordinates.
• Determining the coordinates of a point which is equidistant from two other points.
• Exercise 7.1 class 10 NCERT solutions has over all 10 questions.
Watch videos on
NCERT Solutions For Class 10 Maths Chapter 7 Exercise 7.1 - Coordinate Geometry
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FAQs on NCERT Solutions For Class 10 Maths Chapter 7 Exercise 7.1 - Coordinate Geometry
1. What will I learn in this chapter?
Coordinate Geometry of NCERT Class 10 Maths syllabus will teach you how to find the distance between points with the given coordinates and you shall learn about points on a cartesian plane, distance
formula, section formula, midpoint, points of trisection, the centroid of a triangle and areas from coordinates. NCERT Solutions for Class 10 Maths Chapter 7 consist of 4 exercises which have 33
questions and they cover all the topics from the chapter. These solutions are also drafted in a step by step way to make it easy for the students. In NCERT Class 10 Maths Chapter 7, you will be
taught about the distance between the two points from the given coordinates. You shall see how the area of the triangle is formed with the three given points.
2. Explain about the area of a triangle, polygon and the centroid of a triangle.
Area of a Triangle:
In this part of Coordinate Geometry from Class 10 Maths Chapter 7 of NCERT Solutions, you will be solving the questions based on the below-mentioned formula.
Area of a trapezium = ½ (Sum of parallel sides) (distance between them)
The Centroid of a Triangle:
In this section of Coordinate Geometry of Class 10 Maths, you will be learning about what is the centroid of a triangle and why the three medians of the triangle meet are known as the centroid of a
Area of a Polygon:
In this segment of NCERT CBSE Class 10 Maths Chapter 7, you will be learning how to find the area of a polygon with the help of all the vertices of the polygon.
Formula: A = ½ (x[1]y[2] - y[1]x[2]) + (x[2]y[3] - y[2]x[3]) + …. + (x[n]y[1] - y[n]x[1])
3. Give a brief overview of the distance from the origin.
Here, in NCERT Solutions for Class 10 Maths Chapter 7, you will have to find the distance from the origin to the given point, and the given point is also the origin.
In Coordinate Geometry of NCERT Solutions for Class 10th Maths Chapter 7, you will be given points on the line segment and that line segment will be divided equally in a particular ratio and you
will have to find the coordinate points.
Here in Coordinate Geometry of NCERT Solutions for Class 10 Maths, you will be learning how to find the coordinate points of a given midpoint of the line segment.
4. How do I increase my ranks with Vedantu’s study guide?
You can improve your grades with our help. We at Vedantu have a team of experts who understand the academic needs and requirements of students. The study material provided by Vedantu is crafted as
per the latest syllabus and is made available to everybody through our online portal.
Our solutions and study guide are designed by our subject-matter experts in a step by step way for easy and better understanding of the concept. We made sure that CBSE and NCERT guidelines are
strictly followed while drafting these solutions. All the topics are explained in detail to make it more clear for you.
5. How many questions are there in Exercise 7.1 of Chapter 7 of Class 10 Maths?
Exercise 7.1 of Chapter 7 of Class 10 Maths consists of 10 questions in total. These questions are very well explained in detailed steps in NCERT solutions from Vedantu which will help the student to
learn the concept of the chapter with full understanding. The practice from the solutions will give them confidence. Complex problems are made easy so that the students will not be demotivated.
6. What is Coordinate Geometry according to Chapter 7 of Class 10?
Coordinate geometry may be defined as a study of geometry with the help of a coordinate system. It is a section of geometry using which the positions of given points on a plane can be defined by a
particular pair of numbers referred to as coordinates. Coordinate geometry is an important chapter and students have to learn it properly by paying full attention in class and also practise the
chapter as much as possible.
These solutions are available on Vedantu's official website(vedantu.com) and mobile app free of cost.
7. What is the Distance Formula?
The distance formula in the chapter is used to find the distance between the two points in the plane of XY. The distance of the point from the Y-axis is called the X-coordinate and the distance from
the X-axis is called the Y-coordinate. The point on the X-axis is(x,0) and the point in the Y-axis is(0, y). To find the distance on the plane the theorem of Pythagoras is used in an important
chapter in Class 10.
The distance formula is PQ =√[(x2 – x1)2 + (y2 – y1)2]
Distance = √[(x2 – x1)2 + (y2 – y1)2
8. What are Collinear Points?
Collinear points refer to three or more than three points lying on a line. So, when they join and extend, it forms one straight line. In case the points aren't collinear, they will make a triangle
when joined together. This word is derived from a Latin word Col+Linear. Col means together and linear means a straight line. These concepts have to be clear to solve any type of numericals connected
with the chapter.
9. What are the different formulas for finding Collinear Points?
The three different formulas are the Distance formula, Slope formula and Area Of A Triangle.If A, B and C are the collinear points then AB+BC=AC
We know the distance between the point (x1,y1)and (x2,y2)
Distance =√(X2-X1)2+(Y2--Y1)2
The slope formula is used to measure the steepness of the line.
For triangle area can be found by
(1/2) | [x1(y2 – y3) + x2(y3 – y1) + x3[y1 – y2]| = 0
10. What is covered in Class 10 Chapter 7 Exercise 7.1 of Maths: Coordinate Geometry?
Class 10 Maths Exercise 7.1 Solutions focuses on the basics of coordinate geometry, including understanding the Cartesian plane, plotting points, and using the distance formula to find the distance
between two points.
11. What are the four quadrants of the Cartesian plane in exercise 7.1 class 10th?
Following are the points for class 10 ch 7 maths ex 7.1:
• Quadrant I: Both x and y coordinates are positive (top-right).
• Quadrant II: x is negative, y is positive (top-left).
• Quadrant III: Both x and y coordinates are negative (bottom-left).
• Quadrant IV: x is positive, y is negative (bottom-right).
12. What are some common mistakes to avoid when solving problems in Class 10th maths chapter 7 exercise 7.1?
Coordinate Geometry Class 10 Exercise 7.1 gives:
• Incorrectly plotting points: Ensure accuracy when locating points on the Cartesian plane.
• Sign errors in the distance formula: Pay close attention to the signs of the coordinates when subtracting.
• Arithmetic mistakes: Double-check calculations, especially when squaring numbers and taking square roots.
13. Why is the distance formula important in class 10 ch 7 maths ex 7.1?
The distance formula is crucial in coordinate geometry as it allows you to determine the distance between any two points in the plane, which is fundamental for solving various geometric problems and
understanding the relationships between points. | {"url":"https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10-maths-chapter-7-exercise-7-1","timestamp":"2024-11-07T12:43:35Z","content_type":"text/html","content_length":"665815","record_id":"<urn:uuid:493c1ea3-467b-491a-8375-bb990b37fc32>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00277.warc.gz"} |
5 Maths Gems #178
Welcome to my 178
gems post. This is where I share some of the latest news, ideas and resources for maths teachers.
1. Inverse Proportion
I really like this
. When teaching inverse proportion I always talk about constant products but then I head straight into the formula y = k/x. Catriona suggests a subtle change in method.
2. Graph Transformations
MathsPad has published another brilliant
interactive tool
for the Higher GCSE topic
graph transformations
. It shows how points on the original curves map on to those on the transformed curves. There is a separate section for
quadratic graphs
, which is particularly useful for relating completed square form to graph transformations.
There is also a section for
trigonometric graphs
, where transformations of the graphs of y=sin(x) and y=cos(x) can be explored.
It's great to see a new resource published for teaching graph transformations - I always find that this is the GCSE topic for which it's trickiest to find suitable resources, because many tasks still
include stretches (these were removed from the syllabus back in 2017 - GCSE now just covers reflections and translations).
Do check out MathsPad's full range of
interactive tools
- some of them are free to use even if you don't subscribe (you really should subscribe to MathsPad though!).
3. GeoGebra
It's worth following
to see ideas for interactive
GeoGebra maths resources
that can be used for demonstrations in lessons or student activities. Here are a few recent examples:
📦🐭 GEOGEBRA RESOURCE! 🥡🐱
Try this FREE GeoGebra Practice resource to draw box-and-whisker plots from a set of data. Check it out! https://t.co/VsqOGblJ2P#MTBoS #iteachmath #math #statistics pic.twitter.com/GAz9gbyLAR
— GeoGebra (@geogebra) February 27, 2024
🦾 GEOGEBRA RESOURCE! 🎰
Try this NEW & FREE GeoGebra Exploration resource to convert a number from standard form to expanded form one place value at a time. Check it out! https://t.co/vAIXvL5g48#MTBoS #iteachmath #math
#mathematics pic.twitter.com/UXSN3RHQ69
— GeoGebra (@geogebra) March 4, 2024
4. Pythagoras@nathanday314
shared a clever set of questions on
Pythagoras' Theorem
. There's a lot of challenge here (check out Question 12). Question 3 is deliberately impossible, and note the subtle differences between Questions 3, 4 and 5.
5. Prime Factors
Here are some other things you might have missed:
My school is still seeking an A level teacher to start in September. I can offer 100% A level teaching, or alternatively I can offer a timetable with Year 11 - 13 (or similar) if that's more
desirable. Our students are a delight to teach and my team is awesome. If you live in Surrey or South London, this is an amazing opportunity. Please get in touch (email resourceaholic@gmail.com) if
you're interested - I'm very happy to arrange a tour or call.
I'm presenting at two maths conferences in the next four weeks. The first is
which is near Bristol:And in the Easter holidays I'm presenting at the
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The minimum length of the chord of the circle x2+y2+2x+2y−7=0 w... | Filo
The minimum length of the chord of the circle which is passing through is :
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Question Text The minimum length of the chord of the circle which is passing through is :
Topic Conic Sections
Subject Mathematics
Class Class 11
Answer Type Text solution:1
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ManPag.es -
slasda.f −
subroutine SLASDA (ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
Function/Subroutine Documentation
subroutine SLASDA (integerICOMPQ, integerSMLSIZ, integerN, integerSQRE, real, dimension( * )D, real, dimension( * )E, real, dimension( ldu, * )U, integerLDU, real, dimension( ldu, * )VT, integer,
dimension( * )K, real, dimension( ldu, * )DIFL, real, dimension( ldu, * )DIFR, real, dimension( ldu, * )Z, real, dimension( ldu, * )POLES, integer, dimension( * )GIVPTR, integer, dimension( ldgcol, *
)GIVCOL, integerLDGCOL, integer, dimension( ldgcol, * )PERM, real, dimension( ldu, * )GIVNUM, real, dimension( * )C, real, dimension( * )S, real, dimension( * )WORK, integer, dimension( * )IWORK,
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
Using a divide and conquer approach, SLASDA computes the singular
value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
B with diagonal D and offdiagonal E, where M = N + SQRE. The
algorithm computes the singular values in the SVD B = U * S * VT.
The orthogonal matrices U and VT are optionally computed in
compact form.
A related subroutine, SLASD0, computes the singular values and
the singular vectors in explicit form.
ICOMPQ is INTEGER
Specifies whether singular vectors are to be computed
in compact form, as follows
= 0: Compute singular values only.
= 1: Compute singular vectors of upper bidiagonal
matrix in compact form.
SMLSIZ is INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N is INTEGER
The row dimension of the upper bidiagonal matrix. This is
also the dimension of the main diagonal array D.
SQRE is INTEGER
Specifies the column dimension of the bidiagonal matrix.
= 0: The bidiagonal matrix has column dimension M = N;
= 1: The bidiagonal matrix has column dimension M = N + 1.
D is REAL array, dimension ( N )
On entry D contains the main diagonal of the bidiagonal
matrix. On exit D, if INFO = 0, contains its singular values.
E is REAL array, dimension ( M-1 )
Contains the subdiagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
U is REAL array,
dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
singular vector matrices of all subproblems at the bottom
LDU is INTEGER, LDU = > N.
The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
GIVNUM, and Z.
VT is REAL array,
dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
singular vector matrices of all subproblems at the bottom
K is INTEGER array, dimension ( N )
if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
secular equation on the computation tree.
DIFL is REAL array, dimension ( LDU, NLVL ),
where NLVL = floor(log_2 (N/SMLSIZ))).
DIFR is REAL array,
dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
dimension ( N ) if ICOMPQ = 0.
If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
record distances between singular values on the I-th
level and singular values on the (I -1)-th level, and
DIFR(1:N, 2 * I ) contains the normalizing factors for
the right singular vector matrix. See SLASD8 for details.
Z is REAL array,
dimension ( LDU, NLVL ) if ICOMPQ = 1 and
dimension ( N ) if ICOMPQ = 0.
The first K elements of Z(1, I) contain the components of
the deflation-adjusted updating row vector for subproblems
on the I-th level.
POLES is REAL array,
dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
POLES(1, 2*I) contain the new and old singular values
involved in the secular equations on the I-th level.
GIVPTR is INTEGER array,
dimension ( N ) if ICOMPQ = 1, and not referenced if
ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
the number of Givens rotations performed on the I-th
problem on the computation tree.
GIVCOL is INTEGER array,
dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
of Givens rotations performed on the I-th level on the
computation tree.
LDGCOL is INTEGER, LDGCOL = > N.
The leading dimension of arrays GIVCOL and PERM.
PERM is INTEGER array, dimension ( LDGCOL, NLVL )
if ICOMPQ = 1, and not referenced
if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
permutations done on the I-th level of the computation tree.
GIVNUM is REAL array,
dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
values of Givens rotations performed on the I-th level on
the computation tree.
C is REAL array,
dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
C( I ) contains the C-value of a Givens rotation related to
the right null space of the I-th subproblem.
S is REAL array, dimension ( N ) if
ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
and the I-th subproblem is not square, on exit, S( I )
contains the S-value of a Givens rotation related to
the right null space of the I-th subproblem.
WORK is REAL array, dimension
(6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
IWORK is INTEGER array, dimension (7*N).
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, a singular value did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Definition at line 272 of file slasda.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://manpag.es/SUSE131/3+SLASDA","timestamp":"2024-11-12T13:35:36Z","content_type":"text/html","content_length":"26470","record_id":"<urn:uuid:f2382af6-770d-4e37-a5a4-53b6f0e2810d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00035.warc.gz"} |
RE: st: nbreg - problem with constant?
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
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RE: st: nbreg - problem with constant?
From Simon Falck <[email protected]>
To "[email protected]" <[email protected]>
Subject RE: st: nbreg - problem with constant?
Date Mon, 5 Mar 2012 13:54:45 +0000
...thanks also to Dave Jacobs! :D
-----Original Message-----
From: Simon Falck
Sent: den 5 mars 2012 14:54
To: '[email protected]'
Subject: RE: st: nbreg - problem with constant?
Dear David Hoaglin, Joerg, David Hoaglin and Richard Williams,
Thanks for your insights on the nbreg model!!
I will consider these insights in the forthcoming.
Thank you again,
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of Jacobs, David
Sent: den 4 mars 2012 18:12
To: '[email protected]'
Subject: RE: st: nbreg - problem with constant?
Another simple step is to look at the actual size of your explanatory variables.
I've often found that zero inflated count models have problems converging, but these problems are much diminished if I divide extremely large explanatory variables like state or city populations by a constant such as 10,000 or 100,000 if necessary because, I suppose, this expedient gets everything at about the same numeric level.
Dave Jacobs
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of David Hoaglin
Sent: Friday, March 02, 2012 4:56 PM
To: [email protected]
Subject: Re: st: nbreg - problem with constant?
Hi, Simon.
When you remove the constant from the model, some of the variation in the dependent variable the was accounted for by the constant is then accounted for (to the extent possible) by the predictors that remain in the model. The result is not necessarily to make the coefficients of those predictors larger, but they will generally change.
Consider how removing the constant would work in ordinary multiple regression. If a predictor variable does not have mean 0 (in the data), removing the constant will change its coefficient. You can even see this happen in simple regression when you fit a line that has a slope and an intercept and then fit a line through the origin. It's easy to construct an example in which the two slopes have different signs. One has to keep in mind that the definition of a coefficient in a regression (or similar) model depends on the list of other predictors in the model.
In your negative binomial model, I don't think you want to take exp of the coefficients and interpret the results as if they were the coefficients.
David Hoaglin
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* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2012-03/msg00195.html","timestamp":"2024-11-12T12:30:00Z","content_type":"text/html","content_length":"14382","record_id":"<urn:uuid:3c759765-91af-4eee-9cf4-867786cc8a13>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00752.warc.gz"} |
Non-symmetric Scattering in Light Transport Algorithms
non-gamma corrected pages.
There is also more information on gamma correction.
Non-symmetric scattering is far more common in computer graphics than is generally recognized, and can occur even when the underlying scattering model is physically correct. For example, we show that
non-symmetry occurs whenever light is refracted, and also whenever shading normals are used (e.g. due to interpolation of normals in a triangle mesh, or bump mapping).
We examine the implications of non-symmetric scattering for light transport theory. We extend the work of Arvo et al. into a complete framework for light, importance, and particle transport with
non-symmetric kernels. We show that physically valid scattering models are not always symmetric, and derive the condition for arbitrary models to obey Helmholtz reciprocity. By rewriting the
transport operators in terms of optical invariants, we obtain a new framework where symmetry and reciprocity mean the same thing.
We also consider the practical consequences for global illumination algorithms. The problem is that many implementations indirectly assume symmetry, by using the same scattering rules for light and
importance, or particles and viewing rays. This can lead to incorrect results for physically valid models. It can also cause different rendering algorithms to converge to different solutions (whether
the model is physically valid or not), and it can cause shading artifacts. If the non-symmetry is recognized and handled correctly, these problems can easily be avoided.
Additional information
If you would like more information about the rendering algorithm used to compute the image above, it is described in a different paper.
Make sure to get both the main text and the color plate, if you want the full paper.
If the images are too bright, try the non-gamma corrected pages. All JPEG images were compressed using a quality setting of 90. Return to other recent papers from Stanford
Last modified: June 11, 1996
Eric Veach | {"url":"http://graphics.stanford.edu/papers/non-symmetric/gamma-fixed/","timestamp":"2024-11-14T10:19:10Z","content_type":"text/html","content_length":"4579","record_id":"<urn:uuid:d5bd8ee1-0323-44b7-9015-55bd901dab66>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00385.warc.gz"} |
How to Convert int to double with 2 Decimal Places in C#?
In this C# tutorial, I have explained how to convert int to double with 2 decimal places in C# using different methods.
To convert an integer to a double with two decimal places in C#, you can use the Math.Round method combined with a cast to double. For example:
int intValue = 100;
double doubleValue = Math.Round((double)intValue, 2);
This code snippet casts the integer intValue to a double and rounds it to two decimal places, resulting in doubleValue being 100.00.
Convert int to double with 2 Decimal Places in C#
Now, let us see how to convert int to double with 2 decimal places in C#.
• int: An integer data type that represents whole numbers without any decimal places.
• double: A double-precision floating-point data type that can represent fractional numbers, including those with decimal places.
Method 1: Using ToString and double.Parse
The first method involves converting the integer to a string with the desired format and then parsing it back to a double in C#.
using System;
class Program
static void Main()
int intValue = 100;
string stringValue = intValue.ToString("F2");
double doubleValue = double.Parse(stringValue);
Console.WriteLine(doubleValue.ToString("F2")); // Output: 100.00
In this snippet, "F2" is a format specifier that tells ToString to represent the integer as a fixed-point number with two decimal places. After formatting, double.Parse converts the string back to a
You can see the output in the screenshot below:
Method 2: Mathematical Operation
Another approach is to perform a mathematical operation on the integer to convert it to a double directly in C#.
using System;
class Program
static void Main()
int intValue = 100;
double doubleValue = (double)intValue;
doubleValue = Math.Round(doubleValue, 2);
By dividing the integer by 1.00, we implicitly convert the integer to a double. However, this does not guarantee two decimal places, so we need to round the result.
myDouble = Math.Round(myDouble, 2);
The Math.Round function rounds the double value to the nearest value with two decimal places.
You can see the output in the screenshot below:
Method 3: Using String Interpolation and double.Parse
C# 6 introduced string interpolation, which provides a more readable way to format strings.
using System;
class Program
static void Main()
int intValue = 100;
string stringValue = $"{intValue:F2}";
double doubleValue = double.Parse(stringValue);
Console.WriteLine(doubleValue); // Output: 100.00
The $"{myInt:F2}" syntax is an interpolated string that formats the integer with two decimal places before parsing it to a double.
Method 4: Using Convert Class
The Convert class in C# provides a handy way to perform type conversions. To convert an integer to a double and ensure two decimal places, you can use the Convert.ToDouble method in conjunction with
using System;
class Program
static void Main()
int intValue = 100;
double doubleValue = Convert.ToDouble(intValue);
doubleValue = Math.Round(doubleValue, 2);
Console.WriteLine(doubleValue); // Output: 100.00
Here, Convert.ToDouble changes the integer to a double, and Math.Round ensures the precision of two decimal places.
Method 5: Custom Extension Method
For a more reusable solution, you can create a custom extension method that converts an integer to a double with two decimal places.
using System;
public static class ExtensionMethods
public static double ToDoubleWith2DecimalPlaces(this int value)
return Math.Round((double)value, 2);
class Program
static void Main()
int intValue = 100;
double doubleValue = intValue.ToDoubleWith2DecimalPlaces();
Console.WriteLine(doubleValue); // Output: 100.00
This extension method can be called on any integer to get a double with two decimal places.
Complete Example
Let’s see a full example that demonstrates these methods:
using System;
public class IntToDoubleConverter
public static void Main(string[] args)
int myInt = 123;
// Method 1
string formatted1 = myInt.ToString("F2");
double myDouble1 = double.Parse(formatted1);
Console.WriteLine($"Method 1: {myDouble1}");
// Method 2
double myDouble2 = (double)myInt / 1.00;
myDouble2 = Math.Round(myDouble2, 2);
Console.WriteLine($"Method 2: {myDouble2}");
// Method 3
string formatted3 = $"{myInt:F2}";
double myDouble3 = double.Parse(formatted3);
Console.WriteLine($"Method 3: {myDouble3}");
// Method 4
double myDouble4 = Convert.ToDouble(myInt);
myDouble4 = Math.Round(myDouble4, 2);
Console.WriteLine($"Method 4: {myDouble4}");
// Method 5 (Extension Method)
double myDouble5 = myInt.ToDoubleWith2DecimalPlaces();
Console.WriteLine($"Method 5: {myDouble5}");
Running this code will output:
Method 1: 123.00
Method 2: 123.00
Method 3: 123.00
Method 4: 123.00
Method 5: 123.00
Converting an integer to a double with two decimal places in C# can be achieved through various methods like built-in methods like ToString and Math.Round, or you opt for creating a custom extension
method for reusability, C# provides the flexibility to handle the conversion efficiently.
In this C# tutorial, I have explained how to convert int to double with 2 Decimal Places in C#.
You may also like:
Bijay Kumar is a renowned software engineer, accomplished author, and distinguished Microsoft Most Valuable Professional (MVP) specializing in SharePoint. With a rich professional background spanning
over 15 years, Bijay has established himself as an authority in the field of information technology. He possesses unparalleled expertise in multiple programming languages and technologies such as
ASP.NET, ASP.NET MVC, C#.NET, and SharePoint, which has enabled him to develop innovative and cutting-edge solutions for clients across the globe. Read more… | {"url":"https://aspdotnethelp.com/convert-int-to-double-with-2-decimal-places-in-csharp/","timestamp":"2024-11-05T20:09:40Z","content_type":"text/html","content_length":"152624","record_id":"<urn:uuid:55179904-f532-406a-91fe-3d7536758387>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00342.warc.gz"} |
Utilizing the Structure of the Curvelet Transform with Compressed Sensing
Jul 24, 2021
The discrete curvelet transform decomposes an image into a set of fundamental components that are distinguished by direction and size as well as a low-frequency representation. The curvelet
representation is approximately sparse; thus, it is a useful sparsifying transformation to be used with compressed sensing. Although the curvelet transform of a natural image is sparse, the
low-frequency portion is not. This manuscript presents a method to modify the sparsifying transformation to take advantage of this fact. Instead of relying on sparsity for this low-frequency
estimate, the Nyquist-Shannon theorem specifies a square region to be collected centered on the $0$ frequency. A Basis Pursuit Denoising problem is solved to determine the missing details after
modifying the sparisfying transformation to take advantage of the known fully sampled region. Finally, by taking advantage of this structure with a redundant dictionary comprised of both the wavelet
and curvelet transforms, additional gains in quality are achieved. | {"url":"https://www.catalyzex.com/paper/utilizing-the-structure-of-the-curvelet","timestamp":"2024-11-06T17:41:49Z","content_type":"text/html","content_length":"55908","record_id":"<urn:uuid:544d1812-27e6-4a66-955b-8ada700dfc46>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00501.warc.gz"} |
A line is a one-dimensional figure, which has length but no width. A line is made of a set of points which is extended in opposite directions infinitely. It is determined by two points in a
two-dimensional plane. The two points which lie on the same line are said to be collinear points.
In geometry, there are different types of lines such as horizontal and vertical lines, parallel and perpendicular lines. These lines play an important role in the construction of different types of
polygons. For example, a square is made by four lines of the same lengths, whereas a triangle is made by joining three lines end to end.
What is a Line?
• A line can be defined as a straight set of points that extend in opposite directions
• It has no ends in both directions(infinite)
• It has no thickness
• it is one-dimensional
For More Information On Lines, Watch The Below Video:
Points, Lines and Angles
Points, lines and angles are the basics of geometry which collectively define the shapes of an object.
An example of a combination of points, lines and angles is a rectangle which has four vertices defined by a point, four sides shown by lines and four angles equal to 90 degrees.
Similarly, we can define other shapes such as rhombus, parallelogram, square, kite, cube, cuboid, etc using these three primary figures.
Also, read:
What is a Line segment?
• A line segment is part of a line which has defined endpoints
• It has a beginning point and an ending point
What is a Ray?
• A ray is a part of a line that has one endpoint (i.e. starting point) and it extends in one direction endlessly.
Types of Line
In Geometry, there are basically four types of lines. They are:
Horizontal Lines
When a line moves from left to right in a straight direction, it is a horizontal line.
Vertical Lines
When a line runs from top to bottom in a straight direction, it is a vertical line.
Parallel Lines
When two straight lines don’t meet or intersect at any point, even at infinity, then they are parallel to each other.
Suppose two lines PQ and RS are parallel then it is represented as PQ||RS.
Perpendicular Lines
When two lines meet or intersect at an angle of 90 degrees or at a right angle, then they are perpendicular to each other.
If PQ and RS are two lines which are perpendicular to each other, then it is represented as PQ ⊥ RS.
Some Other types of Lines in Maths
1.Tangent lines
The Tangent is a straight line which just touches the curve at a given point. The normal is a straight line which is perpendicular to the tangent. To calculate the equations of these lines, we shall
make use of the fact that the equation of a straight line passing through the point with coordinates (x[1], y[1]) and having gradient m is given by
We also make use of the fact that if two lines with gradients m[1] and m[2] respectively are perpendicular, then m[1]m[2] = −1.
Example: Suppose we wish to find points on the curve y(x) given by \(x^{3} – 6x^{2} + x + 3 \) where the tangents are parallel to the line y = x + 5.
Solution: If the tangents have to be parallel to the line then they must have the same gradient. The standard equation for a straight line is y = mx + c, where m is the gradient. So what we gain from
looking at this standard equation and comparing it with the straight line y = x + 5 is that the gradient, m, is equal to 1. Thus, the gradients of the tangents we are trying to find must also have
gradient 1.
We know that if we differentiate y(x) we will obtain an expression for the gradients of the tangents to y(x) and we can set this equal to 1. Differentiating and making it equal to 1 we find:
from which
\(3x^{2} – 12x = 0\)
This is a quadratic equation which we can solve by factorisation.
\(3x^{2} – 12x = 0\)
\(3x (x-4) = 0\)
\(\Rightarrow 3x = 0 \;\; or \;\; (x-4) = 0\)
\(\Rightarrow x = 0 \;\; or x = 4\)
Now having found these two values of x we can calculate the corresponding y coordinates. We do this from the equation of the curve: \(y = x^{3} – 6x^{2} + x + 3 \).
when x = 0: y = \((0)^{3} – 6(0)^{2} + 0 + 3 = 3\)
when x = 4: y = \((4)^{3} – 6(4)^{2} + 4 + 3= -25 \)
So the two points are (0, 3) and (4, −25)
These are the two points where the gradients of the tangent are equal to 1, and so where the tangents are parallel to the line that we started out with, i.e. y = x + 5.
2. Secant lines
A line in the plane is a secant line to a circle if it meets the circle in exactly two points. It is also equivalent to the average rate of change, or simply the slope between two points. The average
rate of change of a function between two points and the slope between two points are the same thing.
In the left figure above,
\(\theta =\frac{1}{2}\) (arc A C + arc B D),
while in the right figure,
\(\phi =\frac{1}{2}\) (arc R T – arc S Q),
Where arc AB denotes the angular measure of the arc AB.
Frequently Asked Questions – FAQs
How to define a line?
A line is a figure in geometry, which has only length and no width in a two-dimensional plane and extends infinitely in opposite directions.
What are the types of lines?
In geometry there are four types of lines:
Horizontal Line
Vertical lines
Parallel lines
Perpendicular lines
What is a vertical line?
A vertical line is a line drawn from the bottom to the top in a straight direction.
What is a horizontal line?
A horizontal line is a line drawn from left to right in a straight direction.
What is a line segment?
A line segment is a part of line which has two fixed endpoints. | {"url":"https://mathlake.com/Lines","timestamp":"2024-11-13T08:10:19Z","content_type":"text/html","content_length":"19249","record_id":"<urn:uuid:eb572932-36d7-466b-bdcc-af7e648aab3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00034.warc.gz"} |
A beautiful equation
Can maths be beautiful? Most people understand the beauty of a painting or a piece of music, but what about the squiggles of a mathematical equation? We call great works of art “beautiful” if they
are aesthetically pleasing or express fundamental ideas in a profound way, and mathematicians feel the same way about particularly elegant proofs. Many say the most beautiful result of all is Euler’s
Irrational and imaginary
Let’s start with the numbers. You should be familiar with
The number e is also one you might have met at school. It crops up in many different areas of maths, science, and business, because the function f(x) = e^x has the useful property of being its own
derivative. In other words,
f'(x) = f(x) = e^x.
This comes in handy for modelling processes like radioactive decay or interest payments, where the rate of change is proportional to the value you start with. Like
Numbers that never end are pretty strange, but i is even weirder. Multiplies of i are called “imaginary numbers”, but they’re no less real than the “ordinary” numbers. Mathematicians first used them
to solve a problem with square roots, but they crop up in everything from electrical engineering to computer graphics.
To understand why we need imaginary numbers, work out the square root of 4. The obvious answer is 2, but -2 is also a root of 4 because multiplying two negative numbers gives a positive. But what’s
the square root of -4? It can’t be 2 or -2, because squaring both of these gives 4. To solve the problem, we define the square root of -1 as i, so i^2 = -1. This makes the square root of -4 equal to
2i, the square root of -9 equal to 3i, and so on. A number with both “real” and “imaginary” parts is called a complex number, and written as
a + bi.
By now you might be wondering what any of this has to do with “beauty”. These numbers are clearly useful, but it’s their combination that makes them beautiful. Rearranging Euler’s equation, we get
Euler’s explanation
We know that a calculation like 2^3 can be written as 2 × 2 × 2, but
The Taylor series of function turns a simple f(x) into an infinite sum. Any function that satisfies certain conditions can be expressed as a Taylor series, including
Here n! means “n factorial”, and n! = n × (n-1) × (n-2) × … 2 × 1. If you don’t believe this infinite sum adds up to e^x, try putting x = 1. You can work out each term on a calculator, and you’ll
find that the more terms you add, the closer the result is to 2.71828…
Euler realised that he could use the same formula for e^ix to get a similar result. Powers of i follow a certain pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. When you reach i^5 the pattern
repeats, because i^5 = i^1 × i^4 = i. Using this pattern, we can write:
And by grouping real and imaginary parts we get:
It turns out that the two groups form their own Taylor series. The first group is the Taylor series for cos(x), while the second group is the Taylor series for sin(x), which has been multiplied by i.
This must mean that:
e^ix = cos(x) + i × sin(x).
We’ve cracked it. Now all that remains is to substitute
since cos(
So finally:
Euler’s equation expresses a universal truth, valid in any language or culture. It contains five of the most important numbers in maths: 0, 1, e, i, and | {"url":"https://www.mathscareers.org.uk/beautiful-equation/","timestamp":"2024-11-08T09:05:20Z","content_type":"text/html","content_length":"107361","record_id":"<urn:uuid:369a3e80-d4ff-4f8f-adf4-27d0f6d20d1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00653.warc.gz"} |
You Should Not Always Have Known Better: Understand and Avoid the Hindsight Selection Bias in Probabilistic Forecast Evaluation
This blog post on forecast evaluation discusses common pitfalls and challenges that arise when evaluating probabilistic forecasts. Our aim is to equip Blue Yonder’s customers and anyone else
interested with the knowledge required to setup and interpret evaluation metrics for reliably judging forecast quality.
The hindsight selection bias arises when probabilistic forecast predictions and observed actuals are not properly grouped when evaluating forecast accuracy across sales frequencies. On the one hand,
the hindsight selection bias is an insidious trap that nudges you towards wrong conclusions about the bias of a given probabilistic forecast — in the worst case, letting you choose a worse model over
a better one. On the other hand, its resolution and explanation touch statistical foundations such as sample representativity, probabilistic forecasting, conditional probabilities, regression to the
mean and Bayes’ rule. Moreover, it makes us reflect on what we intuitively expect from a forecast, and why that is not always reasonable.
Forecasts can concern discrete categories — will there by a thunderstorm tomorrow? — or continuous quantities — what will be the maximum temperature tomorrow? We focus here on a hybrid case: Discrete
quantities, which could be, for example, the number of T-shirts that are sold on some day. Such sales number is discrete, it could be 0, 1, 2, 13 or 56; but certainly not -8.5 or 3.4. Our forecast is
probabilistic, do not pretend to exactly know how many T-shirts will be sold. A realistic, yet ambitiously narrow (i.e. precise) probability distribution is the Poisson distribution. We therefore
assume that our forecast produces the Poisson-rate that we believe drives the actual sales process.
A rather mediocre forecast?
Assume the forecast was issued, the true sales have been collected, and the forecast is evaluated via such table:
Data is grouped by the observed sales frequency: We bucket all days into groups in which the T-shirt happened to be sold few (0, 1, or 2), intermediate (3 to 10) or many (more than 10) times. At
first sight, this table shouts unambiguously “slow-sellers are over-forecasted, fast-sellers are under-forecasted.” The forecast is so obviously deeply flawed, we would immediately jump to fixing it,
or wouldn’t we?
In reality, and possibly surprisingly, everything is fine. Yes, slow-sellers are indeed over-forecasted and fast-sellers are under-forecasted, but the forecast behaves just as it should. It is our
expectation — that the columns “mean observed sales” and “mean prediction” ought to be the same — that is flawed. We deal with a psychological problem, with our unrealistic expectation, and not with
a bad forecast! A probabilistic forecast never promised nor will ever fulfill that for each possible group of outcomes, the mean forecast matches the mean outcome.
Let’s explore why that is the case, how to resolve this conundrum satisfactorily, and how to avoid similar biases.
What do we actually ask for?
Let us step back and express in words what the table reveals. The data are bucketed using the actually observed sales, that is, we filter, or condition, the predictions and the observations on the
observations being in a certain range (slow, mid or fast-sellers). The first row contains all days on which the T-shirt was sold 0, 1 or 2 times, its central column provides us with:
i.e. the mean of the observations in the bucket into which we grouped all observations that are 2, 1 or 0 — definitely a number between 0 and 2, which happens to be 0.804. The right-hand column
contains the expected mean prediction for the same bucket of observations,
i.e. for all the observations that are 2 or less, we take the corresponding prediction, and compute the mean over all these predictions.
A priori, there is no reason why the first and the second expressions should take the same value — but we intuitively would like them to: Expecting the mean prediction to equate the mean observation
does not seem too much to ask, does it?
Forward-looking forecast, backward-looking hindsight
Consistent with their etymology, forecasts are forward-looking, and provide us with the probabilities to observe future outcomes,
which is the conditional probability to observe an outcome k, given that the predicted rate is x. Since we have a conditional probability, we consider the probability distribution for the
observations assuming the prediction assumed the value x. For an unbiased forecast, the expectation value of the observation conditioned on a prediction x, that is, the mean observation under the
assumption of a prediction of value x, is:
That’s what any unbiased forecast promises: Grouping all predictions of the same value x, the mean of the resulting observations should approach this very value x. While the distribution could assume
many different shapes, this property is essential.
Let’s have a look back at the table: What we do in the left column is not grouping/conditioning by prediction, but by outcome. The right-hand column thus asks the backward-looking “what has been our
mean prediction, given a certain outcome k” instead of the forward-looking “what will be the mean outcome, given our prediction x.”
To express the backward-looking statement in terms of the forward-looking one, we apply Bayes’ rule,
The backward- and forward-looking questions are different, and so are their answers: Other terms appear, P (prediction = x) and P (observation = k), the unconditional probabilities for a prediction
and outcome. Consequently, the expectation value of the mean prediction, given a certain outcome, becomes:
Minimalistic example
What value does E (prediction | observation = m) assume? Why would it not just simplify to the observation m?
In the vast majority of cases, it holds E (prediction | observation = m) ≠ m. Let’s see why!
Consider a T-shirt that sells equally well every day, following a Poisson distribution with rate 5. The very same predicted rate, 5, applies for every day. The outcome, however, varies. Clearly, 5 is
an overestimate for outcomes 4 and lower, and an underestimate for outcomes 6 and higher. If we group again by outcomes, we encounter:
Again, from this table we conclude that the slow-selling days were over-predicted and the fast-selling days were under-predicted, and were indeed. It holds for every observation E (prediction |
observation = m) = 5, since the prediction is always 5.
The forecast is still “perfect” — the outcomes behave exactly as predicted, they follow the Poisson distribution with rate 5. The impression of under- and over-forecasting is purely a result of the
data selection: By selecting the outcomes above 5, we keep those outcomes that are above the prediction 5 and were under-predicted; by selecting the outcomes below 5, we keep the events below the
prediction 5, which were overpredicted. For a probabilistic forecast, it is unavoidable that some outcomes were under-forecasted and some were over-forecasted. By expecting the forecast to be
unbiased, we expect the under-forecasting and over-forecasting to balanced for a given prediction m. What we cannot expect is that when we actively select the over-forecasted or under-forecasted
observations, that these be not over- or under-forecasted, respectively!
In a realistic situation, we will not deal with a forecast that assumes the very same value for every day, but the prediction itself will vary. Still, selecting “rather large” or “rather small”
outcomes amounts to keeping the under- or over-forecasted events in the buckets. Therefore, we have E (prediction | observation = m) ≠ m in general. More precisely, whenever m is so large that
selecting it is tantamount to selecting under-forecasted events, we’ll have E (prediction | observation = m) < m; when m is sufficiently small that selecting it is tantamount to selecting
over-forecasted events, E (prediction | observation = m) > m.
Deterministic forecasts — you should have known, always!
Why is this puzzling to us? Why do we feel uncomfortable with that discrepancy between mean observation and mean forecast? Our intuition hinges on the equality of prediction and observation that
characterizes deterministic forecasts. In the language of probabilities, a deterministic forecast expresses: P (observation = prediction) = 1 and P (observation ≠ prediction) = 0
The forecaster believes that the observation will exactly match her prediction, i.e. predicted and observed values coincide with probability 1 (or 100%), while all other outcomes are deemed
impossible. That’s a self-confident, not to say, a bold statement. Expressed via conditional probabilities, we can summarize:
in words, whenever we predict to sell k pieces (the condition after the vertical bar), we will sell k pieces. Since determinism does not only imply that every time we predict k we observe k, but also
that every observation k was correctly predicted ex ante to be k, we have:
The determinism makes the distinction between the backward-looking and the forward-looking questions obsolete. With a deterministic forecast, we don’t learn anything new by observing the outcome (we
already knew!), and we would not update our belief (which was already correct).
For such a deterministic forecast, for which all appearing probability distributions collapse to a peak of 100% at the one-and-only-possible outcome, no hindsight selection bias occurs — we pretend
to have known exactly beforehand, so we should have known — always and under all circumstances. If the measurement says otherwise, your “deterministic” forecast is wrong.
Every serious forecast is probabilistic
Probabilistic forecasts make weaker statements than deterministic ones, and for probabilistic forecasts we must abandon the idea that each outcome m was predicted to be m on average — deterministic
forecasts thus seem very attractive. But is it realistic to predict daily T-shirt-sales deterministically? Let’s assume you were able to do so and predict tomorrow’s T-shirt-sales to be 5. That means
that you can name five people that will, no matter what happens (accident, illness, thunderstorm, sudden change of mind…) buy a red T-shirt tomorrow. How can we expect to reach such a level of
certainty? Were you ever that certain that you would buy a red T-shirt on the next day? Even if five friends promised that they would buy one T-shirt tomorrow under all circumstances — how could you
exclude that someone else, within all other potential customers, would buy a T-shirt, too? Apart from certain very idiosyncratic edge-cases (very few customers, stock level is much smaller than true
demand), predicting the exact number of sales of an item deterministically is out of question. Uncertainty can only be tamed up to a certain degree, and any realistic forecast is probabilistic.
Evaluation hygiene
There is an alternative way to refute table 1: By setting up the table, we ask a statistical question, namely whether the forecast is biased or not, and into which direction (let’s ignore the
question of statistical significance for the moment and assume that every signal we see is statistically significant). Just like any statistical analysis, a forecast analysis can suffer from biases.
The way we have selected by the outcomes is a prime example for the selection bias: The events in the group “slow sellers,” “mid sellers,” “fast sellers” are not representative for the entire set of
predictions and observations, but we bucketed them into the under- and over-forecasted ones. Also, we used what is called “future information” in the forecast evaluation: the buckets into which we
grouped predictions and observations are not yet defined at the moment of the prediction, but they are established ex post. Thus, setting up the table as we did violates basic principles for
statistical analyses.
Regression to the mean
The phenomenon that we just encountered — extreme events were not predicted to be as extreme as they turned out to be — directly relates to the “regression to the mean,” a statistical phenomenon for
which we don’t even need a forecast: Suppose you observe a time series of sales of a product that exhibits no seasonality or other time-dependent pattern. When, on a given day, the observed sales are
larger than the mean sales, we can be quite sure that the next day’s observation will be smaller than today’s, and vice versa. Again, by selecting a very large or very small value, due to the
probabilistic nature of the process, we are likely to select a positive or negative random fluctuation, and the sales will eventually “regress to the mean.” Psychologically, we are prone to causally
attributing that regression to the mean — a purely statistical phenomenon — to some active intervention.
Resolution: Group by prediction, not by outcome. Remain vigilant against selection biases.
What is the way out of this conundrum? By grouping by outcomes, we are selecting “rather large” or “rather small” values with respect to their forecast — we are not obtaining a representative sample,
but a biased one. This selection bias leads to buckets that contain outcomes that are naturally “rather under-forecasted” or “rather over-forecasted,” respectively. We suffer from the hindsight
selection bias if we believe that mean prediction and mean observation should be the same within the “slow,” “mid” and “fast” moving items. We must live with and accept the discrepancy between the
two columns. Luckily, we can use Bayes theorem to obtain the realistic expectation value. One solution is thus another column in the table that contains the theoretically expected value of the mean
prediction per bucket, which can be held against the actual mean prediction in that bucket. That is, we can quantify and theoretically reproduce the hindsight selection bias and see whether the
aggregated data matches the theoretical expectation.
A much simpler solution, however, is to ask different questions to the data, namely questions that are aligned with what the forecast promises us. This allows us to directly check whether these
promises are fulfilled, or not: Instead of grouping by outcome buckets, we group by prediction buckets, i.e. by predicted slow-, mid- and fast-sellers. Here, we can check whether the forecast’s
promise (the mean sales given a certain prediction matches that prediction) is fulfilled. For our example, we obtain this table:
Accounting for the total number of measurements, a test of statistical significance would be negative, i.e., show no significant different between the mean observed sales and the mean prediction. We
would conclude that our forecast is not only globally unbiased, but also unbiased per prediction stratum.
In general, you can evaluate a forecast by filtering on any information that is known at prediction time, and the forecast should be unbiased in all tests. However, the filter is not allowed to
contain future information such as the random fluctuations that occur in the observations, on which nature decides only in the future of the prediction point in time.
What should you take away if you made it to this point? (1) When you select by outcome, you don’t have a representative sample. (2) Be skeptic towards your own expectations — very reasonably-looking
intuitive expectations turn out to be flawed. (3) Make your expectations explicit and test them against well-understood cases.
Related posts | {"url":"https://blog.blueyonder.com/zh-hant/you-should-not-always-have-known-better-understand-and-avoid-the-hindsight-selection-bias-in-probabilistic-forecast-evaluation/","timestamp":"2024-11-11T05:05:51Z","content_type":"text/html","content_length":"121736","record_id":"<urn:uuid:58e6b7fb-cbb1-4df0-8b7b-5398ca3aef64>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00716.warc.gz"} |
V / I
30 Aug 2024
V / I & Analysis of variables
Equation: V / I
Variable: V
Impact of Supply Voltage on Resistance Function
X-Axis: -114770.027160901to115230.0273170518
Y-Axis: Resistance Function
Impact of Supply Voltage on Resistance Function
In the field of electrical engineering, understanding the relationship between voltage and resistance is crucial for designing efficient and safe electronic systems. The Ohm’s Law states that voltage
(V) is equal to the product of current (I) and resistance (R), i.e., V = IR. However, when dealing with variable supply voltages, it becomes essential to explore how changes in voltage affect the
system’s resistance characteristics.
The Equation: V / I
Let us consider a simple resistor circuit where we want to determine the relationship between the supply voltage (V) and the current flowing through it. According to Ohm’s Law, we have:
I = V/R
Now, let us isolate R by dividing both sides of the equation with I:
R = V/I
This equation reveals that resistance (R) is directly proportional to the supply voltage (V) and inversely proportional to the current (I). In other words, as the supply voltage increases, the
resistance also increases proportionally.
Variable Supply Voltage: Implications
When the supply voltage varies over time or across different operating conditions, its impact on the resistance function must be taken into account. Let us analyze how changes in V affect R:
• Increasing V: As the supply voltage increases, the resistance (R) also increases, resulting in reduced current flow through the circuit.
• Decreasing V: Conversely, when the supply voltage decreases, the resistance (R) decreases as well, leading to increased current flow.
Real-World Applications
Understanding the impact of supply voltage on resistance is crucial for various applications:
1. Power System Design: In high-power systems, maintaining optimal voltage levels is critical to prevent overheating and ensuring reliable operation.
2. Electronic Device Design: In consumer electronics, designers must consider how changing voltage conditions affect device performance and lifespan.
3. Industrial Automation: In industrial control systems, precise voltage regulation is essential for maintaining desired resistance values in motor circuits.
In conclusion, the equation V / I highlights the inverse relationship between supply voltage and current flow through a resistor circuit. As the supply voltage varies, so does its impact on
resistance function. Understanding this dynamic relationship enables engineers to optimize system design, ensure reliable operation, and prevent potential failures. By considering the implications of
variable supply voltages, designers can create more efficient, safe, and effective electronic systems.
Recommendations for Further Study
For further exploration:
• Investigate how the impact of temperature on resistance function is affected by changes in supply voltage.
• Examine the implications of non-linear resistance characteristics on system design under varying voltage conditions.
Related topics
Academic Chapters on the topic
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DNK 201 - Dynamics
Course Objectives
1. To enable students to construct idealized (particle and rigid body) dynamical models and predict model response to applied forces using Newtonian Mechanics.
2. To teach description and prediction of motion experienced by inertial and non-inertial observers and central force motion.
3. To teach the basic principles of 2-D rigid body motion.
4. To introduce the equations of motion of 3-D rigid bodies.
5. To teach a simple vibration analysis of rigid body.
Course Description
Definitions and principal axioms. Kinematics of particiles. Linear, plane and general motions. Relative motion. Kinetics of particles. Newton’s laws. Impuls and momentum principle. Work and Energy.
Motion with resistance. Central-force Motion systems of particles. Collision. Variablesmass. Kinematics of Rigid Bodies. Kinetics of Rigid Bodies. Work and Energy, Impulse and momentum. Fixed-Axis
rotation of rigit body. Plane motion of rigid body. Vibration of rigid body . Relative motion.
Course Coordinator
Hacı İbrahim Keser
Course Language | {"url":"https://ninova.itu.edu.tr/en/courses/faculty-of-civil-engineering/1856/dnk-201/","timestamp":"2024-11-10T04:35:31Z","content_type":"application/xhtml+xml","content_length":"8018","record_id":"<urn:uuid:995d0a5f-85c7-4619-88f6-774f8394ea25>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00835.warc.gz"} |
Division by Zero and the Derivative
(An archive question of the week)
The indeterminate nature of 0/0, which we looked at last time, is an essential part of the derivative (in calculus): every derivative that exists is a limit of that form! So it is a good idea to
think about how these ideas relate.
Here is a question from 2007:
Derivative Definition and Division by Zero
Hi, I'm having a bit of trouble with the concept behind first principles. Even though one aims to cancel out the denominator, which is technically zero, proofs that show that 1 = 2 may perform the same act (canceling out a denominator that was 0) and these proofs are considered invalid. Why is First Principles considered valid then?
My thoughts are that perhaps the solution lies in the fact that in First Principles, the denominator approaches 0 as opposed to actually being 0, although I figured these two things are more or less equivalent since usually at the end of a First Principles problem one substitutes in 'h' as 0, even though it is stated that h is only approaching 0. Any thoughts?
Adrian never actually stated what he was doing, probably because in his context, “first principles” has always been used in the phrase “finding a derivative from first principles”: that is, directly
applying the definition of the derivative to differentiate a function. We have seen enough such questions to know what he meant; but “first principles” can apply to the fundamental concepts or basic
definitions in any field, so it really should have been stated.
Adrian is comparing the process of finding the limit in a derivative to the sort of false proof explained here in our FAQ (which I should have a post about soon, because that is an interesting
subject of its own):
False Proofs, Classic Fallacies
A typical false proof using division by zero looks like this (from the FAQ):
a = b
a^2 = ab
a^2 – b^2 = ab-b^2
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2 = 1
Here we started with the assumption that two variables a and b are equal, and derived from that the “fact” that 2 = 1; so something must be wrong. The puzzle is to figure out what. The error occurred
on the fifth line, when we divided both sides by (a – b), which, on our assumption, is zero. Dividing both sides of an equation by 0 is invalid, because the result is undefined. What is actually
being done is to remove a common factor from both sides; but we can’t really conclude from \(0x = 0y\) that \(x = y\).
The key idea there is that if you divide by zero in a proof, the result can’t be trusted. So why can it, here?
Doctor Rick took the question:
In order to make sense of what you've written, I have to add some words. You're talking about CALCULATING A DERIVATIVE from first principles (that is, the definition of a derivative as a limit), aren't you?
I just used a word that makes all the difference: the derivative is defined as a LIMIT. We never actually divide by zero; rather, we divide by a very small number, and we find the limit of that quotient as h approaches zero.
This is more or less what Adrian had in mind when he contrasted approaching zero to being zero; but because a typical last step in finding the limit is to actually replace a variable with zero, he is
still unsure. So, what is the difference?
For example, let's find the derivative of f(x) = 2x^2 at x=3:
2(x+h)^2 - 2x^2 4hx + 2h^2
lim --------------- = lim ---------- = lim 4x + 2h = 4x
h->0 h h->0 h h->0
The quantity whose limit we are taking is undefined at h = 0, for just the reason you say. But the LIMIT at h = 0 exists.
If we just plugged in 0 for h at the start, we would get 0/0, which does not have a single defined value; this is why the expression 0/0 is undefined, and we can’t actually let h equal zero in that
expression. But here 0/0 is only a form: a shorthand notation for the fact that the function is a quotient of functions, each of which is approaching zero. In such a case, we have to find an
alternative to evaluating it as it stands. As Doctor Rick has said, it’s a sign saying “bridge out — road closed ahead”, that forces us to take a detour to get to our goal.
In this case our detour is to simplify the expression, dividing numerator and denominator by h, resulting in an expression that can be evaluated at h = 0. What has happened is that we have found a
new function that is defined at h = 0, but is equal to the original function for all other values, so that it has the same limit. But this function is continuous, so that its limit is its value at
that point, and we can just substitute.
For a slightly more complex example of proving a derivative from the definition, see
Proof of Derivative for Function f(x) = ax^n
The key is that the simplification is valid as long as h is not zero; and the limit relates only to values of h other than zero:
Maybe it will help you if you go back to the first-principles definition of a limit: (this is from my old calculus textbook)
Suppose f is a function defined for values of x
near a. (The domain of f need not include a,
although it may.) We say that
L is the limit of f(x) as x approaches a,
provided that, for any epsilon > 0 there
corresponds a deleted neighborhood N of a
such that
L - epsilon < f(x) < L + epsilon
whenever x is in N and in the domain of f.
Notice that the domain of f need not include the point at which the limit is to be taken, and that the condition only needs to hold when x is in the domain of f. This shows that a function, such as our function of h, can have a limit at a point (h=0 in our case) where the function itself is undefined. There is no contradiction there.
Adrian wanted to be sure he understood clearly, so he wrote back:
Thanks Doctor Rick, it is a bit clearer though I still have one small problem. Sticking with the example you gave, you eventually arrive at
lim 4x + 2h = 4x.
What is confusing is that before reaching this step, we were unable to simply sub in h as 0 and I was under the impression that this was because h didn't really equal zero, but rather a number extremely close to zero. However, once we have eliminated the problem of getting 0/0 if we were to sub in h as zero, it seems like it's perfectly alright to treat h as if it is exactly zero, as we can sub it into the equation as zero in order to arrive at 4x. Is it possible to think of 'h' as being BOTH extremely close to zero and exactly zero, and therefore limits to be kind of a way of sidestepping the rule of not being able to divide by zero, or am I totally missing the concept?
Doctor Rick responded by saying something like what I said above (which is worth saying more than once!).
What's going on here is that we have replaced a function f(h) that has no value at h=0, with a CONTINUOUS function g(h) = 4x + 2h. The two functions f(h) and g(h) are equal for every h EXCEPT h=0; therefore they have the same limit at h=0.
The limit at point a of a function g(x) that is CONTINUOUS at a is the value g(a). (That's the definition of a continuous function.) Therefore we easily find the limit of 4x + 2h as h approaches 0, namely 4x; and this must also be the limit of f(h) as h approaches 0.
The original function f had a “hole”, a point where it was not defined. The new function g is exactly the same function with the hole “filled in”. And since the limit is not affected by the hole, it
is the same for both.
Does that help? We are not sidestepping the "rule" (that division by zero is undefined) in the sense of violating it in any way. We are finding the LIMIT of a function with h in the denominator as h approaches 0; the limit, by definition, never invokes division by EXACTLY zero. We evaluate the limit by finding a function that has the same values for every x NEAR zero (but not EXACTLY zero); this function does not involve division by h, so there is nothing wrong with evaluating it at exactly zero.
You could say we are "getting around" division by zero in the literal sense that we are working in the immediate neighborhood of zero without going exactly there. It may be tricky, but it's perfectly legal.
A good case could be made that this “sidestepping” is exactly why limits were invented. In the beginning of calculus, one had to simultaneously think of our h as being zero and yet not being zero —
very small, but still legal to divide by. Limits were the way to allow us to make sense when we talk about derivatives.
2 thoughts on “Division by Zero and the Derivative”
1. With the function y = x^2 consider both x+h and x-h
Then the derivative is {(x+h)^2 – (x-h)^2} / 2h = 4xh / 2h = 2x as the limit
Interestingly, with this function, whatever the value of ‘h’ (bar zero) the slope of the line is always 2x
2. Alternatively consider the result of x+h and x-h taken separately, giving derivatives of 2x+h and 2x-h. In the limit these approach x from both ‘above and below’, which I find satisfying.
If valid, of course!
The first comment is an application of an alternative definition for the derivative, discussed here:
It isn’t always identical to the derivative, but it happens to be very convenient in this specific case, as you end up taking the limit of a constant function. It doesn’t, however, really
bypass the issue under discussion, as you still had to cancel h, and before that you did have the form 0/0.
The symmetry you observe in the left- and right-hand difference quotients is equivalent to your observation about the constancy of the symmetric difference quotient, as the latter is the
average of the former, which is always 2x.
Leave a Comment
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Work, Energy, and Power - Lesson 1 - Basic Terminology and Concepts
The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very
quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a
trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work
in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than
the rock climber.
Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.
Power = Work / time
P = W / t
The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a
Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.
Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other
objects. Thus, the power of a machine is the work/time ratio for that particular
A person is also a machine that has a power rating. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the
same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student's personal power. Despite the diagonal motion
along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the
student is equal to the force that does the work on the student and the height of the staircase is the upward displacement. power rating. It can be assumed that Ben must apply an 800-Newton downward
force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben's body with just enough force to lift his body up the stairs. It can also be assumed that the angle between
the force of the stairs on Ben and Ben's displacement is 0 degrees. With these two approximations, Ben's power rating could be determined as shown below.
Ben's power rating is 871 Watts. He is quite a horse.
Another Formula for Power
The expression for power is work/time. And since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Since the expression for
velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below.
This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity). A powerful car engine is strong and fast. A powerful piece of farm equipment is strong
and fast. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A machine that is strong enough to apply a big force to cause a displacement in a small
mount of time (i.e., a big velocity) is a powerful machine.
Check Your Understanding
Use your understanding of work and power to answer the following questions. When finished, click the button to view the answers.
2. During a physics lab, Jack and Jill ran up a hill. Jack is twice as massive as Jill; yet Jill ascends the same distance in half the time. Who did the most work? ______________ Who delivered the
most power? ______________ Explain your answers.
chin-up, a physics student lifts her 42.0-kg body a distance of 0.25 meters in 2 seconds. What is the power delivered by the student's biceps?
5. Your household's monthly electric bill is often expressed in kilowatt-hours. One kilowatt-hour is the amount of energy delivered by the flow of l kilowatt of electricity for one hour. Use
conversion factors to show how many joules of energy you get when you buy 1 kilowatt-hour of electricity.
6. An escalator is used to move 20 passengers every minute from the first floor of a department store to the second. The second floor is located 5.20 meters above the first floor. The average
passenger's mass is 54.9 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time. | {"url":"https://www.physicsclassroom.com/Class/energy/u5l1e.html","timestamp":"2024-11-13T09:19:29Z","content_type":"application/xhtml+xml","content_length":"219494","record_id":"<urn:uuid:e9060d66-da46-42b4-8521-0aa549c2a7bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00252.warc.gz"} |
Effective Permeability | Ansys Innovation Courses
This lesson covers the concept of flow through porous media, focusing on the effective permeability in a stratified porous medium with non-uniform permeability. It delves into the application of mass
continuity in Darcy's Law and the calculation of pressure equations and velocity profiles. The lesson also discusses the theory behind combining permeabilities in series or parallel, and how to
calculate the effective permeability in such cases. It further explores the concept of continuous permeability variation in radial flow and the use of stream function and potential function in porous
media. For instance, if you have a core where up to a certain length the permeability was k1 and beyond that length, the permeability is k2, the lesson explains how to calculate the effective
Video Highlights
01:03 - Combining permeabilities in series or parallel
02:03 - Calculation of effective permeability
14:37 - Concept of continuous permeability variation in radial flow
25:00 - Use of stream function and potential function in porous media
Key Takeaways
- The effective permeability in a stratified porous medium with non-uniform permeability can be calculated using the principles of mass continuity in Darcy's Law.
- When combining permeabilities in series or parallel, it's important to consider the lengths over which each permeability is valid.
- In cases of continuous permeability variation in radial flow, the permeability is a function of the radius and the pressure profile can be calculated using the given equation.
- Understanding the concepts of stream function, potential function, streamlines, and potential lines is crucial when dealing with flow through porous media. | {"url":"https://innovationspace.ansys.com/courses/courses/mass-continuity-in-porous-media-i/lessons/effective-permeability-lesson-6/","timestamp":"2024-11-15T04:41:21Z","content_type":"text/html","content_length":"176777","record_id":"<urn:uuid:7733efc0-f9e5-43cc-815e-681380c30d5d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00701.warc.gz"} |
Nerd Club
Inbetween hacking CNC drilling machines, and almost as a respite from it, we've been busy preparing some guitar effects to show off at the up-and-coming
Brighton Mini Maker Faire
As well as a variation on the earlier
Fuzz Factory pedal
(made during a BuildBrighton workshop) we've been trying one of these inductorless wah pedal effects -
The final board size is really tiny. This is a perfect candidate for use in some kind of onboard electronics as well as a traditional pedal. Here it is alongside one of our Fuzz Factory boards (wah
above, fuzz below):
When we've got it working, the idea is to replace the 100K linear pot on the wah effect (we'll leave the optional volume control out) with a digital potentiometer and use a microcontroller to set the
wah intensity. This should allow us a couple of different options for activating the wah -
• A wii controller would allow us to rock the guitar up and down, using the actual guitar body as the pedal rocker.
• An ultrasound range finder would allow us to change the wah effect by strumming at different positions along the guitar body
In fact, once we've managed to hook up a microcontroller to control the wah effect, the possibilities are endless!
This is one of the most exciting blog posts for a while. After working on the CNC drilling machine as part of the BuildBrighton £50 CNC challenge for a few weeks, we're actually at the point of
putting software, hardware, nc drill parsing and motor controller all together and actually trying to cut a PCB.
Actually, we don't have the drill part running yet - but this test shows actual movement, and we've used a laser dot in place of a drill head. But it shows a (sort-of) working CNC machine....
Instead of jumping in at the deep end and trying to draw a complex PCB (the printed pattern on the paper) we started off with a simple square. But as you can see from the video, we deliberately drew
(and mounted) the square on an angle, to simulate mounting a PCB on the cutting bed on a wonky angle.
The software takes care of the rotation and follows the dots.
The first few seconds of the video show the machine being calibrated - the software prompts you to place the cutting head over a hole, record this location, then move the head to a second hole. This
is what you see as the head travels diagonally across the board at the start (and the slight delay in the finer movement is us changing some parameters on the PC to reduce the jog step size).
The software then works out the cutting path (in this case, a simple down-across-up type pattern) and sets the motors spinning!
It's interesting to note that the cutting head doesn't necessarily follow the "lines" between the dots (if the dots were on the corners of a square for example), since it is a point-to-point machine,
rather than a line follower. We'll try to demonstrate this more clearly in a later post.
But for now, sit back and enjoy our first CNC test.
It's not bad. It's not perfect - we need to take out any backlash in the gears (the motors themselves have quite a bit of "slop" on the spindle because of the internal gearing) to get greater
accuracy but as an initial test - and particularly the handling of skewed boards - we're quite pleased with progress so far!
Next time we hope to actually cut (or maybe just draw felt-pen dots on) something.....
After an earlier success with our CNC motor testing, we decided to wind down for the night by making a breakout board for our little digipots from Farnell.
Ideally we were after some 10k digital pots that could handle up to 9v (to try out some digital effects pedals) but the nearest we could find were 20k, up to 16v (most digital potentiometers run up
to 5v and we've heard some bad things about using 5v digitpots with a 9v supply). 20k instead of 10k is fine, we'll just use half of the available range (so we'll have 128 steps instead of 256 to
play with). All this would be fine, except the chips themselves are teeny-tiny little things! So we had to make up some breakout boards:
Soic to Dip 14pin
These we easy enough to toner transfer (using a laminator, not a household iron) but a bugger to etch! The traces are a tiny 0.25mm so over-etching/under-cutting was a bit of a problem. From four
boards (we only made up half of the boards in the design) only one was good enough to use without any doctoring!
Using solder paste and a fine-tipped soldering iron (and some blue-tac to hold everything in place while soldering) we got this little chip stuck down and tested in a few minutes. Sadly it was then
too late for any testing, so we'll just have to get this hooked up and tried out some time soon...
There's an often held belief that you can't do CNCs without a parallel port on your PC and cleverly timed move instructions. That may have been the case back in the day, but we've had the nineties
guys, USB is all the rage!
So we're building a USB CNC controller board.
We're quite lucky that these cheap little 5V 28BYJ48 steppers have already been stepped down. They're supposed to be 1/64 but inside they're geared down again. There are loads of places all over the
internet which say that they're stepped down again by 1/32 - meaning you need 64 * 32 = 2048 steps for one complete revolution.
If that's the case, we should be able to get pretty precise movement, even with a massive gear/cog riding on the shaft, and without having to bother with the complexities of micro-stepping. There's
only one way to be sure - and that's spin one around and count the steps!
In-keeping with our NC drill software, we're looking to build a USB controller which we can give a number of steps and have the motor(s) play out those steps. We've no idea at the minute how many
steps we may need to move up to (depending on how many steps per revolution these motors actually need) so we've allowed for a 4-byte value to be sent to our trusty 18F2455 PIC microcontroller.
We're using (as ever) a generic HID device interface and sending data in 8 byte packets.
• The first byte (byte zero) is our "command byte".
• If the value is one, it's a command to set the x motor step count
• If the value is two, it's a command to set the y motor step count
• The second, third, fourth and fifth bytes make up our 4-byte value (0-2,147,483,647)
• The sixth byte (byte 5) is a direction - one is anit-clockwise, zero (or any other value) clockwise.
After sending the x-axis step count (or the y) the controller board stops all motor activity (since if the motors are spinning when new values come in, the x- and y- axis will go out of alignment
with each other, as the earlier axis will be ahead of the later one).
Only once the command byte 254 is sent do the motors actually spin up.
For as long as the x/y step count has a value greater than zero, the motor(s) are given a signal to move them onto the next step. The step-count value is decreased by one each time one of the axis
motors steps. Once both motors have a step-count value of zero, a flag is set to tell the PC that the motors have stopped spinning and the head is now in it's correct position.
Here's a video of some early testing:
What's happening here? Thanks to the autofocus on our camera-phone it's not too clear - but if you squint and stand back from the monitor you might see:
Firstly, the command byte (7th byte) is given the value 1 (set x motor step count), along with the second byte (from the right) set to 16. Since our x count is a 4-byte value, we're setting it to
16*256 = 4096.
We repeat these values with the command byte set to value 2 (set y motor step count) then clear the buffer and send the command value 254 to get the motors spinning.
Giving our control board a value of 4096 makes the motor complete one full rotation.
So there we have it. Our stepper motor has a 1/64 step angle, geared down, not by 1/32 as some other forums suggest, but a full 1/64 again. 64 * 64 = 4096 so this is the number of steps required for
a full rotation.
The video then skips back to a blurry laptop screen, where we enter the same values, but this time setting byte 4 to 1. This is the direction byte. When this is set to one, the motor spins in the
opposite direction.
All in all, we've had quite a successful evening - we've got both axis motors spinning from a custom-built PIC-based USB (HID device) board and some software which we can talk to the board with and
get predictable results. Now we just need to remember how to work with Timer1 to create a 20m/s interrupt on the PIC and we can use this to send servo commands for the z-axis (drill up, drill down
and motor speed).
Tomorrow is another BuildBrighton open evening.
There's even a slim chance that after the beers and pizza, we might actually get something working......
So far we've been working with screen co-ordinates, with the origin (0,0) at the top left. But when reading/parsing NC drill files, the origin seems to follow cartesian co-ordinates with the origin
in the bottom left (like you get when drawing a graph on a piece of paper).
So a quick hack later and our software now starts with the dot nearest the bottom-left-hand corner of the board
It then prompts to move the drilling head to the top-right-hand corner hole
And calculates a cutting path from the current hole (the top-right) back, not necessarily to the origin, but in a way that passes through each hole.
When the cutting head reaches point A, the general rule to follow is "move to the nearest undrilled dot". If we were doing this job manually, we'd probably go from point A to point E, continue
working up the board, then move to the left and work our way back down again.
But that doesn't actually follow the "move to the nearest dot" rule. From point A, the nearest dot is point B, so that's the path that's taken. Now at point B, point E is still further away than one
of the other, non-drilled, points. So the head moves to point C instead. When it gets to point D, however, the nearest undrilled hole is point E, so the cutting head moves there and continues in a
more predictable pattern.
We could probably "iron out" these peculiar movement patterns by looking ahead more than one hole at a time, but that's an awful lot of work for something that we've not even tested yet! For now,
we'll live with a few quirks until we've seen it actually drill a PCB board!
Ever used Mach3 for your CNC machine? Or RouteOut? Or one of those other CNC applications with a myriad of settings so that it can support any type of CNC machine?
If you've ever bought a second-hand CNC machine off eBay or tried to build one from salvaged parts, you know how difficult it is to work out (or guess at) the settings to make it work. Sometimes you
end up shoving numbers in and keep tweaking until the actual output sort-of matches the drawing files you give it.
Well not any more....
Not only has Robot Steve redefined simple CNC design, with his awesome push-fit chassis, but this custom-written software is designed to do away with CNC-hardware-related headaches. Simply load a
file, manually move to head to a start position (origin), move the head to a second position, then hit go.
At the minute the software can read NC drill files.
Here's the app reading a drill file generated from an Eagle PCB
After calculating the scale and rotation for each point (at the minute we're just working on scale, having just read in an nc drill file) the software runs through each point, creating a "cutting
At first, something appears to have gone wrong down the left-hand side. But on further inspection, the software is actually following the simple rule "move to the nearest hole that hasn't been
drilled yet".
Because we set the origin to the top-left-most hole then travel to the bottom-right-most hole, the cutting head should be at the bottom right of our PCB.
From here, it starts with the rule "move to the nearest point" then marks it as "drilled" when it gets there.
Repeating for only holes that haven't been drilled creates the cutting path above.
After a successful couple of hours at the nerd cupboard, we managed to get Steve's CNC design working in two axes. The test board simply uses buttons to turn a connected stepper motor clockwise and
anticlockwise (we've yet to finish our controller software) but proves the concept of moving a drilling head across a gantry and performing a plunge-and-drill operation.
Exhilarating stuff!
We're particularly pleased with the dual-servo control: the first servo is actually an RC motor controller and sets the drilling spindle spinning. Then the z-axis servo plunges the drill head,
pauses, and retracts the head, before stopping the drill spindle. (In code we "detach" the z-axis servo so that it is only powered for as long as is required.)
Surely it's only a matter of time before the y-axis (cutting bed) is in place and we're ready to try out some custom software to drill our first board!
We're still not 100% au fait with the requirements for open source, GNU licences and all that kind of stuff - does everything in the chain have to be open source? Can we use our preferred PIC
microcontrollers to make a USB device when the Microchip software and USB libraries are not open source? Does it really matter?
We're blundering ahead with our software, but have hit upon two potential problems - and hopefully workable answers to them:
The first is converting the NC drill file units (mm or inches) into steps on our CNC machine.
Since a big part of our design process is to make the machine easily repeated and built from spare parts (salvaged printers are a great place to get hold of cheap steppers) we're trying to work from
the assumption that the end user knows nothing (or very little) about their hardware.
Rather than have the user enter all kinds of values to describe their hardware (or worse still, guess at them) we thought we'd have a simple manual calibration routine at the start of the software.
Simply put, the user loads their pre-printed/etched PCB onto the cutting bed and moves the drill head above the top-left-most hole (the software will highlight one if there are multiple
possibilities). This will form the origin (co-ordinate point 0,0) for our cutting routine(s).
The software will then find the bottom-right-most hole (furthest away from the origin) in the drill file, highlight it and prompt the user to move the drill head (maybe using the computer keyboard's
arrow keys or similar) above this second hole. Using these two co-ordinates we can calculate two vitally important things: scale and rotation.
In the NC drill file we have co-ordinates for every point on the board. Using pythagoras theorum (a^2 = b^2 + c^2) we can calculate the distance R1 between the top-left and bottom-right points. On
our actual PCB, this represents the same distances between the same two points. The only difference is the units used to measure this distance.
In the same way the same distance can be described in imperial (inches) and metric (millimetres) by multiplying one set of values by a constant, we can do the same to convert inches or millimetres in
the NC drill files into steps on the CNC machine. Let's pretend both images (above) are exactly the same size. Let's say the distance R1 is measured in millimetres, but we want the same distance R2
in inches. We know that 1 inch = 2.54cm and 1cm=10mm. So we can easily compare R1 (mm) with R2 (inches) by multiplying R2 by 25.4 (or dividing R1 by 2.54 to get R2, the same distance in inches).
We don't know what this constant value is to convert our NC drill measurements (mm/inches) into number of steps, but we can calculate R2, the distance the drill head travelled between two points, in
terms of steps. Knowing the value of R1 (in, say, mm) and the distance of R2 in steps allows us to calculate the scale between the two. Scale = R2/R1
The great thing about using the first hole as our origin point is that it doesn't actually matter at this stage whether or not the PCB board on the CNC machine is dead square. If the PCB board were
not aligned exactly squarely, the difference between these two distances would be exactly the same, since they both describe the radius of a circle with it's centre point at the origin
So now we've worked out the scale (the constant to multiply our NC drill hole positions by to convert distances in mm/inches into number of stepper motor steps) we now need to work out the rotation
of the PCB on the cutting bed
Even if the PCB on the cutting bed were badly skewed, the ratio (scaling) between R1 and R2 would be the same. We know the location of the bottom-right-most hole from the origin according to our NC
drill file, and we know the distance of the same hole on the actual PCB in terms of number of steps travelled in both the x and y axis. What we need to do is calculate by how much the PCB has been
One way to do this is to calculate the angle (from the origin) of the diagonal in the NC drill file (the larger of the three angles above) then calculate the angle from the origin of the current
position of the drilling head.
Since we know the position of the cutting head from the origin in terms of steps (we count the number of steps moved in both the x- and y- axes) we can calculate this angle (the green diagonal) quite
Using simple trigonometry:
If we rotate the bottom triangle 90 degrees clockwise, we can see that we've described a right angled triangle, where the adjacent side is the number of steps travelled in the y axis, the opposite
side is the number of steps travelled in the x-axis, so tanA = stepsY / stepsX
From here we can calculate the angle of the position of the drilling head from the origin.
Using the same principles, and with the values from the NC drill files, we can calculate the angle between the bottom-right-most hole and the origin (top-right-most hole). The opposite side of this
triangle is the difference between the x co-ordinates of the two holes and the adjacent side is the difference between the y co-ordinates of the two holes, so tanA = (y1-y2) / (x1-x2)
Knowing these two angles, we can subtract one from the other to calculate by how much the PCB has been rotated on the cutting bed. Now we know the scaling AND rotation, we can simply apply these to
every point in the NC drill file, to get the number of steps needed to move in both the x- and y- directions to reach in point on the actual PCB on the cutting bed.
Apply rotation to a co-ordinate point can be acheived by using a rotation matrix:
What this scary looking equation boils down to is - the coordinates (x',y') of the point (x,y) after rotation are:
x' = (x * cosA) - (y * sinA)
y' = (x * sinA) + (y * cosA)
Given that we know x and y (from the NC drill file) and we've calculated the rotation of the PCB on the cutting bed, we can work out:
xSteps = ( (x * cosA) - (y * sinA) ) * scaling
ySteps = ( (x * sinA) + (y * cosA) ) * scaling
where x and y are the co-ordinates given in the NC drill file.
And all without knowing how many teeth are on the CNC belt, how many steps the motors turn per full revolution or any of that other junk, nor without any headaches lining up the PCB to get it
absolutely square and accurate on the cutting bed.
In theory, this provides a really nice and easy to use - if not entirely easy to understand for everyone - way of drilling every hole on the PCB, from an NC drill file, given the user has manually
located two holes on the board.
Robot Steve's late entry into the BuildBrighton CNC Drilling Machine competition has in some ways spurred us on and in others caused things to grind to a halt. In some ways, seeing such a simple,
usable design has us wondering whether it's worth continuing with our little laser-cut caddy. The main difference between our approach and Matt's CNC monster was simply scale and cost:
Matt came up with a solid (though possibly over-engineered) design using linear bearings, rods and bolts by the bucketload, expensive steppers and belts - basically blowing the budget to create the
best CNC type machine he could manage.
We stuck to the cheap-as-possible, easy to replicate route, but possibly at the cost of accuracy (we still don't know if our design will actually create a working, functional CNC machine!)
Steve's design fits nicely in the middle.
With 3d printed parts, built from a 3d CAD-based design, he can see his prototype working (virtually) before cutting or casting a single piece of plastic! Yet with minimum part count and easy
push-fit construction, there's no need to worry about bolting plastics edge-to-edge and trying to get everything square. It's a great design.
So is it worth continuing with a slightly shonky design, knowing that eventually we'll probably adopt another in the near future? We'll leave that question, and spend our time constructively on the
one aspect that no-one seems to have addressed just yet: software.
The budget allows for the complete build - including driver board/electronics and controlling software.
The easiest approach would be to get a MACH3 compatible driver board, hook up the steppers and run everything through some milling software like MACH3. But this is an expensive way of going about
things, so we reckon custom software is the way to go.
Also, we're not drilling or milling blank material. Our PCBs with either already be etched, or have tracks and traces already marked on them, ready for etching. So before we start any drilling, we
have to make sure our boards are perfectly lined up to begin with.
In something like mach3 we could do this by moving the drill head to a known position and placing the board underneath it, then moving the head to a second (known) position and rotating the board
until this second point fits under the head. With the board in place, we could lock it down and start the cnc running.
MACH3 also has a myriad of settings, belt-tooth size, leadscrew adjustment values - all things which make it quite complicated and daunting to the untrained user. For our software we want:
• Minimum settings screens - we don't care how big your stepper motor is, the tooth pitch, degrees per step and so on. The software should work with a wide range of machines without any complicated
maths/physics calculations!
• Auto-alignment - placing a PCB exactly squarely on the cutting bed is going to be difficult enough. Cutting the sides square is hard - knowing that you've placed one edge exactly squarely can be
hit-and-miss, and if your edges aren't exactly true and straight, getting the whole thing to line up is almost impossible!
• Auto-scaling for different measurement units - NC drill files are commonly described in imperial (inches) but there is software that uses (and an NC drill command for using) metric (millimetres).
The software should be able to handle mm and inches without the need to re-calculate the drill position data in the NC drill file.
We're not worried about making everything open source, complying with GNU licences and all that - we're just looking to create some software that just works (in Windows at least) using whichever
tools do the job. Of course, details of how the software is created will be explained, should anyone wish to re-create their own, but we're not ruling out any specific technologies just because it's
not "open source" or GNU-a-like or anything like that!
It's a bit late - though not as late as our actual machines are for being ready (they should have been drilling PCBs in time for last August's BuildBrighton Guitar Stompbox Workshop) - but we've had
an exciting last-minute entry from Robot Steve.
Steve's taken the simplicity of our design but coupled it with the super-low cost option (and ease of construction) of a 3D printed solution. We think it looks amazing.
Using a servo for the drill plunge, this design keeps things about as simple as they possibly can be. The rails are structural as well as functional (carrying the carriage for the x/z axes) and the
tiny steppers are mounted directly onto the moving parts. Rather than mess about with linear bearings, Steve has gone for the simpler (and cheaper) design used in many inkjet printers - greased rails
and precision cut nylon blocks sliding along them!
We think the best thing about this design is the relatively low part count. Once you take away the cutting bed (a piece of cheap acrylic or some mdf) and the rails, you're left with a handful of
cheaply produced 3d printed parts.
Steve sent over some early prototype photos. It actually looks better in real life!
Down at BuildBrighton tonight, fellow PIC-programmers were in short supply, so with reluctance we had to crack open an Arduino Duemilanove and learn some crazy Arduino coding. It turns out it's not
as difficult as it looks (but it does still make you feel a bit dirty).
To get things working, we just wanted to be able to turn a stepper motor clockwise and anti-clockwise. Here's the code we came up with:
int state=0;
int dir=0;
void setup(){
pinMode(1, INPUT_PULLUP);
pinMode(2, INPUT_PULLUP);
pinMode(3, OUTPUT);
pinMode(4, OUTPUT);
pinMode(5, OUTPUT);
pinMode(6, OUTPUT);
void loop(){
//read the pushbutton value into a variable
int inputVal1 = digitalRead(1);
int inputVal2 = digitalRead(2);
// Keep in mind the pullup means the pushbutton's
// logic is inverted. It goes HIGH when it's open,
// and LOW when it's pressed.
if (inputVal1 == LOW) {
// turn the motor clockwise
}else if(inputVal2 == LOW) {
// turn the motor anticlockwise
// stop turning the motor
// move the motor
case 0:
// energise coil A
digitalWrite(3, HIGH);
digitalWrite(4, LOW);
digitalWrite(5, LOW);
digitalWrite(6, LOW);
case 1:
// energise coils A+B
digitalWrite(3, HIGH);
digitalWrite(4, HIGH);
digitalWrite(5, LOW);
digitalWrite(6, LOW);
case 2:
// energise coil B
digitalWrite(3, LOW);
digitalWrite(4, HIGH);
digitalWrite(5, LOW);
digitalWrite(6, LOW);
case 3:
// energise coils B+C
digitalWrite(3, LOW);
digitalWrite(4, HIGH);
digitalWrite(5, HIGH);
digitalWrite(6, LOW);
case 4:
// energise coil C
digitalWrite(3, LOW);
digitalWrite(4, LOW);
digitalWrite(5, HIGH);
digitalWrite(6, LOW);
case 5:
// energise coils C+D
digitalWrite(3, LOW);
digitalWrite(4, LOW);
digitalWrite(5, HIGH);
digitalWrite(6, HIGH);
case 6:
// energise coil D
digitalWrite(3, LOW);
digitalWrite(4, LOW);
digitalWrite(5, LOW);
digitalWrite(6, HIGH);
case 7:
// energise coils D+A
digitalWrite(3, HIGH);
digitalWrite(4, LOW);
digitalWrite(5, LOW);
digitalWrite(6, HIGH);
It's a simple state machine - when the motor is turning clockwise, we energise the coils in sequence 1...2...3... etc, when running anti-clockwise we go 7...6...5.... etc
Doing this allows us to quickly and easily make the motor run by pulling an input pin low (pull-up resistors mean the inputs are always high with no input on them).
Here's a video showing the motor in action:
We've had a bit of mixed success at Nerd Towers tonight.
Firstly, we wanted to get our x-axis working on the miniature drilling CNC machine. The y-axis (bed) is easy enough - it's just a plastic bed set on top of an Ikea drawer runner. The x-axis is
altogether more difficult, as it's a gantry-based axis (to keep the size/footprint down).
In the spirit of making everything as cheaply and as repeatable as possible, we're using the same rack-and-pinion approach for our x-axis as we have done on the y.
Simply put, a long toothed edge will run along the top of the gantry, and the stepper motor mounted on top of the moving carriage will pull the carriage along by rotating a cog/pulley along the
tooth-edged strip.
We've decided on this approach as once the parts are designed and proven to work, anyone with access to a laser cutter should be able to make the same thing from our drawings.
The only thing is, we've drawn most of the CNC by eye - so have no idea how far away from the rails or toothed edge our carriage is going to be. So the first thought was to make an adjustable
At the bottom of the picture you can see the side section of our carriage. By adjusting the bolts. the stepper motor (mounted on the top of the carriage section) could be moved closer to and further
away from the toothed edge running along the top.
After bolting all this together, we found that not one single piece of this carriage had been designed properly so we started the whole thing again, this time with a slightly different approach - "
It's a term coined one night at BuildBrighton and fits this approach perfectly.
Basically we make something based on a best guess, then whittle things down until they fit.....
This second carriage (in blue) was designed entirely by eye and with no reference to our frame! (if the frame doesn't fit, we can always tweak that, and the carriage together, until they line up and
mesh together nicely!)
In this instance, the small cog on top of the stepper motor is neither high enough, nor close enough (laterally) to the toothed edge running along the top.
Instead of messing about measuring and re-measuring and cutting and re-cutting, we just decided to make our toothed edge have a bit more play in it. The idea being that we line it up with the
carriage in place and just fix it down!
As we were cutting a new toothed section, we decided to go for a double-sided piece, with the cog on the stepper motor sitting between two rails. Hopefully, this will stop the motor from pushing away
from the toothed edge and skipping steps....
The only thing now, of course, is that we need to make our gantry stands about 3mm higher and a little wider on one side (so they're asymmetrical) so that this new piece doesn't look like it's been
cobbled together and just shoved on as an afterthought. (it has, but there's no need to advertise the fact!)
This new carriage moves along the rails quite nicely.
There's not much play and hopefully this double-rail approach will eliminate this altogether. Here's a quick photo to show the actual size of the gantry. It's really quite small -
So there we have it. A nice sliding gantry with a stepper motor mounted onto it. It's not quite working under it's own steam just yet, but we can't help but feel that we're getting a little bit
Here's how the final thing will sort-of look in place over the y-axis bed. We may even go crazy and lose the lump of scrap wood for a nice piece of red acrylic. That's the kind of colour scheme that
would make Robot Steve have a fit
Well, our CNC drilling machine challenge is certainly generating a bit of interest, both across the 'net and down at BuildBrighton, our local hackspace.
Nowhere in our "rules" does it state that outside help is not allowed - in fact, we even encouraged other people to get involved - so it was great to have a link sent by one of the group asking if it
was of any use. The link was to a laser-cut CNC linear rail on Thingiverse at http://www.thingiverse.com/thing:3554
As we've not really given much thought to our z-axis, we figured we could build this on a gantry over our moving bed (y-axis) to give us an x-axis movement, then simply mount something either on the
underside, or on top, overhanging one edge, with a simple raise/lower mechanism on it for the drilling head.
The first job then was to make up the laser cut linear rails:
The carriage bolted together quite easily, and with the cogs and gears held in place by only a few turns of a screw, we thought we'd try the carrier assembly on the rails:
A neat trick we learned (well, it would have been neat if we'd had the proper tools and not spent four times longer than we needed to trying to cobble it together) was to create the holes in the
acrylic at 2.5mm, not 3mm, to accept the M3 bolts.
Running an M3 tap through the hole (this is where we needed the proper tap-and-die sent, and not just a single tap and a pair of rusty pliers!) created a lovely threaded hole in the acrylic, so all
the bolts held themselves in place and didn't simply fall out when you picked the whole thing up!
The green cog in the centre is the one that will be connected to the stepper motor. As this centre gear turns, it causes the other gears to turn against the rails, causing the entire carrier to move
All we need to do now is stick the lid on and hook up the stepper motor and we're good to go!
And why the garish green/orange colour combo?
Simply put, they were was the first two sheets of acrylic in the pile.
As part of the CNC drilling machine challenge, one of the things we have to be able to do is parse NC drill files. The idea being that using an industry standard file format makes the machine
compatible with a much wider range of PCB layout software.
One of the problems we have is that Eagle sucks.
Yes, that's quite an inflammatory comment but, compared to ExpressPCB, we've seen loads of people have trouble with Eagle-drawn circuit boards.
The first thing is those stupid lozenge shaped pads. And the default hole size seems to be too small. And the pads are ridiculously small. And when you've finally etched your board and drilled it,
it's all too easy to ruin a pad because your 1mm drill bit has ripped up all but the tiniest thread of copper left around the pad (right-most-pad, below). One slight wobble with the drill or
mis-aligned pad and the whole board can be ruined!
Over at Nerd Towers, we defy convention and refuse to get drawn in to the everyone-uses-Eagle-so-we-must argument. Although it's less of an ideological standpoint and more to do with the fact that
it's just so complicated to use when no-one has ever shown you how!
Our tool of choice is ExpressPCB. It's not only free but it's simple to use.
For the hardcore gerber-loving geek crowd, the very things we laud it for may well be it's Achilles Heel too - but it is very simple to use and you can get a PCB thrown together very quickly, all
with 2mm pads with 1mm holes (ok, the default is 0.89mm but what's a tenth of a millimetre between friends?). No worring about mirroring, or not mirroring, or which-do-I-mirror before printing for
toner transfer - just draw on the top (red) layer and print it out!
To produce PCBs for etching, we usually print to a virtual printer, such as CutePDF and make a PDF file for editing in Inkscape but one thing we recently discovered was the "export to DXF" option
This is quite exciting, as it allows us to generate a file which can be parsed and turned into a drill file. The export to dxf option in ExpressPCB can send just the pad data to a single drawing.
Simply loop through the text-based dxf file, find all instances of CIRCLE and write the co-ordinates out to a NC Drill compatible file format!
After a cursory glance at the generated dxf file we can see all our pad data quite easily:
Every pad is a circle entity so we fiddled with a few values and loaded the resulting dxf into Inkscape until we found which entries corresponded to the X and Y co-ordinates. Ultimately it is these
values that we'll be interested in to create our own NC drill file.
(comments in the above image were added one we'd identified which values did what, they were not present in the original, generated dxf)
To try out our idea, we picked a circle and set the X/Y to zero and the radius to 4
Interestingly, Inkscape does not position circles from their centrepoint, but from the bottom left corner of the shape. So we expected to see our shape at -4,-4
Inkscape seems to include the stroke (shape outline) width in the X/Y co-ordinates for each shape. So we reduced the stroke width and indeed the X/Y co-ordinates updated accordingly.
We can only assume that with a stroke width of zero, the shape would indeed line up to -4,-4 and thus prove that the values we changed in our dxf file were indeed the correct x,y and radius values.
With this in mind, we're off to write a simple script to convert metric x/y value pairs into an NC Drill file.....
A few people have been asked about the CNC drill machine challenge and it seems that not everyone has access to the BuildBrighton Google Groups page. So here's the preliminary post, outlining the
basic rules for the challenge:
After making what felt like millions of boards for the midi workshop, we've decided to make a cheap, little drilling machine for making homebrew boards.
As the discussion went along, we decided to see if we could make the smallest, cheapest machine we could. Almost immediately we couldn't agree about a single thing for our machine, so we've decided
to make one each and compare them.
Matt gets loads of stuff cheap off ebay by buying in bulk.
Chris gets stuff cheap by extracting it from old hardware using hammers.
Matt uses Eagle and could create NC Drill files directly from it to drive the machine
Chris uses ExpressPCB and a convoluted print-to-pdf-then-convert-to- svg method of generating drilling data
Matt wants to make a machine that can drill large, panellised pcbs - double eurocard size (200x160).
Chris wants as small a machine as possible to fit on a shelf when not in use, and rarely makes boards bigger than a 50 pence piece.
Chris doesn't get the whole open hardware movement and still prefers PIC microcontrollers over AVR.
Matt thinks an Arduino-based machine would be much more open to hacking if other people used our plans to make their own boards.
Matt can get cheap linear bearings and has experience of making a gantry-based cnc device
Chris thinks he can cobble something together out of ball-bearings and sellotape
The list goes on and on....
So, we've come up with our own hacker challenge. Hopefully this could be the first of many - little projects that we can work on in small groups or individually, then present back to the group.
It would be really cool if other people joined in (or maybe even joined forces) to come up with alternative ideas for this first challenge.
We'd love to see some alternative ideas to make a drilling machine - we've come up with some parameters (it's not really a contest, so it's not fair to call them rules) but here's what we've
eventually agreed to work towards. Because it's a machine that we're hoping others will want to try and make (assuming either design eventually works) our first rule is about cost and availability of
It's easy to hit ebay and Rapid and Farnell and blow a small fortune, but that puts it out of reach for a lot of people. Similarly, it's possible to smash open some old hardware and salvage some cool
stuff but no-one else might be able to get hold of the same equipment, and would need to find alternatives. So the rules are
• If buying all new components, not more than £50 on the entire build (just think about it, a cnc based device for under £50!)
• If using salvaged hardware (stepper motors from old printers, for example) not more than £20 total build cost
• As we're building a small machine, the footprint of the device, when put away, should not exceed A4 size (210x297mm)
• But it'd be useless if it could only drill tiny pcbs, so it has to be able to drill up to half eurocard sized boards (100x80mm)
• Any bought components should be accessible to everyone and you should reasonably expect to be able to purchase the same components from the same, or alternative suppliers 12 months from now (so
you can't win a job lot of motors from some bloke off ebay and put cost of materials down as 50p)
• Overseas suppliers can be used to keep costs down, as their price per unit is often much less than UK suppliers.
• The device is to be platform independent - it can use any microcontroller and any software can be used to control the machine, including homemade software/drivers.
• It has only to work on any one platform, not all
• For accuracy, the drilling machine should be able to drill a 1mm hole in a max-sized 2mm pad and leave a complete copper ring intact around the hole
• For building materials, assume a price for acrylic sheets as £1 per mm thickness, per A4 sheet
• Cheaper alternatives (such as mdf or laser ply) can be used instead of acrylic for the chassis if required
• Trivial components and the cost of pcb etching, running a laser cutter etc and consumables will not be used to calculate total build cost
• The price for delivery of all components bought online will be included in the final cost
More rules may be added as the build goes on, but only to clarify any points that may arise, not to act as a restrictive force to deter alternative ideas.
So there we have it. Another stupid challenge that probably won't get finished (or much beyond the planning stages) But there's nothing like seeing someone making more progress on a project than you,
to act as a catalyst and get things moving along. So who else fancies joining in?
The prize? We'll use the "best" design for a "proper version" of the machine for use in the hackspace for all members making their own pcbs. Better than money or other tangible goods, you get to win
At an epic BuildBrighton session last night, there was loads going on. Including a new design for our CNC drilling machine challenge.
One of the things we're keen to stick to is repeat-ability (the other is, of course, a shoestring budget). So instead of expensive (and as a few people have pointed out, extremely accurate,
precision-made) pulleys and belts, we're going all out with laser-cut acrylic and little, cheap stepper motors.
The previous platform we built was nice and sturdy, but during it's construction we came across a little problem. It's not a major problem, and can surely be remedied easily by anyone with the proper
tools and a bit of time and effort, but it does stand in the way of repeat-ability. Basically, getting the two runners exactly parallel is a little bit difficult.
If the runners are not exactly equi-distant, along their entire length, the platform "binds" as it reaches the end of its travel in both directions. This isn't really a major project if the length of
travel is kept to about half the total length of the bed (in fact, this is about the maximum we're expecting our bed to travel) but there's just something about the fact that it's so easy to mess
this part up that we're not happy with.
So we're going with a much easier (and some would say, slightly shonky) design:
It's a single runner with the platform bolted along the centre-line. The single runner can be mounted onto a scrap of wood (exactly what we used here!) and it's not really important how square this
mounting is (we mounted it along a centre-line along the wood for neatness, but it's not critical).
The only bit that's really important is getting the gantry mounts on the sides square with the runner.
BUT - having mounted the runner onto the board, the runner is fixed nice and securely. Using a set-square the gantry mount can be placed and fitted from the same side of the board. In our previous
design, having to bolt everything together meant lifting the whole assembly up, and this is where things slipped out of alignment.
With this design, it's quite easy to get the runner and gantry at 90 degrees.
With the two gantry supports fitted, the top/cross piece just clips into place.
In theory, everything should be nice and square. Next we need to fit the stepper motors and actually get something moving to see if this design actually works ;-)
One thing that isn't immediately obvious from these photos is just how teeny tiny this little machine is. It's almost comical. In fact, a few BuildBrighton regulars have decided that it's not only
comical but unusable - and have already started a book on how long before this design is abandoned and v3.0 started. We'll show 'em....
Last night's BuildBrighton meeting was pretty cool, with a great turnout, and people actually getting on and making stuff (standing around drinking beer, talking nerd and eating pizza may be cool,
but the real cool kids just get on with it!)
Seb was busy trying out his stepper-driven drawing machine.
This is just a proof-of-concept to try out the software and hardware kit, but the final idea is to scale this up and use bike chains/sprockets and draw across a 5m surface!
After a successful test, Seb tweaked the software and started producing some real works of art....
It can even draw images - albeit slightly abstract ones...
No, the machine didn't suddenly start drawing upside-down.
Some of us probably need some remedial how-to-use-your-smartphone lessons. Still, a great looking print, even if the photo is a bit wonky!
Here's Joel from the earlier BuildBrighton Stompbox workshop getting busy with a new enclosure for a variation on the Fuzz Factory pedal we made. The box is actually an old (and he tells us, quite
crappy) effect pedal, but with space for four knobs and a nice big stomping area, it's the perfect box to put his modified fuzz factory in.
The first job, of course, is to clean up the artwork and design a custom decal sticker. Joel attacks his enclosure with a wire brush mounted on an angle grinder....
Showing us how to do it properly - note the dust mask and safety goggles
It's been a long, exhausting, but ultimately satisfying day, after running the BuildBrighton guitar stompbox workshop today. There was a sell-out crowd (plus a couple of last-minute attendees,
squeezed in on the day) and we had a fantastic success rate - of 12 boxes, 10 were (eventually) working by the end of the workshop!
The boxes were based on the Dallas Fuzz Factory schematic, with a couple of tweaks. Firstly, as only two pots were to be exposed on the box (drive and volume) the others used were trimmer pots
(compression and gate) so each person could create their own unique sound and capture it inside the box!
Fuzz Factory Schematic
As well as making the circuits on stripboard, everyone had a go at drilling their alloy enclosures in the BuildBrighton "dirty workshop" as well as fitting all the hardware and wiring everything up.
As well as creating a circuit board from the schematics using stripboard (veroboard), everyone had to wire up all the pots, footswitch and sockets and it was amazing to get ten fully-working pedals
after about six hours hard work!
Jamie Matthews was the first successful candidate to rock out with his fuzz pedal
Note how you hear the tone/flavour of the pedal changing as Tom tweaks the compression and gate trimmer pots. These pots allowed us to tailor each pedal without having to swap out lots of different
It didn't take long for others to follow:
Though not everyone was successful on their first attempt. Jon (in this video) actually managed not only to make a crackly pedal, but a direct short across the drive pot terminals in this pedal
actually caused it to smoke (shortly after this video ended). When a tiny flame appeared, it was decided to replace his pot and capacitor, just to be sure!
Not so much smoke on the water as "smoke in the pedal"
But it was great to see genuine smiles break out - the real ones that you just can't hide - as more and more pedals fuzzed and buzzed into life
For that authentic rock sound, we were using 2 x germanium transistors in each pedal. Germanium transistors are not as stable nor consistent as silicone ones, so some have a higher gain than others.
Real pedal enthusiasts have been known to sit and try out loads of different transistor combinations, to get that "just right" tone. We included sockets for our transistors to all people to do this
if they wanted. As time was tight, Jason spent a few hours testing all our germanium transistors and sorting them into (relatively) high and low gain. We used a low gain for Q1 (see schematic) and a
high gain for Q2.
Joel, from GAK specifically asked for two high gain transistors to make really "hot" pedal. You can hear his pedal going crazy when pushed to the top end of the range
Here's the documentation that everyone was following. Use it to create your own fuzz-factory clone!
Stompbox Workshop | {"url":"https://nerdclub-uk.blogspot.com/2012/08/","timestamp":"2024-11-13T08:44:08Z","content_type":"application/xhtml+xml","content_length":"257418","record_id":"<urn:uuid:8f0da9db-673d-4d00-acc8-45461f924085>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00605.warc.gz"} |
On non Artinian cofinite generalized local cohomology modules
F. Vahdanipour
Let R be a commutative Noetherian ring and I ⊆ J be ideals of R. Let M and N be finitely generated R-modules such that pd[R](M) < ∞. In this paper, we study the supremum of non Artinian I-cofinite
modules H^i[I](M,N) over a commutative Noetherian ring where i ≤ 1 and give a bound for q[J](M,N) by using q[I](M,N). We show that q[J](M,N) ≤ q[I](M,N) + cd[J](M,N/IN).
Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15(3) (2022), pp. 45-51 | {"url":"https://tcms.org.ge/Journals/ASETMJ/Volume15/Volume15_3/Abstract/abstract15_3_4.html","timestamp":"2024-11-04T01:20:51Z","content_type":"text/html","content_length":"1497","record_id":"<urn:uuid:f1d09756-f6e6-4251-822a-4fbe6cf46876>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00193.warc.gz"} |
CLONE - trigonometric functions in attribute draw:formula (19.171, part1 ODF1.2) specify angle in degree but widely implemented as radian
• Type:
• Status: New
• Priority:
• Resolution: Unresolved
• Affects Version/s: ODF 1.2
• Proposal:
Change the text "degree" to "radian".
Current text:
atan2(x,y) returns the angle in degrees of the <..> with the x-axis
Proposed text:
atan2<..> returns the angle in radians of the vector (x,y) with the x-axis
I have skipped "(x,y)" here, because that is in issue [DEL:OFFICE-3822:DEL].
Change the text "degree" to "radian". Current text: sin returns the trigonometric sine of n, where n is an angle specified in degrees cos returns the trigonometric cosine of n, where n is an
angle specified in degrees tan returns the trigonometric tangent of n, where n is an angle specified in degrees atan returns the arc tangent of n in degrees atan2(x,y) returns the angle in
degrees of the <..> with the x-axis Proposed text: sin returns the trigonometric sine of n, where n is an angle specified in radians cos returns the trigonometric cosine of n, where n is an angle
specified in radians tan returns the trigonometric tangent of n, where n is an angle specified in radians atan returns the arc tangent of n in radians atan2<..> returns the angle in radians of
the vector (x,y) with the x-axis I have skipped "(x,y)" here, because that is in issue OFFICE-3822 .
For the trigonometric functions sin, cos, tan, atan, and atan2 the angle is specified to be in degree. But the applications Apache OpenOffice, Symphony, LibreOffice, Calligra Stage, PowerPoint
Preview 2013 have implemented it to be in radian.
Does anyone know an application, that has implemented the angle to be in degree?
If not, I suggest to change it in ODF1.2 Errata. | {"url":"https://issues.oasis-open.org/browse/OFFICE-3972","timestamp":"2024-11-12T13:46:32Z","content_type":"text/html","content_length":"70108","record_id":"<urn:uuid:3a6388f5-32b3-44bb-a123-2981e517232c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00874.warc.gz"} |
Discrete Mathematics with Applications, 4th edition
English | 2010 | ISBN: 978-0-495-39132-6 | 993 Pages | PDF | 10 MB
Susanna Epp’s DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract
concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to
think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability,
automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp’s
emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses.
Download from free file storage
Resolve the captcha to access the links! | {"url":"https://scanlibs.com/discrete-mathematics-with-applications-4th-edition/","timestamp":"2024-11-11T05:00:58Z","content_type":"text/html","content_length":"39614","record_id":"<urn:uuid:9bca010e-17e6-4446-9915-1f934ef5a09c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00323.warc.gz"} |
\dfrac{{{m^{ - 1}}}}{{{m^{ - 1}} + {n^{ - 1}}}}$ is equal to
Hint:We know that whenever a number or variable has its power as $ - 1$, it can be converted into fraction form as:${a^{ - 1}} = \dfrac{1}{a}$.We will use this step in the above fraction in order to
simplify it. Further, we will solve the converted fraction step by step to attain the required solution.
Complete step-by-step answer:
We are given a fraction, $\dfrac{{{m^{ - 1}}}}{{{m^{ - 1}} + {n^{ - 1}}}}$ in the question. Our aim will be to simplify this fraction in order to reach a solution. But right now, we can’t see any
arithmetic operation that can be applied. It can be observed that each variable(s) in the numerator as well as denominator has power $ - 1$.
We know that the number with power $ - 1$ is equal to reciprocal of the number (${a^{ - 1}} = \dfrac{1}{a}$).
We will use this to convert our fraction.
$\dfrac{{{m^{ - 1}}}}{{{m^{ - 1}} + {n^{ - 1}}}} = \dfrac{{\dfrac{1}{m}}}{{\dfrac{1}{m} + \dfrac{1}{n}}}$
Now, addition by using LCM can be carried out in the denominator part
$\dfrac{{\dfrac{1}{m}}}{{\dfrac{1}{m} + \dfrac{1}{n}}} = \dfrac{{\dfrac{1}{m}}}{{\dfrac{{n + m}}{{mn}}}}$
Since the denominator is also a fraction, we will simplify it as
\dfrac{{\dfrac{1}{m}}}{{\dfrac{{m + n}}{{mn}}}} = \dfrac{{\left( {mn} \right)\left( {\dfrac{1}{m}} \right)}}{{m + n}} \\
= \dfrac{{n \times m \times \dfrac{1}{m}}}{{m + n}} = \dfrac{n}{{m + n}} \\
This is the required simplified fraction.
So, the correct answer is “Option D”.
Note:The student should not be confused while handling variables since they cannot be added or subtracted directly. The student should carefully add the two fractions by first calculating their Least
Common Multiple (LCM). Always remember that we can approach the solution in these type of questions by using a conversion step ( here, ${a^{ - 1}} = \dfrac{1}{a}$) that will make it easier to solve. | {"url":"https://www.vedantu.com/question-answer/the-fraction-dfracm-1m-1-+-n-1-is-equal-to-a-m-b-class-10-maths-cbse-5f5c3cf28a2fd7303bea85f6","timestamp":"2024-11-14T00:22:49Z","content_type":"text/html","content_length":"166636","record_id":"<urn:uuid:0e6a743a-a3a6-4585-9e20-ba457b789634>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00711.warc.gz"} |
Lesson 10
Angles, Arcs, and Radii
• Let’s analyze relationships between arc lengths, radii, and central angles.
Problem 1
Here are 2 circles. The smaller circle has radius \(r\), circumference \(c\), and diameter \(d\). The larger circle has radius \(R\), circumference \(C\), and diameter \(D\). The larger circle is a
dilation of the smaller circle by a factor of \(k\).
Using the circles, write 3 pairs of equivalent ratios. Find the value of each set of ratios you wrote.
Problem 2
Tyler is confident that all circles are similar, but he cannot explain why this is true. Help Tyler explain why all circles are similar.
Problem 3
Circle B is a dilation of circle A.
1. What is the scale factor of dilation?
2. What is the length of the highlighted arc in circle A?
3. What is the length of the highlighted arc in circle B?
4. What is the ratio of the arc lengths?
5. How does the ratio of arc length compare to the scale factor?
Problem 4
Kiran cuts out a square piece of paper with side length 6 inches. Mai cuts out a paper sector of a circle with radius 6 inches, and calculates the arc length to be \(4\pi\) inches. Whose paper is
larger? Explain or show your reasoning.
(From Unit 7, Lesson 9.)
Problem 5
A circle has radius 3 centimeters. Suppose an arc on the circle has length \(4\pi\) centimeters. What is the measure of the central angle whose radii define the arc?
(From Unit 7, Lesson 9.)
Problem 6
A circle with a shaded sector is shown.
1. What is the area of the shaded sector?
2. What is the length of the arc that outlines this sector?
(From Unit 7, Lesson 8.)
Problem 7
The towns of Washington, Franklin, and Springfield are connected by straight roads. The towns wish to build an airport to be shared by all of them.
1. Where should they build the airport if they want it to be the same distance from each town’s center? Describe how to find the precise location.
2. Where should they build the airport if they want it to be the same distance from each of the roads connecting the towns? Describe how to find the precise location.
(From Unit 7, Lesson 7.)
Problem 8
Chords \(AC\) and \(DB\) intersect at point \(E\). Select all pairs of angles that must be congruent.
angle A D B and angle A C B
angle A D B and angle C A D
angle D E A and angle C E B
angle C A D and angle C B D
angle B C A and angle C B D
(From Unit 7, Lesson 2.) | {"url":"https://curriculum.illustrativemathematics.org/HS/students/2/7/10/practice.html","timestamp":"2024-11-03T07:45:02Z","content_type":"text/html","content_length":"91442","record_id":"<urn:uuid:6980afc6-5d85-4f97-9c0f-37c174136817>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00516.warc.gz"} |
shannon wiener index calculator
Shannon index - abbreviated H in the literature. The Shannon-Wiener diversity index is a heterogeneity measure that incorporates species richness and evenness (Hollenbeck and Ripple, 2007). Here each
column is one community, each row is one species, and the numbers are either probabilities or numbers found for each species. ii) Shannon maximum . Using a joint dataset retrieved from Foursquare
API, … Additionally it also provides Species Richness and Evenness in a single button click. Shannon index - abbreviated H in the literature. This app is designed to simplify the calculation process
of species diversity to get Shannon-Wiener Diversity Index. Let's use R to calculate H' for the two communities in the example above. Shannon wiener diversity index calculation | birds |
biodiversity. Partiamo da una popolazione X ha una caratteristica con n modalità (m1,..,mn). Simpsons diversity index. It measures the rarity and commonness of species in a community. The maximum
value of H is. Reply. Shannon Entropy (Information Content) Calculator. Simpson's index of diversity - abbreviated 1-D in the literature. The Shannon index has been a popular diversity index in the
ecological literature, where it is also known as Shannon's diversity index, the Shannon–Wiener index, the Shannon–Weaver index and the Shannon entropy. Species Richness (S) Simpson's Index (D) Index
Of Similarity (1 - D) Reciprocal Index (1/D) Shannon-Wiener Index (H) Evenness (E) Glossary of Biodiversity Measures Species Richness - The number of different species found in a particular
environment. I have 11 locations (paired fields) for which I already defined (with QGIS) and calculated the proportions of 5 landscape factors (crops, grassland, semi-natural, forest and others)
within a … In previous descriptions, the Shannon Wiener index was mostly calculated using multiple degrees of data, and in some papers coverage was used. Select the number of categories or classes
(between 2 and 20) and input your sample data (e.g. Example \(\PageIndex{3}\):Calculating Shannon-Weiner Index. The free online Shannon's diversity index calculator. First, enter the number of
species, and then enter the name you wish to give the species, if available, and the given populations for each of the species—in any given order. shannon weiner index for meadowa Posted on
2013-10-21 2017-09-04 Author bpmsg Categories Online Tools , Other Articles Tags Berger Parker , calculation , Excel , Shannon Entropy , Shannon Weaver Index , Shannon-Wiener Index , Simpson Index
Species richness - abbreviated S in the literature. When all species in the data set are equally common, all pi values = 1/R and the Shannon-Weiner index equals ln(R). Shannon-Wiener Diversity Index.
Shannon wiener species diversity index calculator â. Diversity index wikipedia. Szukaj projektów powiązanych z Shannon diversity index calculator lub zatrudnij na największym na świecie rynku
freelancingu z ponad 19 milionami projektów. 1 15 1 Shannon-Wiener Diversity Index is a term in biology/ecology. To understand the basic concept of diversity, you might watch my video here; it
explains how diversity can be characterized using diversity indices – like the Simpson index– taking into account richness and evenness. Shannon-Wiener Diversity Index is a term in biology/ecology.
The Shannon-Wiener diversity index for each sample is calculated. These notes will show you how to conduct the Hutcheson t-test and so get a statistical significance of the difference in Shannon
diversity between two samples. Sugar Maple. When calculating Shannon’s index, how do you standardize by the number of quadrats sampled when unequal sample unit size occurs among sites? In plainer
(but longer) language, here’s the Shannon-Wiener index: To practice, calculate the Shannon-Wiener diversity index (H) for one of the five samples from Activity 1 (Sample _____). Shannon index
calculation (in google sheets or excel) youtube. Moreover, inverse Simpson is asymptotically equal to rarefied species richness in sample of two individuals, and Fisher's $\alpha$ is … This tutorial
explains how to calculate the Shannon Wiener diversity index and Evenness. Where pi = the proportion of individuals in the species. It measures the rarity and commonness of species in a community.
The user only have to enter the number of individuals recorded for each species present. There are two methods to compute the Shannon-Wiener diversity index in the surveyed region, which is
calculated based on each sample data (Method 1) or based on the pooled data of the survey region (Method 2). Read more on aquatic biodiversity and calculations and download instructional documents.
35. Biodiversity Calculations made simple! shannon wiener diversity index explanation, From what I have read, the Index calculates the probability that any 2 randomly selected organisms belong to the
same species, and it is thought that the closer the result is to zero, the more diverse the region is. A diversity index (also called phylogenetic or Simpson's Diversity Index) is a quantitative
measure that reflects how many different types (such as species) there are in a dataset (a community) and that can simultaneously take into account the phylogenetic relations among the individuals
distributed among those types, such as richness, divergence or evenness. Simpson's reciprocal index - abbreviated 1/D in the literature. Additionally it also provides Species Richness and Evenness in
a single button click. How to calculate a biodiversity index | amnh. Let’s compute the Shannon-Weiner diversity index for the same hypothetical community in the previous example. Note that lower
values indicate more diversity while … Identification plates by feeding strategies, Identification plates by taxonomic groupings, Counting organisms in photographs with ImageJ (Fiji), Water sampling
and handling of water samples. This index is borrowed from information science, and is calculated as follows: € H'=−p i lnp i i=1 S ∑ Where p i is the relative abundance of species i, S is the total
number of species present and ln is the natural log. Ogni modalità ha una probabilità di verificarsi P(mi). Shannon-Wiener Index (H’) Most commonly used index of diversity in ecological studies;
Values range from 0 to 5, usually ranging from 1.5 to 3.5; Calculated: Where: n i = number of individuals or amount (e.g., biomass or density) of each species (the i th species) 11 1 1 The
Shannon index has been a popular diversity index in the ecological literature (Tandon et al. This app is designed to simplify the calculation process of species diversity to get Shannon-Wiener
Diversity Index. Relative to other diversity indices, such as Simpson’s index, it is considered sensitive to the addition of rare species (Krebs, 1989). Diversity indices like the Shannon entropy
("Shannon-Wiener index") and the Gini-Simpson index are not themselves diversities.They are just indices of diversity, in the same way that the diameter of a sphere is an index of its volume but is
not itself the volume. Shannon-Weiner index (H) - This diversity measure came from information theory and measures the order (or disorder) observed within a particular system. The Shannon-Weiner
index is most sensitive to the number of species in a sample, so it is usually considered to be biased toward measuring species richness. Shannon index - abbreviated H in the literature. How many
categories? Also known as the Shannon-Wiener or Shannon-Weaver index. First, let us calculate the sum of the given values. The calculation is performed using a natural logarithm. I would like to
calculate Shannon's diversity index for habitat diversity, i.e. The key is the formula that determines the variance of the Shannon index. Shannon-Wiener diversity index is commonly used in measuring
the community species diversity or assessing the ecological status of marine environments. Shannon-Wiener Index This diversity measure is based on information theory; simply, the measure of order (or
disorder) within a particular system. CALCULATION USING THE SHANNON-WIENER INDEX JESÚS LÓPEZ BAEZA*1,3, DAMIANO CERRONE†2, KRISTJAN MÄNNIGO‡3 1University of Alicante, Spain 2Tampere University of
Technology, Finland 3SPIN Unit, Estonia ABSTRACT This study will compare the results of measuring Urban Complexity using the Shannon-Wiener index in two different methods. Formula: H = -SUM[(pi) * ln
(pi)] E=H/H max Where, SUM = Summation pi= Numbe of individuals of species i/total number of samples S = Number of species or species richness H max = Maximum diversity possible E= Eveness=H/H max
The samples of 5 species are 60,10,25,1,4. Species richness - abbreviated S in the literature. About image and VIRTUE, Short video lessons on biofilms and biodiversity. The calculation is performed
using the natural logarithm. The calculator uses the following formula to calculate the Shannon-Wiener diversity index: H = - S p i ln ( p i) . Sea water in constant motion. Please let us know your
suggestions and comments. Stack Exchange Network. Rejestracja i składanie ofert jest darmowe. What I can't understand, is the math behind the index; for example: why is it multiplying n by n-1?
Sample Values (S) = 60,10,25,1,4 number of species (N) = 5. Ecology how can i calculate shannon-wiener diversity index in. Thanks for using our software! where n i is the number of observations from
the sample in the i th of k (non-empty) categories and n is the sample size. For a population (or a non-random sample), we can use Brillouin’s index of diversity, instead of Shannon’s
index.Brillouin’s index is defined as. Shannon-Wiener Index (H’) Most commonly used index of diversity in ecological studies; Values range from 0 to 5, usually ranging from 1.5 to 3.5 ; Calculated:
Where: n i = number of individuals or amount (e.g., biomass or density) of each species (the i th species) N = total number of individuals (or amount) for the site, and ln = the natural log of the
number. Evenness (E): E = H / ln( S ) . In ecological studies, this order is characterized by the number of individuals observed for each species in the sample plot (e.g., biofilm on a plexiglass
disc). Also known as the Shannon-Wiener or Shannon-Weaver index. Der Shannon-Index (häufig auch Shannon-Wiener-oder Shannon-Weaver-Index) ist eine mathematische Größe, die in der Biometrie für die
Beschreibung der Diversität (vgl. View Shannon-Wiener Excel sheet_AMK.pdf from BIOL 220W at Pennsylvania State University. Als ein Maß für die Biodiversität, bzw. This calculator is free to use and
is designed for biologists, ecologists, teachers, and students needing to quickly calculate the biodiversity indexes of an ecosystem. Reply. After testing for normalization using the
Kolmogorov–Smironov test, the Shannon - Wiener index was used as the dependent variable, and spectral values from original and synthetic bands from different processing on the ETM + data were used as
the independent variables. Simpson's reciprocal index - abbreviated 1/D in the literature. Shannon Wiener Species Diversity Index Formula - Data Analysis. Shannon–Wiener index of diversity
(information index) A measure used by ecologists when a system contains too many individuals for each to be identified and examined.A small sample is used; the index (D) is the ratio of the number of
species to their importance values (e.g. The Simpson index is a dominance index because it gives more weight to common or dominant species. Biodiversity calculator for the simpson and shannon
indexes. Here p i is the proportion of total number of species made up of the ith species. Mannigfaltigkeit der Arten wird häufig der Shannon-Wiener-Index Hs verwendet, welcher oft fälschlicherweise
als Shannon-Weaver-Index bezeichnet wird, da Shannon seinen Beitrag zusammen mit einem weiteren Artikel von Weaver in Buchform publizierte. Calculating racial/ethnic diversity using the
shannon-wiener index. The calculation is performed using a natural logarithm. use the Shannon-Wiener diversity index, or H'. Richness (S): Total number of species in the community The Simpson index
is based on the arithmetic mean, in the general concept of diversity it corresponds to a “true” diversity of order two. In particular, for a random sample, we can use Shannon’s index of diversity
(aka as Shannon-Weiner’s index), which is defined as. Species richness - abbreviated S in the literature. How to calculate shannon index for plankton diversity? Shannon-Wiener Diversity Index. It
measures the rarity and commonness of species in a community. An equivalent formula is. Copyright © 2020 University of Gothenburg, Reports and project work about / with VIRTUE, Cooperation partners
and financial support. 150 0 2 Species diversity index to calculate, one of: Shannon index - abbreviated H in the literature. The BPMSG diversity online calculater allows you to calculate
diversity indices from your sample input data. the diversity of landscape factors. … Does anybody know the formula, or how this can be performed in Excel? Calculate the Shannon diversity index and
Evenness for these sample values. In the Shannon-Wiener diversity index, you have to add one pi lnpi term for EACH species that you counted. SDR-IV Help Measuring and understanding biodiversity by R
M H Seaby and P A Henderson SDR is designed to help you measure and understand the variation in … What biodiversity measure(s) would you like to have computed? the following definition for Shannon's
entropy: Here pi is the probability for the i-th event. The calculation is performed using a natural logarithm. where n i is the number of observations from the sample in the i th of k (non-empty)
categories and n = is the sample size. Here p i is the proportion of total number of species made up of the ith species. If abundance is primarily concentrated into one species, the index will be
close to zero. Species. of individuals. The user only have to enter the number of individuals recorded for each species present. This calculator is free to use and is designed for biologists,
ecologists, teachers, and students needing to quickly calculate the biodiversity indexes of an ecosystem. There is also a spreadsheet calculator, that you can download and use for … Photo and
Microscopy for VIRTUE-s projects, Understanding what we see. ある．前者ではShannon-Wiener 指数，後者ではSimpson の多様度指数がよく用いられる．この2つは個体数に一定の分布型を仮定しないノンパラメトリッ
クな指数である． Shannon-Wiener 指数H’は， H' pilnpi i 1 s で表される．Sは群集内の種数，pi は群集内の全種の個体 Biodiversität) eingesetzt wird.Er beschreibt die Vielfalt in betrachteten Daten und
berücksichtigt dabei sowohl die Anzahl unterschiedlicher Datenkategorien (z. Charles. The more unequal the abundance of species, the larger the weighted geometric mean of the pi values, the smaller
the index. Charles says: March 13, 2018 at 9:59 am Mary, Perhaps I don’t understand your question, but Shannon’s index doesn’t assume that numbers need to be equal. Simpson's index of diversity -
abbreviated 1-D in the literature. The Shannon-Wiener species diversity index per sample plot was calculated. Using cover data for vegetation. Also known as the Shannon-Wiener or Shannon-Weaver
index. 2 0 33. In particular, the exponent of the Shannon index is linearly related to inverse Simpson (Hill 1973) although the former may be more sensitive to rare species. where c = INT(n/k)
and d = MOD(n, k).. Shannon wiener index (Shannon-Weiner index) is used to describe the disorder and uncertainty of individual species. Simpson's index of diversity - abbreviated 1-D in the
literature. 香農多樣性指數（Shannon's diversity index）用來估算群落多樣性的高低，亦稱為(Shannon-Weiner diversity index)。公式如下： ′ = − ∑ = 其中S表示總的物種數，pi表示第i個種占總數的比例(Pielou
1975)。 Simpson Diversity Index (D): D = S pi2. Effects of currents, tides and waves. Shannon index. The Shannon-Wiener Diversity Index, often symbolized by H' (or, H-prime), is a measure of species
diversity that takes into consideration not only the number of species present, but includes their relative abundance in the population. I'm trying to find out how to perform Hutcheson's t-test for
significance on Shannon-Wiener Indices. The value of the Shannon-Wiener Index usually lays between 1.5 and 3.5 for ecological data and rarely exceeds 4.0. In the Shannon index, p is the proportion (n
/N) of individuals of one particular species found (n) divided by the total number of individuals found (N), ln is the natural log, Σ is the sum of the calculations, and s is the number of species.
sum = (60+10+25+1+4) = 100 Shannon's entropy or information content is an important concept that bridges physical entropy and information theory. Simpson's reciprocal index - abbreviated 1/D in the
literature. Simpson's reciprocal index - abbreviated 1/D in the literature. This calculator uses The Shannon equitability index is simply the Shannon diversity index divided by the maximum diversity
\( E_{H} = \frac{H}{\log(k)} \) This normalizes the Shannon diversity index to a value between 0 and 1. For our uses, this order could be characterized by the number of species and/or the number of
individuals in each species, within our sample plot. p1-p10 ARE ONLY AN … Temperature impact on the shannon-wiener biodiversity index (bdi. Diversity calculator online (bpmsg). The calculator uses
the following formula to calculate the Shannon-Wiener diversity index: Here pi is the proportion of total number of species made up of the ith species. 33 44 11 (May 1975)[2]. Simpson's index of
diversity - abbreviated 1-D in the literature. No. Laformula dell'indice di Shannon-Wienerè la seguente: In genere è indicata con la lettera H o H'. Where do I find observations, reports and
projects? The calculator uses the following formula to calculate the Shannon-Wiener diversity index: H = - S p i ln ( p i) . In general the concept of diversity can be formulated using the power
mean. For the environment (B) demonstrated in table 4 the diversity index calculated by Shannon-Wiener diversity index model w as 1.6094 with evenness of 1. , 2007; Pandey and Kulkarni, 2006; Price,
1975), where it is known as Shannon’s diversity index, the Shannon -Wiener index, the Shannon- Weaver index and the Shannon entropy Also known as the Shannon-Wiener or Shannon-Weaver index. Effective
number of species . H = -SUM [ (pi) * ln (pi)] E=H/H max Where, SUM = Summation pi= Numbe of individuals of species i/total number of samples S = Number of species or species richness H max = Maximum
diversity possible E= Evenness=H/H max. Bpmsg diversity online calculator – bpmsg. First, enter the number of species, and then enter the name you wish to give the species, if available, and the
given populations for each of the species—in any given order. The higher the uncertainty, the higher the diversity. Shannon-Wiener Diversity Index is a term in biology/ecology. ( m1,.., mn ) X ha una
probabilità di verificarsi p ( mi ) can be using... Shannon-Index ( häufig auch Shannon-Wiener-oder Shannon-Weaver-Index ) ist eine mathematische Größe, in. Seguente: in genere è indicata con la
lettera H o H ' how can i Shannon-Wiener! User only have to enter the number of species in a community indicata con la lettera H H! Shannon-Weiner index physical entropy and information theory, let
us calculate the Shannon index |... Same hypothetical community in the Shannon-Wiener diversity index: H = - S p i is the proportion total! And calculations and download instructional documents let
us calculate the sum of the ith species / ln ( S:! Biodiversity measure ( S ) the following formula to calculate diversity indices from your sample input.. Is it multiplying n by n-1 una
caratteristica con n modalità ( m1,.., mn ) diversity. Higher the uncertainty, the higher the diversity i calculate Shannon-Wiener diversity index calculation ( in sheets! Read more on shannon wiener
index calculator biodiversity and calculations and download instructional documents by n-1 sample is calculated one pi lnpi for... Virtue, Short video lessons on biofilms and biodiversity know the
formula, or how this be! The number of categories or classes ( between 2 and 20 ) and input your sample (. Would you like to have computed physical entropy and information theory read more aquatic!,
or H ' what biodiversity measure ( S ) la lettera H o H ' google sheets or )! Primarily concentrated into one species, the smaller the index status of marine environments a community on biofilms
biodiversity. Abbreviated 1-D in the example above Richness ( S ) = 5 simplify the calculation process of species a. Ca n't understand, is the math behind the index will be close to.... Used in
measuring the community species diversity to get Shannon-Wiener diversity index wikipedia è indicata con la lettera H H! Indices from your sample data ( e.g lnpi term for each species present can.
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Selecting integration methods
Selecting integration methods¶
The description of an integration method on a whole mesh is done thanks to the structure getfem::mesh_im, defined in the file getfem/getfem_mesh_im.h. Basically, this structure describes the
integration method on each element of the mesh. One can instantiate a getfem::mesh_im object as follows:
getfem::mesh_im mim(mymesh);
where mymesh is an already existing mesh. The structure will be linked to this mesh and will react when modifications will be done on it (for example when the mesh is refined, the integration method
will be also refined).
It is possible to specify element by element the integration method, so that element of mixed types can be treated, even if the dimensions are different.
To select a particular integration method on a given element, one can use:
mim.set_integration_method(i, ppi);
where i is the index of the element and ppi is the descriptor of the integration method. Alternative forms of this member function are:
void mesh_im::set_integration_method(const dal::bit_vector &cvs,
getfem::pintegration_method ppi);
void mesh_im::set_integration_method(getfem::pintegration_method ppi);
which set the integration method for either the convexes listed in the bit_vector cvs, or all the convexes of the mesh.
The list of all available descriptors of integration methods is in the file getfem/getfem_integration.h. Descriptors for integration methods are available thanks to the following function:
getfem::pintegration_method ppi = getfem::int_method_descriptor("name of method");
where "name of method" is to be chosen among the existing methods. A name of a method can be retrieved with:
std::string im_name = getfem::name_of_int_method(ppi);
A non exhaustive list (see Appendix B. Cubature method list or getfem/getfem_integration.h for exhaustive lists) of integration methods is given below.
Examples of exact integration methods:
• "IM_NONE()": Dummy integration method (new in getfem++-1.7).
• "IM_EXACT_SIMPLEX(n)": Description of the exact integration of polynomials on the simplex of reference of dimension n.
• "IM_PRODUCT(a, b)": Description of the exact integration on the convex which is the direct product of the convex in a and in b.
• "IM_EXACT_PARALLELEPIPED(n)": Description of the exact integration of polynomials on the parallelepiped of reference of dimension n
• "IM_EXACT_PRISM(n)": Description of the exact integration of polynomials on the prism of reference of dimension n
Examples of approximated integration methods:
• "IM_GAUSS1D(k)": Description of the Gauss integration on a segment of order k. Available for all odd values of k <= 99.
• "IM_NC(n,k)": Description of the integration on a simplex of reference of dimension n for polynomials of degree k with the Newton Cotes method (based on Lagrange interpolation).
• "IM_PRODUCT(a,b)": Build a method doing the direct product of methods a and b.
• "IM_TRIANGLE(2)": Integration on a triangle of order 2 with 3 points.
• "IM_TRIANGLE(7)": Integration on a triangle of order 7 with 13 points.
• "IM_TRIANGLE(19)": Integration on a triangle of order 19 with 73 points.
• "IM_QUAD(2)": Integration on quadrilaterals of order 2 with 3 points.
• "IM_GAUSS_PARALLELEPIPED(2,3)": Integration on quadrilaterals of order 3 with 4 points (shortcut for "IM_PRODUCT(IM_GAUSS1D(3),IM_GAUSS1D(3))").
• "IM_TETRAHEDRON(5)": Integration on a tetrahedron of order 5 with 15 points.
Note that "IM_QUAD(3)" is not able to integrate exactly the base functions of the "FEM_QK(2,3)" finite element! Since its base function are tensorial product of 1D polynomials of degree 3, one would
need to use "IM_QUAD(7)" (6 is not available). Hence "IM_GAUSS_PARALLELEPIPED(2,k)" should always be preferred over "IM_QUAD(2*k)" since it has less integration points.
An alternative way to obtain integration methods:
getfem::pintegration_method ppi =
getfem::classical_exact_im(bgeot::pgeometric_trans pgt);
getfem::pintegration_method ppi =
getfem::classical_approx_im(bgeot::pgeometric_trans pgt, dim_type d);
These functions return an exact (i.e. analytical) integration method, or select an approximate integration method which is able to integrate exactly polynomials of degree <= d (at least) for convexes
defined with the specified geometric transformation.
Methods of the mesh_im object¶
Once an integration method is defined on a mesh, it is possible to obtain information on it with the following methods (the list is not exhaustive). | {"url":"https://getfem.readthedocs.io/en/latest/userdoc/binteg.html","timestamp":"2024-11-09T20:31:15Z","content_type":"text/html","content_length":"25767","record_id":"<urn:uuid:c08e744b-3c12-4115-bded-c6be29e06b0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00831.warc.gz"} |
Interview Questions And Answers Forum
From: Chichawatni
Registered: 2014-10-13
Posts: 8,355
Minimum Value:
If x and y are two digits of a number such that the number 6538xy is divisible by 80 then minimum value of sum of x and y is:
Option A):
Option B):
Option C):
Option D):
Correct Answer is Option C):
Failure is the first step towards seccess.
2015-05-23 05:49:12
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Solving boundary value problems matlab software
You either can include the required functions as local functions at the end of a file as done here, or you can save them as separate, named files in a directory on the. In a boundary value problem
bvp, the goal is to find a solution to an ordinary differential equation ode that also satisfies certain specified boundary conditions. Boundary value problems for engineers with matlab solutions.
Boundary value problems bvps are ordinary differential equations that are subject to boundary conditions. In this section well define boundary conditions as opposed to initial conditions which we
should already be familiar with at this point and the boundary value problem. You either can include the required functions as local functions at the end of a file as done here, or you can save them
as separate, named files in a directory on the matlab path. Introduction to numerical ordinary and partial differential. We will also work a few examples illustrating some of the interesting
differences in using boundary values instead of initial conditions in solving differential equations. To extract the solution, the final two lines of code in the main program. Later chapters focus on
solving mixed boundary value problems using a variety of mathematical techniques. Oct 21, 2011 a boundary value problem is a system of ordinary differential equations with solution and derivative
values specified at more than one point.
Pdf solving linear boundary value problem using shooting. After a brief section on the threedimensional graphical capabilities of matlab, chapter 11 introduces partial differential equations based on
the model proble heat flomw o anf d steadystate distribution. Currently i have implemented the following basis functions. This video describes how to solve boundary value problems in matlab, using
the bvp4c routine. Download it once and read it on your kindle device, pc, phones or tablets. Solve boundary value problems for ordinary differential equations. Using ad to solve bvps in matlab acm
transactions on.
The text begins with an overview and history of mixed boundary value problems. This tutorial shows how to formulate, solve, and plot the solutions of boundary value problems bvps for ordinary
differential equations. These topics are usually taught in separate courses of length one semester each, but solving odes with matlab provides a sound treatment of all three in about 250 pages.
Numerical solutions of boundaryvalue problems in odes. While solving boundary value problems using bvp4c the graph is plotted as the command plot x,bs1. How do you use matlab for solving boundary
value problems with. Partial differential equation toolbox extends this functionality to generalized problems in 2d and 3d with dirichlet and neumann boundary conditions. Solving odes with matlab
each chapter begins with examples of the topic, and progresses to the development of numerical methods, focusing on the most widely used approaches. Solving boundary value problems for ordinary di
erential equations in matlab with bvp4c lawrence f. You either can include the required functions as local functions at the end of a file as done here, or save them as separate, named files in a.
Unlike initial value problems, a bvp can have a finite solution, no solution, or infinitely many solutions. Solving boundary value problem for piecewise defined. To solve this equation in matlab, you
need to write a function that represents the equation as a system of firstorder equations, a function for the boundary conditions, and a function for the initial guess.
Matlab boundaryvalue odes matlab has two solvers bvp4c and bvp5c for solving boundaryvalue odes bvp5c. The bvp4c and bvp5c solvers work on boundary value problems that have twopoint boundary
conditions, multipoint conditions, singularities in the solutions, or. The book has basically emerged from the need in the authors lectures on advanced numerical methods in biomedical engineering at
yeditepe university and it is aimed to assist the students in solving. An important part of the process of solving a bvp is providing a guess for the required solution. Then the bvp solver uses these
three inputs to solve the equation. The initial guess of the solution is an integral part of solving a bvp, and the quality of the guess can be critical for the solver performance or even for a
successful computation. I would like to put a builtin solver for boundary value differentialalgebraic equations on the whish list for future releases. Solving bvp for dae in matlab matlab answers
matlab central. Matlab includes bvp4c this carries out finite differences on systems of odes sol bvp4codefun,bcfun,solinit odefun defines odes bcfun defines boundary conditions solinit gives mesh
location of points and guess for solutions guesses are constant over mesh. To make solving bvps as easy as possible, the default in bvp4c is to approximate these derivatives with finite differences.
Standard, chebyshev, laguerre, legendre, and hermite. This tutorial shows how to write the functions files that describes the problem. Boundary value problems jake blanchard university of wisconsin
madison spring 2008. Tutorial on solving bvps with bvp4c file exchange matlab.
The matlab program bvp4c solves twopoint boundary value problems bvps of considerable generality. Most commonly, the solution and derivatives are specified at just two points the boundaries defining
a twopoint boundary value problem. Learn more about differential equations, piecewise matlab. You either can include the required functions as local functions at the end of a file as done here, or
save them as separate, named files in a directory on the matlab path. Finally, the chapters end with a tutorial that presents how to solve example problems using matlab and the symbolic math toolbox.
To solve this system of equations in matlab, you need to code the equations, boundary conditions, and initial guess before calling the boundary value problem solver bvp5c. Matlab is used to solve
numerous application examples in the book. The tutorial introduces the function bvp4c available in matlab 6. Solve bvp with multiple boundary conditions matlab. Solving boundary value problems for
ordinary differential. Solve boundary value problem fifthorder method matlab bvp5c. Python package for solving initial value problems ivp and twopoint boundary value problems 2pbvp using the
collocation method with various basis functions.
This package is free software which is distributed under the gnu general public license, as part of the r open source software project. Background information, solver capabilities and algorithms, and
example summary. The example function twoode has a differential equation written as a system of two firstorder odes. Unlike initial value problems, a boundary value problem can have no solution, a
finite number of solutions, or infinitely many solutions. I use matlab commands ode23 and ode45 for solving systems of differential equations and this program involves an. The r package bvpsolve for
the numerical solution of boundary value problems bvps is presented. Solving boundary value problems for ordinary di erential. This matlab function integrates a system of differential equations of
the form y. I am a bit dissapointed that this is the end of the road. To solve this equation in matlab, you need to code the equation, initial conditions, boundary conditions, and event function,
then select a suitable solution mesh before calling the solver pdepe. Boundary value problems 15859b, introduction to scientific computing paul heckbert 2 nov. Unlike ivps, a boundary value problem
may not have a solution, or may have a finite. The numerical method requires partial derivatives of several kinds. Bvp4c is a program that allows one to solve boundary value problems in matlab.
Boundary value problems for engineers springerlink. A 1d pde includes a function ux,t that depends on time t and one spatial variable x. How do you use matlab for solving boundary value problems.
More generally, one would like to use a highorder method that is robust and capable of solving general, nonlinear boundary value problems. While solving boundary value problems using bvp4c the. Fast
algorithms and matlab software for solution of the. Classes of problems the test problems that can be solved using the r package bvpsolve can be categorized into the following classes. Solve boundary
value problem fifthorder method matlab. The initial guess of the solution is an integral part of solving a bvp. To solve this system of equations in matlab, you need to code the equations, boundary
conditions, and initial guess before calling the boundary value problem solver bvp4c. Matlab can handle some singular bvps look at the documentation for bvp4c and the singularterm option in bvpset so
you need to bring your equation in the form that matlab can handle.
The sbvppackage contains functions for solving boundary value problems for. I know that the equations work because i have tested them in matlab. Use features like bookmarks, note taking and
highlighting while reading boundary value problems for engineers. Solve boundary value problem fourthorder method matlab. It includes some well known codes to solve boundary value problems of
ordinary dierential equations odes and dierential algebraic equations daes. The initial guess of the solution is an integral part of solving. Pdf solving boundary value problems in the open source.
For twopoint boundary value conditions like the ones in this problem, the boundary conditions function should have the signature res. Matlab boundary value problem ii two equation youtube. From the
matlab command line or any matlab program, sbvp is called by.
Kierzenka solving boundary value problems for ordinary differential equations in matlab with bvp4c. The boundary conditions specify a relationship between the values of the solution at two or more
locations in the interval of integration. This keeps the the spectrum of the book rather focussed. I encountered some complications solving a system of nonlinear 3 equations odes boundary value
problems numerically using the shooting method with the runge kutta method in matlab. Emphasis is placed on the boundary value problems that are often met in these fields. Solving boundary value
problems using matlab duration. Solving boundary value problems in the open source software r. | {"url":"https://tiltegambsnow.web.app/383.html","timestamp":"2024-11-12T15:34:48Z","content_type":"text/html","content_length":"15789","record_id":"<urn:uuid:86eb12fa-34af-4198-8e3c-ef7fc9f850c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00219.warc.gz"} |
The Seven Pillars of Statistical Reasoning
Originally published in The Reasoner Volume 10, Number 8– August 2016
Statistician Stephen Stigler put forward in the 1980’s the amusing Law of Eponymy which bears his name(!). According to Stigler’s Law, the vast majority (some say all) of scientific discoveries are
not named after those who actually made the discovery. Wikipedia lists a rather impressive number of instances of Stigler’s Law, featuring the Higgs Boson, Halley’s comet, Euler’s formula, the
Cantor-Bernstein-Schroeder theorem, and of course Newton’s first two laws of mechanics. Of particular interest is the case of Gauss, who according to this list, has his name mistakenly attached to
three items.
Rather coherently his recent book, S. Stigler (2016: The Seven Pillars of Statistical Wisdom, Harvard University Press), presents the fascinating edifice of statistics by giving more emphasis to the
key ideas on which its foundations rest, rather than to the figures who came up with them. The seven pillars are: Aggregation, or how to discard information to make things clearer; Information
measurement, or why not all pieces of information are equally important; Likelihood, or how probability plays a fundamental role in the calibration of statistical inference; Intercomparison, or why
the internal variation in data sets is fundamental in statistical comparisons; Regression, explaining why tall parents tend to have, on average, children who are shorter than themselves; Design, or
why asking well-posed questions is fundamental in statistics; and Residual, or how to simplify the analysis of complicated phenomena by abstracting from the effects of known causes. To each of the
seven pillars, Stigler devotes a chapter which outlines the history of the idea, and illustrates its relevance with many examples, ranging from astronomy to biology to medicine – as the saying goes
statisticians do really get to play in everyone’s backyard. In the concluding chapter Stigler identifies “the site” for the eight pillar, which is nonetheless still waiting for someone to be wrongly
credited with its introduction.
Interestingly, logicians also have some merit in the construction of the seven pillars of statistical wisdoms. In the chapter devoted to Design, for instance, Stigler points out that C.S. Peirce
explicitly theorised on the key concept of randomisation in his criticism to the then emerging theory of just noticeable differences in psychophysiology. In the 1885 essay On Small Differences in
Sensation, published in Memoirs of the National Academy of Sciences, 3, 73-83, C. Peirce and J. Jastrow, argued experimentally against the existence of a discrete threshold past which a detectable
stimulus ceases to be so. The crux of their argument, as reported by Stigler, consists in the extremely careful experimental design, of which Peirce and Jastrow give ample documentation, aimed at
ensuring the most rigorous randomisation in their lifted weights experiment. Stigler suggests that Peirce was aware of the methodological importance of randomisation well beyond the specific case of
this experiment. To this effect it is recalled that Peirce had defined “induction” as “reasoning from a sample taken at random to the whole lot sampled”. | {"url":"https://sites.unimi.it/hosni/2016/08/the-seven-pillars-of-statistical-reasoning/","timestamp":"2024-11-05T23:13:44Z","content_type":"text/html","content_length":"56951","record_id":"<urn:uuid:dc4a9e96-36d4-474c-9a08-bc5a928ddfa7>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00434.warc.gz"} |
2D Inverse Kinematics (IK) | Package Manager UI website
2D Inverse Kinematics (IK)
The 2D Inverse Kinematics (IK) package allows you to apply 2D IK to the bones and Transforms of your characters’ animation skeletons. 2D IK automatically calculates for the positions and rotations of
the a chain of bones moving towards a target position. This makes it easier to pose and animate character limbs for animation, or to manipulate a skeleton in real-time, as manually keyframing the
chain of bones is not required.
The following workflow continues from the 2D Animation package animation workflow, and demonstrates how to apply 2D IK to your character skeletons.
1. Refer to the hierarchy of bones created with the 2D Animation package's Bone Editor (refer to the 2D Animation package documentation for further information).
2. Add the IK Manager 2D component to the GameObject at the top of the hierarchy. This is usually the main root bone of the entire character skeleton.
3. Add to the IK Solvers list by selecting which type of IK Solver to use. The IK Solvers are also added as additional GameObjects in the hierarchy.
4. With an IK Solver selected, create and set the Effector and Target for the IK Solver.
5. Position bones by moving the Target's position to move the chain of bones with IK applied.
IK Solvers
The IK Solver calculates the position and rotation the Effector and its connected bones should take to achieve their Target position. Each type of IK Solver has its own algorithm that makes them
better suited to different kinds of conditions.
The following are properties are available to all Solvers:
Property Description
Effector Define the bone or Transform the IK Solver solves for.
Target The Transform which is used to indicate the desired position for the Effector.
Constrain Rotation This constrains the rotation of the Effector to the rotation of the Target.
Restore Default Pose Enable to restore the bones to their original positions before 2D IK is applied. Disable to apply 2D IK in relation to the Effector’s current position and
Weight Use the slider to adjust the degree the IK Solver’s solution affects the original Transform positions. At the lowest value of 0, the IK solution is
ignored. At the maximum value of 1 the IK solution is fully applied. This value is further influenced by the IK Manager's master Weight setting.
The following properties are only available -
to Chain (CCD) and Chain (FABRIK)
Chain Length The number of bones/Transforms (starting from the Effector) in the chain that the IK solution is applied to.
Iterations The number of times the algorithm runs.
Tolerance The threshold where the Target is considered to have reached its destination position, and when the IK Solver stops iterating.
This is a standard two bone Solver that is ideal for posing joints such as arms and legs. This Solver’s chain length is fixed to three bones - starting from the Effector bone/Transform and including
up to two additional bones in its chain.
Chain (CCD) - Cyclic Coordinate Descent
This IK Solver uses the Cyclic Coordinate Descent algorithm,which gradually becomes more accurate the more times thealgorithm is run. The Solver stops running once the set tolerance or number of
iterations is reached.
The following property is only available to the Chain (CCD) IK Solver:
Property Description
Velocity The speed the IK algorithm is applied to the Effector until it reaches its destination.
Chain (FABRIK) - Forward And Backward Reaching Inverse Kinematics
This IK Solver uses the Forward And Backward Reaching Inverse Kinematics (FABRIK) algorithm. It is similar to Chain (CCD) as its solution becomes more accurate the more times its algorithm is run.
The Solver stops running once the set tolerance or number of iterations is reached.
The Chain (FABRIK) Solver generally takes less iterations to reach the Target's destination compared to Chain (CCD), but is slower per iteration if rotation limits are applied to the chain. This
Solver is able to adapt quickly to if the bones are manipulated in real-time to different positions.
IK Manager 2D
The IK Manager 2D component controls the IK Solvers in the hierarchy. Add the Manager component to the highest bone in the hierarchy, commonly referred to as the Root bone.
1. In this example, add the component to PlunkahG as it is the Root bone in the hierarchy:
2. To add an IK Solver, click the + symbol at the bottom right of the IK Solvers list (see below).
3. A drop-down menu then appears with three options - Chain (CCD), Chain (FABRIK), and Limb. Each type of IK Solver uses a different algorithm to solve for the position of Effectors.
IK Solvers are iterated in descending order, with Solvers lower in the list referring to the positions set by the Solvers higher in the list. The order of Solvers usually reflects the order of bones/
Transforms in the skeleton hierarchy.
For example, if the arm bone is the child of the torso bone, then the torso's IK Solver should be set above the arm’s Solver in the list. Rearrange the Solvers by dragging the leftmost edge of a row
up or down.
Weight measures the degree that a Solver’s solution affects the positions of the bones/Transforms in the chain. The IK Manager 2D has a master Weight property that affects all Solvers it controls. It
is applied in addition to the Solver’s individual Weight settings.
Restore Default Pose
Click this to reset all bones and Transforms back to their original positions.
Creating an Effector and its Target
After creating an IK Solver, the next step is to set the Effector and its Target. A Target is a Transform that represents the target position the Effector attempts to reach. As the Effector moves
towards the Target position, the IK Solver calculates for the position and rotation of the Effector and the chain of bones it is connected to.
Follow the steps below to set a Target:
1. Select the last bone in the chain.
2. Create an empty Transform (right-click > Create Empty). It is automatically created as a child of the highlighted bone.
3. Move the position of the Transform to the tip of the last bone in the chain.
4. Select the IK Solver. With its Inspector window open, drag the Transform from the hierarchy onto the Effector field
5. Click the Create Target button. A Target is created at the Transform's position.
If the Create Target button appears inactive, ensure that the Chain Length value is set to one or greater.
1. The Target is created as a child of the IK Solver. It appears as a circle gizmo in the Scene view. Move the Target to manipulate the connected chain of bones.
Scripting API Reference
Adding New Solvers
You can add your own solver by extending from the class Solver2D. Your extended class will then show up as a new solver under the solver menu in the IKManager2D component.
This is the base class for all IK Solvers in this package. IKManager2D will detect all classes extending this and accept it as a Solver it can control. Implement or override the following methods to
create your own IK Solver:
• protected abstract int GetChainCount()
This function returns the number of IK chains the solver owns. Use this to return the number of IK chains your solver owns.
• public abstract IKChain2D GetChain(int index)
This function returns the IKChain2D at the given index. Use this to return the IKChain2D your solver owns at the given index.
• protected virtual bool DoValidate()
This function does validation for all parameters passed into the solver. Use this to check if your solver is set up correctly with all inputs.
• protected virtual void DoInitialize()
This function initializes the solver and builds the IK chains owned by the solver. This is called whenever the solver is invalid after changing the target of the solver or other parameters of the
solver. Use this to do initialize all the data from the parameters given to the solver, such as the IK chains owned by the solver.
• protected virtual void DoPrepare()
This function prepares and converts the information of the Transforms (position, rotation, IK parameters etc) to structures which can be used by the IK algorithms. Use this to do any work to gather
data used by your solver when updating the IK positions.
• protected abstract void DoUpdateIK(List effectorPositions)
This function calculates and sets the desired IK positions for the Transforms controlled by the solver given a list of effector positions for each chain owned by the solver. The effector positions
may be overridden by user positions if manipulated from the SceneView.
• protected virtual Transform GetPlaneRootTransform()
This function returns the transform whose localspace XY plane is used to perform IK calculations. Use this to define the Transform used.
This is the class which stores the transforms involved in an IK chain. When a chain is set up with a target and a transform count, initializing the Solver will populate the chain with the right
transforms if valid.
• Target - The transform which is used as the desired position for the target.
• Effector - The transform to perform IK on to reach a desired position.
• TransformCount - The number of transforms involved in the IK solution starting from the target. This is generally equivalent to ChainLength in solvers.
• Transforms - All transforms involved in the chain. In general, the last transform in this is the target transform and the first transform is considered the root transform for the chain.
• Lengths - The lengths between each transform in the chain.
This attribute allows you to tag your Solver2D with a different name under the IKManager2D. Use this if you do not want to use the name of the class of the Solver2D.
Example when giving the LimbSolver2D the name 'Limb' in the menu: [Solver2DMenuAttribute("Limb")] | {"url":"https://docs.unity.cn/Packages/com.unity.2d.ik@1.2/manual/index.html","timestamp":"2024-11-06T20:53:24Z","content_type":"text/html","content_length":"20431","record_id":"<urn:uuid:32dcbc9e-e534-4a28-85ea-7917e77cc123>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00016.warc.gz"} |
Factors and Multiples | Curious Toons
Table of Contents
Introduction to Factors
Definition of Factors
Let’s start with the definition of factors. In math, a factor is a whole number that can be divided evenly into another number. In simpler terms, if you take a number and divide it by one of its
factors, you’ll get a whole number as a result, with no remainder. For example, consider the number 12. The factors of 12 are numbers that you can multiply together to get 12. This means that 1, 2,
3, 4, 6, and 12 are all factors of 12 because:
• (1 \times 12 = 12)
• (2 \times 6 = 12)
• (3 \times 4 = 12)
You can think of factors as building blocks of a number. Every integer has at least two factors: 1 and itself. Factors are important because they help in understanding how numbers relate to each
other, and they play a critical role in various areas of math, including fractions, divisibility, and algebra. Recognizing factors allows us to simplify expressions and solve problems more
Identifying Factors of a Number
Now that we know what factors are, let’s talk about how to identify them. To find the factors of a specific number, we can start by listing all the integers from 1 up to that number. Then, we check
which of these numbers can divide our chosen number evenly, meaning there’s no remainder left over. For example, if we want to find the factors of 10, we would check each number from 1 to 10:
• (1) divides (10) evenly: (10 \div 1 = 10)
• (2) divides (10) evenly: (10 \div 2 = 5)
• (3) does not divide (10) evenly.
• (4) does not divide (10) evenly.
• (5) divides (10) evenly: (10 \div 5 = 2)
• (6) does not divide (10) evenly.
• (7) does not divide (10) evenly.
• (8) does not divide (10) evenly.
• (9) does not divide (10) evenly.
• (10) divides (10) evenly: (10 \div 10 = 1)
So the factors of 10 are 1, 2, 5, and 10. Identifying factors is a simple yet powerful skill that enhances our overall number sense and empowers us to solve a variety of mathematical problems.
Introduction to Multiples
Definition of Multiples
Let’s start by understanding what multiples are! A multiple of a number is the result of multiplying that number by any whole number. For instance, if we take the number 3, its multiples would be
obtained by multiplying 3 by 0, 1, 2, 3, and so on. So, the multiples of 3 are 0 (3 × 0), 3 (3 × 1), 6 (3 × 2), 9 (3 × 3), and it continues indefinitely. In general, for any whole number ( n ), the
multiples are ( 0, n, 2n, 3n, ) etc. This means that you can think of multiples as “skip counting” by that number. It’s also important to note that every number has an infinite number of multiples
because you can always keep multiplying it by higher whole numbers. Multiples can be positive (like 3, 6, 9) or negative (like -3, -6, -9) because multiplication can work with negative numbers as
well. Understanding multiples is crucial because they help us in various areas of math, including finding least common multiples, working with fractions, and solving problem scenarios in real life.
Identifying Multiples of a Number
Now that we know what multiples are, let’s focus on how to identify them! To find the multiples of a specific number, you can follow a simple process. Start with the number you want to find the
multiples of (let’s call it ( n )). Begin multiplying ( n ) by whole numbers; this will give you the first few multiples. For example, if we want to identify the multiples of 4, you can do the
• 4 × 0 = 0
• 4 × 1 = 4
• 4 × 2 = 8
• 4 × 3 = 12
• 4 × 4 = 16
So, the first five multiples of 4 are 0, 4, 8, 12, and 16. You can continue this process to find as many multiples as you need. Another way to identify multiples is to look for patterns: for
instance, multiples of 5 always end in 0 or 5. This can make it easier to identify multiples in larger numbers. You can also use divisibility rules, where if a number divides evenly into another
number (with no remainder), that larger number is a multiple of the smaller one. Practice identifying multiples to strengthen this skill, as it will be very useful throughout your math journey!
Prime and Composite Numbers
Definition of Prime Numbers
Let’s start by defining prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In simpler terms, it means that a prime number cannot
be formed by multiplying two smaller natural numbers. For example, the number 2 is prime because its only divisors are 1 and 2. Similarly, 3 is prime since its only factors are 1 and 3. Other
examples include 5, 7, 11, and 13.
One special case is the number 1, which is not considered a prime number because it has only one divisor (itself), not two. An interesting thing about prime numbers is that they are the building
blocks of all natural numbers; every number can be expressed as a product of prime numbers, a concept known as prime factorization. For instance, the number 28 can be expressed as 2 × 2 × 7, or 2² ×
7 in its prime factorization form. Understanding prime numbers is crucial as they play a significant role in various fields, including cryptography, computer science, and number theory.
Understanding Composite Numbers
Now, let’s explore composite numbers. A composite number is a natural number greater than 1 that has more than two positive divisors. This means that composite numbers can be divided evenly by
numbers other than just 1 and themselves. For instance, the number 4 is composite because it can be divided by 1, 2, and 4. Another example is 6, which has divisors 1, 2, 3, and 6.
Unlike prime numbers, composite numbers can be broken down into smaller factors. For example, the number 12 can be factored into 2 × 6 or 3 × 4, or further into its prime factorization: 2² × 3. It’s
important to note that all even numbers greater than 2 are composite since they can be divided by 2. This understanding is not just foundational for identifying numbers; it’s also essential for many
areas of mathematics, such as finding the greatest common divisors or the least common multiples. Recognizing the difference between prime and composite numbers will enhance your ability to work with
numbers in more complex mathematical scenarios.
Greatest Common Factor (GCF)
Finding the GCF
The Greatest Common Factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. To find the GCF, there are several methods we can use. One common technique is the
prime factorization method. This involves breaking down each number into its prime factors. For instance, let’s take the numbers 18 and 24. The prime factorization of 18 is 2 × 3 × 3 (or (2 \times 3^
2)), and for 24, it is 2 × 2 × 2 × 3 (or (2^3 \times 3)).
Next, we look for the common prime factors. Both numbers have the prime factors 2 and 3. Now we find the lowest power of each common factor: for 2, it’s (2^1) and for 3, it’s (3^1). Therefore, the
GCF is (2^1 \times 3^1 = 6). Another method to find the GCF is using the Euclidean algorithm, which involves repeated division. By employing these methods, you can efficiently find the GCF of any set
of numbers, which is essential for simplifying fractions and solving complex problems in higher mathematics.
Applications of GCF
Understanding the GCF has practical applications in many areas of math and everyday life. One of the primary uses of GCF is in simplifying fractions. When you wish to simplify a fraction, finding the
GCF of the numerator and the denominator allows you to reduce the fraction to its simplest form. For example, to simplify (\frac{18}{24}), we find the GCF, which we previously determined to be 6.
Dividing both the numerator and the denominator by 6 gives us (\frac{3}{4}), a simpler equivalent fraction.
GCF also plays a critical role in solving word problems involving grouping or distributing objects. For instance, if you have 36 apples and 48 oranges, GCF can help you determine how to group these
fruits into baskets evenly. By using GCF, you can create the largest possible baskets without any left over. Additionally, GCF is useful when finding common denominators in adding or subtracting
fractions. Being adept at finding GCF will not only make you more confident in your math skills but also better prepared for real-life applications and advanced mathematical concepts.
Least Common Multiple (LCM)
Finding the LCM
Finding the Least Common Multiple (LCM) of two or more numbers is an essential skill in math, especially when working with fractions or solving problems involving multiples. The LCM is the smallest
multiple that is common to each number in a given set. There are several methods to find the LCM. One popular method is listing multiples: write out multiples of each number until you find the
smallest one that appears in all lists. For example, to find the LCM of 3 and 4, list their multiples:
• Multiples of 3: 3, 6, 9, 12, 15, …
• Multiples of 4: 4, 8, 12, 16, …
You’ll notice that 12 is the smallest common multiple. Another efficient method is using prime factorization: break each number down into its prime factors, then take the highest power of each prime
that appears. For example, ( 3 = 3^1 ) and ( 4 = 2^2 ), so the LCM is ( 2^2 \times 3^1 = 12 ). Knowing how to find the LCM helps in solving many real-world problems, ensuring that we can add or
subtract fractions with different denominators and more!
Applications of LCM
The Least Common Multiple (LCM) has various practical applications that extend beyond the classroom, making it a valuable concept in real life. One common application is in scheduling events. For
instance, if two events occur every 6 days and 8 days respectively, determining when they coincide requires finding the LCM of 6 and 8, which is 24. This means both events will occur together every
24 days.
Another application of LCM can be found in managing fractions. When adding or subtracting fractions with different denominators, the LCM helps in finding a common denominator, allowing for
straightforward calculations. Furthermore, LCM is essential in solving problems in number theory, such as finding solutions to systems of equations involving multiples. In fields ranging from
engineering to data analysis, LCM helps streamline processes and optimize solutions, ensuring efficiency and effectiveness. Understanding the LCM not only supports mathematical computations but also
equips you with skills applicable in everyday scenarios, fostering better problem-solving techniques.
As we wrap up our exploration of factors and multiples, let us step back and consider the broader implications of what we’ve learned. At first glance, factors and multiples may seem like mere
numbers, but they serve as the building blocks of mathematical understanding. They connect diverse areas of math, from algebra to number theory, and even to real-world applications like cryptography
and computer science.
Think of factors as the friends who help you break down a problem into manageable pieces, allowing you to see the structure of numbers clearly. On the other hand, multiples represent the pathways
that lead to greater understanding, expanding our numerical horizons. This interplay between breaking down and building up isn’t just a mathematical concept; it reflects the way we approach
challenges in life.
Consider how understanding factors and multiples empowers you to tackle more complex problems, both in math and beyond. As you continue on your mathematical journey, remember that every equation can
be simplified, every problem has multiple solutions, and each of you has the power to explore new territories of knowledge. Keep questioning, keep engaging with numbers, and let curiosity guide your
learning. The world of math is vast—welcome the adventure! | {"url":"https://curioustoons.in/factors-and-multiples/","timestamp":"2024-11-09T21:00:17Z","content_type":"text/html","content_length":"106928","record_id":"<urn:uuid:91e0ae2f-ec23-4969-8f82-fb2ee4b20ba9>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00220.warc.gz"} |
Go Math Grade 6 Exercise 8.2: Understanding Percents Solutions - Math Book Answers
Go Math Grade 6 Exercise 8.2: Understanding Percents Solutions
Go Math! Practice Fluency Workbook Grade 6 California 1st Edition Chapter 8 Percents
Page 51 Problem 1 Answer
Given: The percent was given as 0.17
Find: Here, Write each decimal as a percent.
Solution: To convert a decimal to a percent, move the decimal point two places to the right and then write the % symbol after the number.
Therefore: 0.17=17%
The result of converting a decimal to a percent,17%
Page 51 Problem 2 Answer
The given decimal number is 0.56 .We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
0.56 is written as 56%.
Page 51 Problem 3 Answer
The given decimal number is 0.04. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
0.04 is written as 4%.
Page 51 Problem 4 Answer
The given decimal number is 0.7.We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
⇒0.7×100=7/10×100 = 70%.
0.7 is written as 70%.
Page 51 Problem 5 Answer
The given decimal number is 0.025. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
0.025 is written as 2.5%.
Page 51 Problem 6 Answer
The given decimal number is 0.803. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
0.803 is written as 80.3%.
Page 51 Problem 7 Answer
The given decimal number is 1.3. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
⇒1.3×100 = 13/10×100.
1.3 is written as 130%.
Page 51 Problem 8 Answer
The given decimal number is 2.10. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a number from decimal to percent then add a percent sign %.
We have to multiply the given decimal number by 100 to convert it into percent and then add the percent sign.
2.1 is written as 210%.
Page 51 Problem 9 Answer
The given fraction is 13/50. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
13/50 is written as 26%.
Page 51 Problem 10 Answer
The given fraction is 3/5. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
3/5 is written as 60%.
Page 51 Problem 11 Answer
The given fraction is 3/20. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
3/20 is written as 15%.
Page 51 Problem 12 Answer
The given fraction is 127/100.We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
127/100 is written as 127%.
Page 51 Problem 13 Answer
The given fraction is 5/8. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
5/8 is written as 62.5%.
Page 51 Problem 14 Answer
The given fraction is 45/90. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
45/90 is written as 50%.
Page 51 Problem 15 Answer
The given fraction is 7/5. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
7/5 is written as 140%.
Page 51 Problem 16 Answer
The given fraction is 19/25. We have to write the given decimal number in percentage form.
Multiply by 100 to convert a fraction to percent then add a percent sign %.
We have to multiply the given fraction by 100 to convert it into percent and then add the percent sign.
19/25 is written as 76%.
Page 51 Problem 17 Answer
Three numbers are given: 0.3,19/50,22%. We have to arrange the given numbers in increasing order.
We can convert all the given numbers either as fractions or decimal numbers or as a percent.
The given first number is 0.3. To convert the decimal number in percent, we have to multiply it by 100 and then add percent sign, “%”.
So, 0.3 is 30%.
The second given number is 19/50.
To convert the fraction in percent, we have to multiply it by 100 and then add percent sign, “%”.
So, 19/50 is 38%.
The given third number is 22% which is already a percent. Now, on comparing all the three percents, we get
The final order of the numbers from the least to greatest is 22%<0.3<19/50.
Page 51 Problem 18 Answer
Three numbers are given: 11%,1/8,2/25. We have to arrange the given numbers in increasing order.
We can convert all the given numbers either as fractions or decimal numbers or as a percent.
The given first number 11% is already a percent. So, we have to convert the remaining two fractions, 1/8&2/25 to percent now.
∴To convert the fraction in percent, we have to multiply it by 100 and then add percent sign, “%”.
and 2/25×100
Now, on comparing all the three percents, we get
Page 51 Problem 19 Answer
Three numbers are given: 5/8,0.675,5%.
We have to arrange the given numbers in increasing order.We can convert all the given numbers either as fractions or decimal numbers or as a percent.
The first given number is 5/8. To convert the fraction in percent, we have to multiply it by 100 and then add the percent sign, “%”.
So, 5/8 is 62.5%.
The given second number is 0.675. To convert the decimal number in percent, we have to multiply it by 100 and then add the percent sign, “%”.
So, 0.675 is 67.5%.
The given third number is 5% which is already a percent. Now, on comparing all the three percents, we get
The final order of the numbers from the least to greatest is 5%<5/8<0.675.
Page 51 Problem 20 Answer
Three numbers are given: 1.25,0.51,250% . We have to arrange the given numbers in increasing order.
We can convert all the given numbers either as fractions or decimal numbers or as a percent.
The first two given numbers are 1.25&0.51 which are decimal numbers.
To convert the decimal number into percent, we have to multiply it by 100 and then add the percent sign, “%”.
The given third number is 250% which is already a percent. Now, on comparing all the three percents, we get
The final order of the numbers from the least to greatest is 0.51<1.25<250%.
Page 51 Problem 21 Answer
Three numbers are given: 350/100,0.351,27%. We have to arrange the given numbers in increasing order.
We can convert all the given numbers either as fractions or decimal numbers or as percent.
The first given number is 350/100.
To convert the fraction in percent, we have to multiply it by 100 and then add the percent sign, “%”.
So, 350/100 is 350%.
The given second number is 0.351.
To convert the decimal number into percent, we have to multiply it by 100 and then add the percent sign, “%”.
So, 0.351/ is 35.1%.
The given third number is 27% which is already a percent. Now, on comparing all the three percents, we get
The final order of the numbers from the least to greatest is 27%<0.351<350/100.
Page 51 Problem 22 Answer
Given: – There are three given numbers 4/8,0.05,51%. We have to order these number from the least to greatest.
We can order the numbers by simplifying the fractions and percentages until it becomes a number with a decimal point which makes comparison easier.
Simplifying the first fraction 4/8
The second number is already in its decimal representation. There is no further simplification possible.
The third number is in its percentage form. Using the definition of percentage, we can express the number as 51%=51/100
Now, comparing the three numbers, we have 0.51>0.5>0.05
This implies 51%>4/8>0.05
The final order of the numbers from the least to greatest is 0.05<4/8<51%.
Page 51 Problem 23 Answer
Given: – During one hour, 6 out of 25 cars were travelling above the speed limit.
We have to find what percent of the cars were traveling above the speed limit.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
The fraction form for the cars travelling above the speed limit is 6/25.
Multiplying the numerator and denominator with 4, we get
The numerator of the fraction with denominator 100 is the required percentage. 6/25=24%
The percentage of the cars that were travelling above the speed limit is 24%.
Page 51 Problem 24 Answer
Given: – At Oaknoll School, 90 out of 270 students own computers.
We have to find the percentage of students who do not own computers.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
The number of students who do not use computers is 270−90=180.
The fraction form for the number of students who don’t use the computers is 180/270.
Simplifying the fraction, we get 90⋅2/90⋅3=2/3
The numerator of the fraction with denominator 100 is the required percentage. After rounding off to the nearest tenth, the percentage will be 2/3=66.7%
The percent of students at Oaknoll School that do not own computers (rounded to the nearest tenth) is 66.7%.
Page 52 Exercise 1 Answer
Given: – The given number in decimal form is 0.34.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 3.
After shifting the decimal point two places to the right, the number becomes 034.0
Now adding the percentage symbol, the final value in percentage form is 34%.
Writing the decimal as a percent, we get 34%.
Page 52 Exercise 2 Answer
Given: – The given number in decimal form is 0.06.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 0.
After shifting the decimal point two places to the right, the number becomes 006.0 (or) 6.
Now adding the percentage symbol, the final value in percentage form is 6%.
Writing the decimal as a percent, we get 6%.
Page 52 Exercise 3 Answer
Given: – The given number in decimal form is 0.93.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 9.
After shifting the decimal point two places to the right, the number becomes 093.0 (or) 93.
Now adding the percentage symbol, the final value in percentage form is 93%.
Writing the decimal as a percent, we get 93%.
Page 52 Exercise 4 Answer
Given: – The given number in decimal form is 0.57.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 5.
After shifting the decimal point two places to the right, the number becomes 057.0 (or) 57.
Now adding the percentage symbol, the final value in percentage form is 57 %.
Writing the decimal as a percent, we get 57%.
Page 52 Exercise 5 Answer
Given: – The given number in decimal form is 0.8.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 8.
After shifting the decimal point two places to the right, the number becomes 080.0 (or) 80.
Now adding the percentage symbol, the final value in percentage form is 80%.
Writing the decimal as a percent, we get 80%.
Page 52 Exercise 6 Answer
Given: – The given number in decimal form is 0.734.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 7.
After shifting the decimal point two places to the right, the number becomes 073.4 (or) 73.4.
Now adding the percentage symbol, the final value in percentage form is 73.4 %.
Writing the decimal as a percent, we get 73.4%.
Page 52 Exercise 7 Answer
Given: – The given number in decimal form is 0.082.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 0.
After shifting the decimal point two places to the right, the number becomes 008.2 (or) 8.2.
Now adding the percentage symbol, the final value in percentage form is 8.2%
Writing the decimal as a percent, we get 8.2%.
Page 52 Exercise 8 Answer
Given: – The given number in decimal form is 0.225.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 2.
After shifting the decimal point two places to the right, the number becomes 022.5 (or) 22.5.
Now adding the percentage symbol, the final value in percentage form is 22.5%
Writing the decimal as a percent, we get 22.5%.
Page 52 Exercise 9 Answer
Given: – The given number in decimal form is 0.604.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 6.
After shifting the decimal point two places to the right, the number becomes 060.4 (or) 60.4.
Now adding the percentage symbol, the final value in percentage form is 60.4%.
Writing the decimal as a percent, we get 60.4%.
Page 52 Exercise 10 Answer
Given: – The given number in decimal form is 0.09.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 0.
After shifting the decimal point two places to the right, the number becomes 009.0 (or) 9.
Now adding the percentage symbol, the final value in percentage form is 9%.
Writing the decimal as a percent, we get 9%.
Page 52 Exercise 11 Answer
Given: – The given number in decimal form is 0.518.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 0 and 5.
After shifting the decimal point two places to the right, the number becomes 051.8 (or) 51.8.
Now adding the percentage symbol, the final value in percentage form is 51.8 %.
Writing the decimal as a percent, we get 51.8%.
Page 52 Exercise 12 Answer
Given: – The given number in decimal form is 1.03.
We have to find the percentage form of this decimal.
By appropriately shifting the decimal point, we can express a decimal in its percentage form.
The given number has its decimal point in between 1 and 0.
After shifting the decimal point two places to the right, the number becomes 103.0 (or) 103.
Now adding the percentage symbol, the final value in percentage form is 103%.
Writing the decimal as a percent, we get 103%.
Page 52 Exercise 13 Answer
Given: – The given fraction is 3/10.
We have to express the fraction as a percentage.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
Multiply the numerator and the denominator with 10.
The numerator of the fraction with denominator 100 is the required percentage.
Writing the fraction as a percent, we get 30%.
Page 52 Exercise 14 Answer
Given: – The given fraction is 2/50.
We have to express the fraction as a percentage.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
Multiply the numerator and denominator with 2.
The numerator of the fraction with denominator 100 is the required percentage.
Writing the fraction as a percentage, we get 4%.
Page 52 Exercise 15 Answer
Given: – The given fraction is 7/20.
We have to express the fraction as a percentage.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
Multiply the numerator and denominator by 5.
The numerator of the fraction with denominator 100 is the required percentage.
Writing the fraction as a percentage, we get 35%.
Page 52 Exercise 16 Answer
Given: – The given fraction is 1/5.
We have to express the fraction as a percentage.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
Multiply the numerator and denominator with 20.
The numerator of the fraction with denominator 100 is the required percentage.
Writing the fraction as a percentage, we get 20%.
Page 52 Exercise 17 Answer
Given: – The given fraction is 1/8.
We have to express the fraction as a percentage.
Multiplying the numerator and denominator with a number that can make the denominator equal to 100.
The corresponding numerator is the required percentage.
The numerator of the fraction with denominator 100 is the required percentage.
Writing the fraction as a percentage, we get 12.5%.
Page 52 Exercise 18 Answer
Given: The fraction is 3/25
To find: Express the fraction as a percent.Summary: To convert a fraction into a percent, we divide the numerator by the denominator.
Multiply the decimal with 100. Another way is to multiply the numerator and the denominator with some number so that the denominator becomes 100.
The corresponding numerator is the required percentage.
Multiply the numerator and the denominator with 4.
The numerator of the fraction with denominator 100 is the required percentage.
We can write the fraction 3/25 as 12%.
Page 52 Exercise 19 Answer
Given: The fraction is 3/4
To find: Express the fraction as a percent.Summary: To convert a fraction into a percent, we divide the numerator by the denominator.
Multiply the decimal thus obtained with 100.
Divide the numerator by the denominator.
Multiply the decimal with 100.
We can write the fraction 3/4 as 75%.
Page 52 Exercise 20 Answer
Given: The fraction is 23/50.
To find: Express the fraction as a percent.
Summary: We can convert a fraction into a percent, by first dividing the numerator by the denominator and then multiplying the decimal with 100.
Another way is to multiply the numerator and the denominator with some number so that the denominator becomes 100, and the numerator is the percentage.
Multiply the numerator and the denominator with 2.
The numerator of the fraction with denominator 100 is the required percentage.
Converting the fraction into percentage we get, 23/50
Page 52 Exercise 21 Answer
Given: The fraction is 11/20.
To find: Express the fraction as a percent.
Summary: To convert a fraction into a percent, we divide the numerator by the denominator.
Multiply 100 to the decimal. Or else, multiply the numerator and the denominator with some number so that the denominator becomes 100.
And the corresponding numerator is the required percentage.
Multiply the numerator and the denominator with 5.
The numerator of the fraction with denominator as 100 is the percentage.
Page 52 Exercise 22 Answer
Given: The fraction is 43/50
To find: Express the fraction as a percent.
Summary: First we divide the numerator by the denominator. Multiply the decimal with 100.
Another way is to multiply the numerator and the denominator with some number so that the denominator becomes 100.
Then the corresponding numerator becomes the required percentage.
Multiply the numerator and the denominator with 2.
The numerator of the fraction with denominator as 100 is the required percentage.
Converting the fraction into percentage, we get, 43/50=86%.
Page 52 Exercise 23 Answer
Given: The fraction is 24/25
To find: Express the fraction as a percent.Summary: In order to convert a fraction into a percent, divide the numerator by the denominator.
Multiply the decimal with 100. Alternate way is to multiply the numerator and the denominator with some number so that the denominator becomes 100, so that the corresponding numerator is the required
Multiply both numerator and the denominator with 4.
The required percentage is the numerator of the fraction with denominator as 100.
Converting the fraction to the percentage we get,
Page 52 Exercise 24 Answer
Given: The fraction is 7/8.
To find: Express the fraction as a percent.
Summary: To convert a fraction into a percent, first we divide the numerator by the denominator.
Then multiply the decimal with 100. Another way is to multiply the numerator and the denominator with some number so that the denominator becomes 100.
The corresponding numerator is the required percentage.
Divide the numerator by the denominator.
Multiply the decimal with 100.
We can write the given fraction 7/8 as 87.5%
Go Math Answer Key
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AP Statistics Chapters 3 & 4 Measuring Relationships Between 2 Variables. - ppt download
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Degree Of Polynomial Worksheet
Degree Of Polynomial Worksheet. In each case, find the degree of the given polynomial. We get a smooth and continuous line.
Graphs Of Polynomial Functions Worksheet worksheet from novenalunasolitaria.blogspot.com
(i) a + b + c. Web degree of polynomial interactive and downloadable worksheets. In each case, find the degree of the given polynomial. | {"url":"http://studydblamb123.s3-website-us-east-1.amazonaws.com/degree-of-polynomial-worksheet.html","timestamp":"2024-11-04T02:04:10Z","content_type":"text/html","content_length":"24874","record_id":"<urn:uuid:61c14c40-52d6-47ec-8be1-2b5c530c9512>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00239.warc.gz"} |
QuiverTools. If you use it for your research, please cite it using .
I am really excited to announce something that has been in the making for quite some time, and which is the result of the hard work with
Many questions in the representation theory and algebraic geometry of quivers and their moduli of representations can be answered using very algorithmic methods. This is what QuiverTools is about. We
do not deal with actual representations (that is what QPA is for), rather our focus is on the geometry of moduli spaces.
An example
Let us illustrate some things. One of my favourite quiver moduli is the 6-dimensional Kronecker moduli space, which Hans and I considered in
On Chow rings of quiver moduli
. Let us define it in QuiverTools:
from quiver import *
Q = KroneckerQuiver(3)
X = QuiverModuliSpace(Q, (2, 3))
Then we can compute some of its basic invariants as follows:
X.is_projective() # True
X.is_smooth() # True
X.dimension() # 6
X.picard_rank() # 1
X.index() # 3
Just like in the Hodge diamond cutter you can compute Betti numbers:
X.betti_numbers() # [1, 0, 1, 0, 3, 0, 3, 0, 3, 0, 1, 0, 1]
We also support the recent work involving Chow rings and rigidity questions for quiver moduli:
eta = -Q.canonical_stability_parameter(d) / 3
X.degree(eta) # 57
X.if_rigidity_inequality_holds() # True
This is only a very quick tour of what it can do, more explanations are to come!
If you have problems installing it, you can also run it inside your browser using this interactive binder notebook.
Some comments
• This release is really only a v1.0. Let us know if you find issues, or have feature requests!
• We are also working on a Julia version, focusing more on performance. Stay tuned for more information.
• We are also working on more detailed instructions: for now there is the documentation taken from the docstrings, which should be good enough if you have some experience with Sage and quiver
moduli, but a more user-friendly guide is in the works. | {"url":"https://pbelmans.ncag.info/blog/2024/07/07/quivertools/","timestamp":"2024-11-11T17:19:44Z","content_type":"text/html","content_length":"24650","record_id":"<urn:uuid:e574e73a-47df-4cb7-a4f5-faf79da0794d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00894.warc.gz"} |
Alternate Solution to Example Problem on Series Circuits
Here is an alternate way of solving the example problem. Instead of first finding all the resistances, we can begin the problem by finding the emf voltage first.
By Ohm's Law, we know that the emf is equal to the product of the total current and the total resistance.
\(\varepsilon = I R\)
\(\varepsilon = (1.0)(30) = 30V\)
Figure 1: Example Problem, with given data
Now that we know the emf voltage, we also know the total voltage. Since \(\varepsilon =V\), where V is the total voltage, then \(V=30 V\).
Also, the total voltage, \(V\), is equal to the sum of the voltages across the resistors in this circuit (because this is a series circuit).
\(V = V_1 + V_2 + V_3 + V_4\)
We also know all the voltages across the resistors except for \(V_4\). So,
\(V_4 = V - (V_1 + V_2 + V_3)\)
\(V_4 = 30 - (5.0 + 8.0 + 7.0)\)
\(V_4 = 10V\)
The only unknowns now are the resistances. We know the voltages across each resistor. Also, because this is a series circuit, the current through each resistor is the same. Then, by Ohm's Law, we can
find the resistances of each resistor:
\(R_1 = \frac {V_1}{I}, R_2 = \frac {V_2}{I}, R_3 = \frac{V_3}{I}\)
\(R_1 = \frac {5.0}{1.0} =5.0 \quad \Omega \quad R _2 = \frac {8.0}{1.0} = 8.0 \quad \Omega \quad R_3 = \frac{7.0}{1.0} = 7.0 \Omega\) | {"url":"https://www.physics.uoguelph.ca/alternate-solution-example-problem-series-circuits","timestamp":"2024-11-14T13:45:06Z","content_type":"text/html","content_length":"58148","record_id":"<urn:uuid:f251178c-1357-41ac-b26f-5ed1db1aa115>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00546.warc.gz"} |
Blog Archives
Variation is also known as "Variability", "Dispersion", "Spread", and "Scatter". (5 names for one thing is one more example why statistics is confusing.) Variation is 1 of 3 major categories of
measures describing a Distribution or data set. The others are Center (aka "Central Tendency") with measures like Mean, Mode, and Median and Shape (with measures like Skew and Kurtosis). Variation
measures how "spread out" the data is.
There are a number of different measures of Variation. This compare-and-contrast table shows the relative merits of each.
• The Range is probably the least useful in statistics. It just tells you the highest and lowest values of a data set, and nothing about what's in between.
• The Interquartile Range (IQR) can be quite useful for visualizing the distribution of the data and for comparing several data sets -- as described in a recent post on this blog.
• Variance is the square of the Standard Deviation, and it is used as an interim step in the calculation of the latter. This squaring overly emphasizes the effects very high or very low values.
Another drawback is that it is in units of the data squared (e.g. square kilograms, which can be meaningless). There is a Chi-Square Test for the Variance, and Variances are used in F tests and
the calculations in ANOVA.
• The Mean Absolute Deviation is the average (unsquared) distance of the data points from the Mean. It is used when it is desirable to avoid emphasizing the effects of high and low values
• The Standard Deviation, being the square root of the Variance, does not overly emphasize the high and low values as the Variance does. Another major benefit is that it is in the same units as
the data.
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Alpha is the Significance Level of a statistical test. We select a value for Alpha based on the level of Confidence we want that the test will avoid a False Positive (aka Alpha aka Type I) Error. In
the diagrams below, Alpha is split in half and shown as shaded areas under the right and left tails of the Distribution curve. This is for a 2-tailed, aka 2-sided test.
In the left graph above, we have selected the common value of 5% for Alpha. A Critical Value is the point on the horizontal axis where the shaded area ends. The Margin of Error (MOE) is half the
distance between the two Critical Values.
A Critical Value is a value on the horizontal axis which forms the boundary of one of the shaded areas. And the Margin of Error is half the distance between the Critical Values.
If we want to make Alpha even smaller, the distance between Critical Values would get even larger, resulting in a larger Margin of Error.
The right diagram shows that if we want to make the MOE smaller, the price would be larger Alpha. This illustrates the Alpha - MOE see-saw effect. But what if we wanted a smaller MOE without making
Alpha larger? Is that possible? It is -- by increasing
, the Sample Size. (It should be noted that, after a certain point, continuing to increase
yields diminishing returns. So, it's not a universal cure for these errors.)
If you'd like to learn more about Alpha, I have 2 YouTube videos which may be of interest:
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Quadratic forms and applications in algebraic geometry
Tuesday, 05/30/2023, 9 am
Friday, 06/02/2023, 6 pm
RWTH Aachen University:
Pontdriesch 14/16
Background Material
Lecture Notes from the course on algebraic number theory (Nebe)
For a more elaborate version we recommend the book by J. Neukirch on Algebraische Zahlentheorie.
Lecture Notes from the course on quadratic forms (Nebe)
We recommend the book by M. Kneser on Quadratische Formen.
As a preparation for the algebraic geometric talks read Wolfgang Ebeling Lattices and Codes Chapter 1 which treats the classification of root lattices. This is also useful if you want to have
examples for integral quadratic forms and thus lattices over the Dedekind domain of rational integers.
The lectures on automorphisms of K3 surfaces follow the survey article Shigeyuki Kondō - A survey of finite groups of symplectic automorphisms of K3 surfaces, 2018 J. Phys. A: Math. Theor. 51 053003
For further reading on K3 surfaces I recommend the introduction/survey Harder, A., Thompson, A. (2015). The Geometry and Moduli of K3 Surfaces.
In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY.
OSCAR documentation and installation
Lecture Notes
Lecture Notes for the lecture on Tuesday morning (Nebe)
Slides of Markus Kirschmers talk on Lattices over Dedekind domains (Tuesday afternoon)
Material for the talk on Lattices in Oscar
Slides for the talk on sums of squares in number fields (Krasensky)
Notes for the lecture on Friday morning (Nebe) and Friday early afternoon (Breuer) | {"url":"http://www.math.rwth-aachen.de/~Gabriele.Nebe/SummerSchool2023/download.html","timestamp":"2024-11-10T18:51:22Z","content_type":"application/xhtml+xml","content_length":"4853","record_id":"<urn:uuid:f3f2426d-56f8-4f5c-b5c6-67a91af6012e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00708.warc.gz"} |
In which condition intersection of two sets is equal to the union of two sets?
In which condition intersection of two sets is equal to the union of two sets?
There are not three consecutive statement. There is an equivalence between two equalities. You do the former by showing that if the intersection equals the union then every element that is in A must
be in B, and by symmetry that all in B must be in A.
What is the union of two equal sets?
The union of two sets contains all the elements contained in either set (or both sets). The union is notated A ⋃ B. The intersection of two sets contains only the elements that are in both sets.
What is the intersection of two sets?
The intersection of two or more given sets is the set of elements that are common to each of the given sets. The intersection of sets is denoted by the symbol ‘∩’. In the case of independent events,
we generally use the multiplication rule, P(A ∩ B) = P( A )P( B ).
Can intersection and union be equal?
14.4 Union and intersection (EMA7Z) The union is written as A∪B or “A or B”. The intersection of two sets is a new set that contains all of the elements that are in both sets. In the final column the
union, A∪B, is equal to A and the intersection, A∩B, is equal to B since B is fully contained in A.
Is the empty set a subset of the empty set?
No set is a proper subset of itself. The empty set is a subset of every set. The empty set is a proper subset of every set except for the empty set.
What is the intersection of two equal sets?
Intersection of two given sets is the largest set which contains all the elements that are common to both the sets. To find the intersection of two given sets A and B is a set which consists of all
the elements which are common to both A and B. The symbol for denoting intersection of sets is ‘∩’.
Is the empty set equal to a set containing the empty set?
The power set of a set is defined to be the set which contains all of the subsets of the set and nothing more. The only subset of the empty set is the empty set itself. Hence, the power set of the
empty set is the set containing only the empty set.
What is empty set intersection empty set?
Intersection With the Empty Set The empty set is the set with no elements. If there are no elements in at least one of the sets we are trying to find the intersection of, then the two sets have no
elements in common. In other words, the intersection of any set with the empty set will give us the empty set. | {"url":"https://profoundqa.com/in-which-condition-intersection-of-two-sets-is-equal-to-the-union-of-two-sets/","timestamp":"2024-11-05T06:11:26Z","content_type":"text/html","content_length":"60556","record_id":"<urn:uuid:c22fa620-e0de-4153-9f75-d2d275e5791b>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00345.warc.gz"} |
Lower bounds on individual sequence regret
In this work, we lower bound the individual sequence anytime regret of a large family of online algorithms. This bound depends on the quadratic variation of the sequence, Q [T], and the learning
rate. Nevertheless, we show that any learning rate that guarantees a regret upper bound of O(√Q [T]) necessarily implies an Ω(√Q [T]) anytime regret on any sequence with quadratic variation Q [T] .
The algorithms we consider are linear forecasters whose weight vector at time t + 1 is the gradient of a concave potential function of cumulative losses at time t. We show that these algorithms
include all linear Regularized Follow the Leader algorithms. We prove our result for the case of potentials with negative definite Hessians, and potentials for the best expert setting satisfying some
natural regularity conditions. In the best expert setting, we give our result in terms of the translation-invariant relative quadratic variation. We apply our lower bounds to Randomized Weighted
Majority and to linear cost Online Gradient Descent. We show that bounds on anytime regret imply a lower bound on the price of "at the money" call options in an arbitrage-free market. Given a lower
bound Q on the quadratic variation of a stock price, we give an Ω(√Q) lower bound on the option price, for Q < 0.5. This lower bound has the same asymptotic behavior as the Black-Scholes pricing and
improves a previous Ω(Q) result given in [4].
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 7568 LNAI
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 23rd International Conference on Algorithmic Learning Theory, ALT 2012
Country/Territory France
City Lyon
Period 29/10/12 → 31/10/12
Funders Funder number
Google Inter-university center for Electronic Markets and Auctions
Israeli Ministry of Science
Maryland Ornithological Society
United States-Israel Binational Science Foundation
Israel Science Foundation
Israeli Centers for Research Excellence 4/11
Dive into the research topics of 'Lower bounds on individual sequence regret'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/lower-bounds-on-individual-sequence-regret-2","timestamp":"2024-11-12T10:07:26Z","content_type":"text/html","content_length":"54506","record_id":"<urn:uuid:86de2b1e-f7e9-4bc1-9010-e54fb56126cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00433.warc.gz"} |
Water (H₂O) - Definition, Structure, Preparation, Uses, Benefits
Water stands as a paramount covalent compound within the realm of chemistry. This molecule is composed of two hydrogen atoms bonded to a single oxygen atom through covalent bonds, a configuration
that renders it essential for myriad biological processes and ecological systems. Its unique properties, such as its capability to dissolve a vast array of substances, high specific heat capacity,
and surface tension, make water indispensable not only to life on Earth but also in numerous chemical reactions and industrial applications. As a covalent compound, water exemplifies the fundamental
principles of chemical bonding, showcasing the intricate interplay of elements that defines the chemical world.
What is Water ?
Water is symbolized as H₂O, is a simple yet vital substance made up of two hydrogen atoms bonded to one oxygen atom. It’s found everywhere on Earth, from vast oceans to tiny droplets in the air. This
clear, tasteless, and odorless liquid is crucial for all forms of life, supporting plants, animals, and humans by helping in processes like drinking, cleaning, and growing food. Water also takes on
different forms; it can be solid ice, liquid, or gas (steam), changing with the temperature. Its ability to dissolve many substances makes it an essential part of nature and daily life.
Chemical Names and Formulas
Property Value
Formula H2O
Name Water
Alternate Names Dihydrogen Monoxide, Distilled Water, Hydrogen Hydroxide, Oxidane
Types Of Water (H₂O)
Water (H₂O) is a remarkable substance that exists in three primary states: liquid, solid, and gaseous. Each form exhibits unique characteristics and plays a crucial role in the Earth’s ecosystems and
human activities.
Liquid Water
Liquid water is the most familiar state to us, covering about 71% of the Earth’s surface as oceans, rivers, lakes, and rain. This form is indispensable for all known forms of life, serving as a
medium for biological reactions, transportation of nutrients, and regulation of body temperature. Liquid water is unique for its high specific heat capacity, surface tension, and ability to dissolve
more substances than any other liquid, making it known as the “universal solvent.”
Solid Water (Ice)
When water freezes at 0°C (32°F), it becomes solid ice. Ice is less dense than liquid water, which is why it floats — a property crucial for aquatic life to survive under ice-covered waters in
winter. Ice forms in various structures, from the vast ice sheets in Antarctica to the delicate frost patterns on a window. This solid state of water is also essential for activities such as ice
skating and preserving food.
Gaseous Water (Steam/Vapor)
Water vapor, the gaseous form of water, is produced when water evaporates or boils. Steam is water in its gaseous state at temperatures above 100°C (212°F), where it becomes visible, like the steam
from a boiling kettle. Water vapor, on the other hand, is invisible and a significant component of the Earth’s atmosphere, contributing to the greenhouse effect and the water cycle. This cycle,
through evaporation and condensation, distributes heat and moisture around the globe, influencing weather patterns and climate.
Structure Of Water (H₂O)
The structure of water (H₂O) is both simple and fascinating, making it one of the most important substances on Earth. At its core, a water molecule consists of two hydrogen atoms and one oxygen atom.
These atoms are not just randomly attached; the two hydrogen atoms form a V-shape, or an angle, as they bond with the oxygen atom. This happens because the oxygen atom shares electrons with each
hydrogen atom through a type of bond known as a covalent bond. This shared bonding gives water its liquid form at room temperature, unlike other similar molecules that might be gases.
What makes water really special is the way these molecules interact with each other. The side of the water molecule where the hydrogen atoms are is slightly positive, while the opposite side, near
the oxygen, is slightly negative. This causes water molecules to be attracted to each other like tiny magnets, a property called hydrogen bonding. These bonds are strong enough to hold water
molecules close together but flexible enough to let them move around freely, leading to water’s unique properties, such as its ability to dissolve many substances, its surface tension, and its
unusual behavior when it freezes, expanding instead of contracting. This structure and the interactions it enables are crucial for life as we know it, affecting everything from the way our cells
function to the weather patterns that shape our world.
Preparation of Water (H₂O)
Synthesizing Water from Hydrogen and Oxygen
The direct synthesis of water involves the reaction of hydrogen gas (H₂) with oxygen gas (O₂) to produce water. This process can be summarized by the chemical equation:
This reaction is highly exothermic, meaning it releases a significant amount of energy in the form of heat and light. In a controlled laboratory environment, this can be demonstrated by igniting a
mixture of hydrogen and oxygen gases, where the resulting “explosion” produces water vapor. For safety and control, this reaction is usually carried out using a spark to ignite the gases in a
controlled explosion chamber designed to contain the reaction’s energy.
Electrolysis of Water
Another method to prepare water, though seemingly counterintuitive, is the electrolysis of water itself. This process uses electrical energy to split water into its constituent elements, hydrogen and
oxygen gas. The reverse process can then recombine these gases to form water, demonstrating both the decomposition and synthesis of water:
In this case, electrolysis serves as an educational tool, showing the reversible nature of chemical reactions and the conservation of matter.
Physical Properties of Water (H₂O)
Property Description
Molecular Weight 18.01528 g/mol
State at Room Temperature Liquid
Boiling Point 100°C (212°F) at 1 atmosphere (atm) pressure
Freezing Point 0°C (32°F) at 1 atm pressure
Density 1 g/cm³ at 4°C (maximum density)
pH 7 (neutral) at 25°C
Heat Capacity High (about 4.186 joule/gram °C), meaning it can absorb a lot of heat before it gets hot.
Surface Tension High (72.8 milliNewtons/m at 20°C), allowing it to form droplets and flow in narrow spaces.
Solvent Properties Excellent solvent due to its polarity, capable of dissolving many substances.
Thermal Conductivity 0.6065 W/(m·K) at 25°C, making it a good conductor of heat compared to other liquids.
Viscosity 0.890 mPa·s at 25°C, indicating its resistance to flow.
Dielectric Constant High (about 78.5 at 25°C), making it an effective medium for electrochemical reactions.
Chemical Properties of Water (H₂O)
Water molecules are polar, with the oxygen atom carrying a slight negative charge and the hydrogen atoms carrying slight positive charges. This polarity enables water to form hydrogen bonds with
other molecules.
Cohesion and Adhesion
• Cohesion: Water molecules stick together due to hydrogen bonding.
• Adhesion: Water molecules adhere to other types of surfaces.
Heat Capacity
Water has a high specific heat capacity, allowing it to absorb or release large amounts of heat with minimal temperature change. The specific heat of water is approximately 4.186J/g∘C.
Water ionizes slightly into hydronium and hydroxide ions:
Density Anomaly
Water reaches its maximum density at 4°C. As it freezes, it expands:
This expansion decreases its density, allowing ice to float on liquid water.
Solvent Capabilities
Water’s ability to dissolve many substances is due to its polarity, though no single equation defines this property. It’s the basis for many chemical reactions, such as salt dissolution:
Water can be decomposed into oxygen and hydrogen gases by electrolysis, a process that requires the passage of an electric current, demonstrating its role in electrochemical reactions.
Neutral pH
Pure water has a neutral pH of 7, indicating it’s neither acidic nor basic. The pH is determined by the concentration of hydronium ions:
Water (H₂O) Chemical Compound Information
Chemical Identifiers
Property Value
CAS Registry Number 7732-18-5
PubChem Compound ID 962
PubChem Substance ID 24851683
SMILES Identifier O
InChI Identifier InChI=1/H2O/h1H2
RTECS Number ZC0110000
MDL Number MFCD00011332
Uses of Water (H₂O)
Drinking and Cooking
Water is essential for hydration and is used in cooking processes like boiling, steaming, and as an ingredient in countless recipes.
It is crucial for irrigating crops, sustaining livestock, and providing the necessary moisture for plant growth.
Water’s solvent properties make it effective for cleaning dishes, clothes, and surfaces, removing dirt and contaminants.
Industrial Processes
Used in manufacturing processes, cooling machinery, and as a solvent in a wide range of chemical productions.
Energy Production
Water is a key component in hydroelectric power plants, steam generation in thermal power plants, and cooling in nuclear reactors.
Rivers, canals, and oceans serve as pathways for the transportation of goods and people.
Water bodies are central to recreational activities like swimming, boating, and fishing.
Lakes, rivers, and oceans provide habitats for a diverse range of flora and fauna, supporting biodiversity.
Chemical and Biological Reactions
Water is a medium for countless chemical reactions in laboratories and biological processes within living organisms.
Climate Regulation
The distribution and cycling of water influence weather patterns and help regulate the Earth’s temperature through processes like evaporation and precipitation.
Benefits of Water (H₂O)
Water is essential for maintaining body hydration, crucial for nearly all bodily functions, including regulation of body temperature, nutrient transport, and waste removal.
Nutrient Transport
Water acts as a solvent, facilitating the transport and absorption of vitamins, minerals, and other nutrients in biological systems.
It helps in the elimination of waste products and toxins from the body through urine, sweat, and other excretory paths, supporting kidney function and overall detoxification.
Digestive Health
Water aids in digestion by dissolving nutrients and helping to move food through the gut, preventing constipation and promoting healthy gastrointestinal function.
Skin Health
Adequate hydration keeps the skin moisturized, elastic, and healthy, reducing the risk of dry skin, wrinkles, and other skin issues.
Temperature Regulation
Water has a high heat capacity, enabling it to play a critical role in regulating body temperature through sweating and respiration.
Cognitive Function
Proper hydration is linked to improved concentration, memory, and mood, as water comprises a significant portion of the brain.
Joint Lubrication
Water lubricates and cushions joints, reducing the risk of joint pain and wear over time, important for movement and comfort.
Weight Management
Drinking water can aid in weight management by increasing satiety and enhancing metabolic rate, helping to control hunger and burn more calories.
Ecosystem Support
Water supports ecosystems by providing habitats for diverse wildlife, facilitating nutrient cycles, and sustaining plant life, essential for biodiversity.
Agricultural Productivity
Water is vital for agriculture, enabling the growth of crops and sustaining livestock, which are foundational to food security.
Economic Development
Access to water boosts economic activities by supporting industries, agriculture, and energy production, crucial for development and prosperity.
How Much Water Should You Drink Daily?
Adults should aim for 8-10 cups (2-2.5 liters) of water daily, though needs vary by activity level, climate, and health.
What Does Water Do for Your Body?
Water hydrates, aids in nutrient transport, supports digestion, regulates temperature, and helps remove body wastes efficiently.
What Is the Full Name of Water?
The chemical name for water is dihydrogen monoxide, represented as H₂O, comprising two hydrogen atoms and one oxygen atom.
What Is the Density of Water?
The density of water is 1 gram per cubic centimeter (g/cm³) at its maximum density at 4°C (39.2°F). | {"url":"https://www.examples.com/chemistry/water.html","timestamp":"2024-11-13T22:45:34Z","content_type":"text/html","content_length":"130762","record_id":"<urn:uuid:82842775-84a6-4040-b426-0b461b7d2685>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00581.warc.gz"} |
Equilibrium statistical theory for nearly parallel vortex filaments
The first mathematically rigorous equilibrium statistical theory for three-dimensional vortex filaments is developed here in the context of the simplified asymptotic equations for nearly parallel
vortex filaments, which have been derived recently by Klein, Majda, and Damodaran. These simplified equations arise from a systematic asymptotic expansion of the Navier-Stokes equation and involve
the motion of families of curves, representing the vortex filaments, under linearized self-induction and mutual potential vortex interaction. We consider here the equilibrium statistical mechanics of
arbitrarily large numbers of nearly parallel filaments with equal circulations. First, the equilibrium Gibbs ensemble is written down exactly through function space integrals; then a suitably scaled
mean field statistical theory is developed in the limit of infinitely many interacting filaments. The mean field equations involve a novel Hartree-like problem with a two-body logarithmic interaction
potential and an inverse temperature given by the normalized length of the filaments. We analyze the mean field problem and show various equivalent variational formulations of it. The mean field
statistical theory for nearly parallel vortex filaments is compared and contrasted with the well-known mean field statistical theory for two-dimensional point vortices. The main ideas are first
introduced through heuristic reasoning and then are confirmed by a mathematically rigorous analysis. A potential application of this statistical theory to rapidly rotating convection in geophysical
flows is also discussed briefly.
ASJC Scopus subject areas
• General Mathematics
• Applied Mathematics
Dive into the research topics of 'Equilibrium statistical theory for nearly parallel vortex filaments'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/equilibrium-statistical-theory-for-nearly-parallel-vortex-filamen","timestamp":"2024-11-08T08:10:50Z","content_type":"text/html","content_length":"55185","record_id":"<urn:uuid:9903fe05-eece-47ca-8893-1b8dd8e6071e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00200.warc.gz"} |
Kilograms to Ounces Converter
โ Switch toOunces to Kilograms Converter
How to use this Kilograms to Ounces Converter ๐ ค
Follow these steps to convert given weight from the units of Kilograms to the units of Ounces.
1. Enter the input Kilograms value in the text field.
2. The calculator converts the given Kilograms into Ounces in realtime โ using the conversion formula, and displays under the Ounces label. You do not need to click any button. If the input
changes, Ounces value is re-calculated, just like that.
3. You may copy the resulting Ounces 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 Kilograms to Ounces?
The formula to convert given weight from Kilograms to Ounces is:
Weight[(Ounces)] = Weight[(Kilograms)] × 35.27396195
Substitute the given value of weight in kilograms, i.e., Weight[(Kilograms)] in the above formula and simplify the right-hand side value. The resulting value is the weight in ounces, i.e., Weight
Calculation will be done after you enter a valid input.
Consider that a premium electric car's battery pack weighs 450 kilograms.
Convert this weight from kilograms to Ounces.
The weight of battery pack in kilograms is:
Weight[(Kilograms)] = 450
The formula to convert weight from kilograms to ounces is:
Weight[(Ounces)] = Weight[(Kilograms)] × 35.27396195
Substitute given weight of battery pack, Weight[(Kilograms)] = 450 in the above formula.
Weight[(Ounces)] = 450 × 35.27396195
Weight[(Ounces)] = 15873.2829
Final Answer:
Therefore, 450 kg is equal to 15873.2829 oz.
The weight of battery pack is 15873.2829 oz, in ounces.
Consider that a luxury home gym equipment set weighs 100 kilograms.
Convert this weight from kilograms to Ounces.
The weight of home gym equipment set in kilograms is:
Weight[(Kilograms)] = 100
The formula to convert weight from kilograms to ounces is:
Weight[(Ounces)] = Weight[(Kilograms)] × 35.27396195
Substitute given weight of home gym equipment set, Weight[(Kilograms)] = 100 in the above formula.
Weight[(Ounces)] = 100 × 35.27396195
Weight[(Ounces)] = 3527.3962
Final Answer:
Therefore, 100 kg is equal to 3527.3962 oz.
The weight of home gym equipment set is 3527.3962 oz, in ounces.
Kilograms to Ounces Conversion Table
The following table gives some of the most used conversions from Kilograms to Ounces.
Kilograms (kg) Ounces (oz)
0.01 kg 0.3527 oz
0.1 kg 3.5274 oz
1 kg 35.274 oz
2 kg 70.5479 oz
3 kg 105.8219 oz
4 kg 141.0958 oz
5 kg 176.3698 oz
6 kg 211.6438 oz
7 kg 246.9177 oz
8 kg 282.1917 oz
9 kg 317.4657 oz
10 kg 352.7396 oz
20 kg 705.4792 oz
50 kg 1763.6981 oz
100 kg 3527.3962 oz
1000 kg 35273.9619 oz
A kilogram is the base unit of mass in the International System of Units (SI). The kilogram (kg) is used as a unit of mass in various fields and applications globally like science, healthcare,
education, commerce and trade, agriculture, etc.
The ounce is a unit of weight in the imperial system and the avoirdupois system. One ounce is approximately equal to 28.3495 grams. Ounces are commonly used for measuring small quantities of items,
particularly in cooking and food-related contexts.
Frequently Asked Questions (FAQs)
1. How do I convert kilograms to ounces?
Multiply the number of kilograms by 35.274 to get the equivalent in ounces. For example, 2 kg ร 35.274 = 70.548 ounces.
2. What is the formula for converting kg to ounces?
The formula is: ounces = kilograms ร 35.274.
3. How many ounces are in a kilogram?
There are approximately 35.274 ounces in 1 kilogram.
4. Is 1 kg equal to 35.274 ounces?
Yes, 1 kilogram is approximately equal to 35.274 ounces.
5. How do I convert ounces to kilograms?
Divide the number of ounces by 35.274 to get the equivalent in kilograms. For example, 70 ounces รท 35.274 โ 1.986 kilograms.
6. What is the difference between kilograms and ounces?
Kilograms and ounces are both units of mass, but kilograms are part of the metric system while ounces are part of the imperial system. One kilogram is approximately equal to 35.274 ounces.
7. How many ounces are there in half a kilogram?
Half a kilogram is approximately 17.637 ounces because 0.5 kg ร 35.274 = 17.637 ounces.
8. How many ounces are in 0.75 kilograms?
0.75 kg ร 35.274 = approximately 26.455 ounces.
9. How do I use this kg to ounce converter?
Enter the value in kilograms that you want to convert, and the converter will automatically display the equivalent in ounces.
10. Why do we multiply by 35.274 to convert kg to ounces?
Because there are approximately 35.274 ounces in 1 kilogram, so multiplying by 35.274 converts kilograms to ounces.
11. What is the SI unit of mass?
The SI unit of mass is the kilogram.
12. Are kilograms larger than ounces?
Yes, kilograms are larger than ounces. One kilogram equals approximately 35.274 ounces.
13. How many ounces are in 2 kilograms?
2 kg ร 35.274 = approximately 70.548 ounces.
14. How to convert 3.5 kg to ounces?
3.5 kg ร 35.274 = approximately 123.459 ounces.
Weight Converter Android Application
We have developed an Android application that converts weight between kilograms, grams, pounds, ounces, metric tons, and stones.
Click on the following button to see the application listing in Google Play Store, please install it, and it may be helpful in your Android mobile for conversions offline. | {"url":"https://convertonline.org/unit/?convert=kg-ounce","timestamp":"2024-11-02T18:13:25Z","content_type":"text/html","content_length":"100230","record_id":"<urn:uuid:1605e092-fe1d-43c8-9267-66674bfc9298>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00170.warc.gz"} |
Theoretical study of reactive and nonreactive turbulent coaxial jets
The hydrodynamic properties and the reaction kinetics of axisymmetric coaxial turbulent jets having steady mean quantities are investigated. From the analysis, limited to free turbulent boundary
layer mixing of such jets, it is found that the two-equation model of turbulence is adequate for most nonreactive flows. For the reactive flows, where an allowance must be made for second order
correlations of concentration fluctuations in the finite rate chemistry for initially inhomogeneous mixture, an equation similar to the concentration fluctuation equation of a related model is
suggested. For diffusion limited reactions, the eddy breakup model based on concentration fluctuations is found satisfactory and simple to use. The theoretical results obtained from these various
models are compared with some of the available experimental data.
NASA STI/Recon Technical Report N
Pub Date:
August 1976
□ Coaxial Flow;
□ Jet Flow;
□ Turbulent Boundary Layer;
□ Turbulent Mixing;
□ Chemical Reactions;
□ Diffusion;
□ Free Flow;
□ Hydrodynamics;
□ Mathematical Models;
□ Reaction Kinetics;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1976STIN...7629539G/abstract","timestamp":"2024-11-03T19:56:22Z","content_type":"text/html","content_length":"35633","record_id":"<urn:uuid:b3a86d7f-8efd-4ed9-8b79-6c51b0d96005>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00094.warc.gz"} |
Signed and Unsigned Binary Numbers
Understanding Number Representation Techniques
1. Integers can be represented in signed and unsigned ways.
2. Signed numbers use a sign flag to distinguish between positive and negative values.
3. Unsigned numbers store only positive numbers.
4. Techniques include Binary, Octal, Decimal, and Hexadecimal.
5. Binary Number System is a popular technique used in digital systems.
6. Binary System represents binary quantities with two possible states.
7. Binary numbers are indicated by an 0b prefix or a 2 suffix.
8. Unsigned binary numbers lack a sign bit, while signed binary numbers use a sign bit to distinguish between positive and negative numbers.
Sign-Magnitude form
Sign-magnitude is one way to represent signed numbers in digital logic. In this form, a fixed number of bits are dedicated to representing the sign and the remaining bits represent the magnitude
(absolute value) of the number. Here's a breakdown:
Key points:
• Sign bit: The most significant bit (MSB) is used to represent the sign. 0 indicates positive, and 1 indicates negative.
• Magnitude representation: Remaining bits represent the absolute value of the number, using the same format as unsigned numbers.
• Range: For n bits, the representable range is - (2^(n-1) - 1) to + (2^(n-1) - 1), meaning both positive and negative numbers can be represented within the same format.
Example (8-bit representation):
• +43: 00101011
• -43: 10101011
• Inefficient: Two representations exist for zero (positive 0 and negative 0), wasting space.
• Complex arithmetic: Addition and subtraction require different logic depending on the signs, making them more complex than other methods like 2's complement.
• Overflow detection: Detecting overflow conditions is more challenging compared to other representations.
Comparison with other forms:
• 1's complement: Similar to sign-magnitude but uses an inverted version of the magnitude for negative numbers. Less complex addition/subtraction but suffers from negative zero and overflow issues.
• 2's complement: Adds 1 to the 1's complement representation of negative numbers. Eliminates negative zero, simplifies arithmetic, and offers efficient overflow detection. This is the most common
representation in modern digital systems.
While not widely used in modern digital logic due to its limitations, sign-magnitude has some historical significance and niche applications:
• Simple educational tool to understand signed number representation.
• Specialized applications where simplicity is valued over efficiency (e.g., low-power systems).
A number is represented inside a computer with the purpose of performing some calculations using that number. The most basic arithmetic operation in a computer is the addition operation. That’s why a
computer can also be called as an adder.
When adding two numbers with the same signs, add the values and keep the common sign.
Example 1
Add the numbers (+5) and (+3) using a computer. The numbers are assumed to be represented using 4-bit SM notation.
111 <- carry generated during addition
0101 <- (+5) First Number
+ 0011 <- (+3) Second Number
1000 <- (+8) Sum
Let’s take another example of two numbers with unlike signs.
Example 2
Add the numbers (-4) and (+2) using a computer. The numbers are assumed to be represented using 4-bit SM notation.
000 <- carry generated during addition
1100 <- (-4) First number
+ 0010 <-(+2) Second Number
1110 <- (-2) Sum
Here, the computer has given the wrong answer of -6 = 1110, instead of giving the correct answer of -2 = 1010.
1's Complement
By inverting each bit of a number, we can obtain the 1's complement of a number. The negative numbers can be represented in the form of 1's complement. In this form, the binary number also has an
extra bit for sign representation as a sign-magnitude form.
2's Complement
By inverting each bit of a number and adding plus 1 to its least significant bit, we can obtain the 2's complement of a number. The negative numbers can also be represented in the form of 2's
complement. In this form, the binary number also has an extra bit for sign representation as a sign-magnitude form | {"url":"https://www.anuupdates.org/2024/02/signed-and-unsigned-binary-numbers.html","timestamp":"2024-11-11T14:36:50Z","content_type":"application/xhtml+xml","content_length":"199894","record_id":"<urn:uuid:9556af7b-43b2-43ac-aabb-ee9c54d6e430>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00720.warc.gz"} |
Bridge of Konigsberg Problem
• Konigsberg is the former name of a German city that is now in Russia.
• The following picture shows the inner city of Konigsberg with the river Pregel.
• The river Pregel divides the city into four land areas A, B, C and D.
• In order to travel from one part of the city to another, there exists seven bridges.
Konigsberg Bridge Problem-
Konigsberg Bridge Problem may be stated as-
│“Starting from any of the four land areas A, B, C, D, is it possible to cross each of the seven bridges│
│ │
│exactly once and come back to the starting point without swimming across the river?” │
Konigsberg Bridge Problem Solution-
In 1735,
• A Swiss Mathematician Leon hard Euler solved this problem.
• He provided a solution to the problem and finally concluded that such a walk is not possible.
Euler represented the given situation using a graph as shown below-
In this graph,
• Vertices represent the landmasses.
• Edges represent the bridges.
Euler observed that when a vertex is visited during the process of tracing a graph,
• There must be one edge that enters into the vertex.
• There must be another edge that leaves the vertex.
• Therefore, order of the vertex must be an even number.
Based on this observation, Euler discovered that it depends on the number of odd vertices present in the network whether any network is traversable or not.
Euler found that only those networks are traversable that have either-
• No odd vertices (then any vertex may be the beginning and the same vertex will also be the ending point)
• Or exactly two odd vertices (then one odd vertex will be the starting point and other odd vertex will be the ending point)
• Since the Konigsberg network has four odd vertices, therefore the network is not traversable.
• Thus, It was finally concluded that the desired walking tour of Konigsberg is not possible.
│NOTE │
│ │
│If the citizens of Konigsberg decides to build an eighth bridge from A to C, then- │
│ │
│ • It would be possible to walk without traversing any bridge twice. │
│ • This is because then there will be exactly two odd vertices. │
│ │
│However, adding a ninth bridge will again make the walking tour once again impossible.│
To gain better understanding about Konigsberg Bridge Problem,
Next Article- Handshaking Theorem
Get more notes and other study material of Graph Theory.
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Math 21
Math 218: Partial Differential Equations
Winter 2001
John Hunter
e-mail: jkhunter@ucdavis.edu
Phone: (530) 752-3189
Office: 654 Kerr Hall
Office hours: M F 1:00 2:00 p.m., W 3:00 4:00 p.m.
Lectures: MWF 2:10 3:00 p.m., Wellman 208
Texts for Math 218
The recommended text for Math 218 is:
L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, Volume 19, AMS.
Some other references are the following.
An introduction to elliptic and parabolic PDEs is in:
N. V. Krylov, Lectures on Elliptic and Parabolic Partial Differential Equations in Holder Spaces , Graduate Studies in Mathematics 12, AMS Providence RI, 1996.
A standard reference on elliptic PDEs is:
D. Gilbarg, and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , Springer-Verlag, New York, 1983.
For a readable account of maximum principles, see:
M. H. Protter, and H. F. Weinberger, Maximum Principles in Differential Equations , Prentice-Hall, Englewood Cliffs, 1967.
Spherical means are discussed in:
F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations , Springer-Verlag, New York, 1981.
For the Tychonoff solution of the heat equation, and Lewy's example of a PDE without solutions, see:
F. John, Partial Differential Equations , 4th ed., Springer-Verlag, New York, 1982.
The Cauchy-Kowalewski theorem is proved in:
P. Garabedian, Partial Differential Equations , John Wiley & Sons, New York, 1964.
Characteristic surfaces of hyperbolic PDEs are described in:
R. Courant, and D. Hilbert, Methods of Mathematical Physics , Vol. 2, Interscience, New York, 1961.
For a thorough account of distribution theory and the Fourier transform, see:
L. Hormander, The Analysis of Linear Partial Differential Operators , Vol. 1, Springer-Verlag, Berlin, 1983.
For the solution of linear initial value problems by the Fourier transform, and the Hadamard-Petrowsky dichotomy, see:
J. Rauch, Partial Differential Equations , Springer-Verlag, New York, 1991.
For an introduction to the physical aspects of Burger's equations, see:
G. Whitham, Linear and Nonlinear Waves , Wiley-Interscience, New York, 1973.
An account of the mathematical theory of conservation laws is in
J. Smoller, Shock Waves and Reaction-Diffusion Equations , Springer-Verlag, New York, 1983.
For the numerical solution of conservation laws, see
R. J. LeVeque, Numerical Methods for Conservation Laws , Birkhauser, Basel, 1992.
Homework Assignments
Latex and postscript files for the problem sets can be found here.
Set 1:
latex file
postscript file
Set 2:
latex file
postscript file
Set 3:
latex file
postscript file
Set 4:
latex file
postscript file
Set 5:
latex file
postscript file
Set 6:
latex file
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Efficient estimation of integrated volatility in presence of infinite variation jumps
We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero.
The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with
respect to locally "stable" Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical
characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators
contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic
function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.
• Central limit theorem
• Integrated volatility
• Itô semimartingale
• Quadratic variation
ASJC Scopus subject areas
• Statistics and Probability
• Statistics, Probability and Uncertainty
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Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis
School of Automation and Information Engineering, Xi’an University of Technology, Xi’an 710048, China
Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi’an University of Technology, Xi’an 710048, China
Author to whom correspondence should be addressed.
Submission received: 9 August 2022 / Revised: 16 September 2022 / Accepted: 22 September 2022 / Published: 26 September 2022
Fuzzy dispersion entropy (FuzzDE) is a very recently proposed non-linear dynamical indicator, which combines the advantages of both dispersion entropy (DE) and fuzzy entropy (FuzzEn) to detect
dynamic changes in a time series. However, FuzzDE only reflects the information of the original signal and is not very sensitive to dynamic changes. To address these drawbacks, we introduce
fractional order calculation on the basis of FuzzDE, propose $FuzzDE α$, and use it as a feature for the signal analysis and fault diagnosis of bearings. In addition, we also introduce other
fractional order entropies, including fractional order DE ($DE α$), fractional order permutation entropy ($PE α$) and fractional order fluctuation-based DE ($FDE α$), and propose a mixed features
extraction diagnosis method. Both simulated as well as real-world experimental results demonstrate that the $FuzzDE α$ at different fractional orders is more sensitive to changes in the dynamics of
the time series, and the proposed mixed features bearing fault diagnosis method achieves 100% recognition rate at just triple features, among which, the mixed feature combinations with the highest
recognition rates all have $FuzzDE α$, and $FuzzDE α$ also appears most frequently.
1. Introduction
Entropy, as a measure of time series disorder and predictability, can evaluate the complexity of the signal [
]. The greater the entropy value, the higher the complexity of signal [
]. In recently years, entropy has been widely applied in mechanical fault diagnosis and has shown excellent performance [
Dispersion entropy (DE) divides time series into integer series by introducing different mapping criteria for the first time [
], which enables it to capture more amplitude information than permutation entropy (PE) and sample entropy (SE) [
]. Some scholars have made every attempt to study the improved version of DE to further enhance its performance as a complexity index. Fluctuation-based DE (FDE) and reverse DE (RDE) have also been
proposed by introducing fluctuation information and distance information between time series and white noise [
]. In 2021, by combining the fluctuation information of FDE and the distance information of RDE [
], the reverse DE (FRDE) based on fluctuation is proposed, which has better stability and discrimination ability for different types of time series.
Fuzzy dispersion entropy (FuzzDE) is a new method proposed in 2021 [
], which combines the advantages of fuzzy entropy (FuzzEn) as well as DE by replacing the round mapping function of DE with fuzzy membership function in FuzzEn, by which the dynamic changes of time
series can be retained to a greater extent and the problem of missing useful information brought about by round mapping function can be alleviated. Nevertheless, the FuzzDE still suffers from the
same problem of single feature as common entropies, which cannot characterize the time series from multiple fractional orders.
To address the problem of single fractional order, in recent years, many scholars have conducted research on the application of fractional order calculation to entropy [
]. In 2019, the fractional fuzzy entropy algorithm was proposed and used for physiological and biomedical analysis of EEG signals [
]. In 2020, generalized refined composite multiscale fluctuation-based fractional dispersion entropy (
$GRCMFDE α$
) combined refined composite multiscale dispersion entropy (RCMDE) as well as fractional order calculation and was applied for bearing signal fault diagnosis with good results [
]. In 2022, fractional order calculation was introduced to slope entropy to effectively diagnose the location and severity of faults in rolling bearings [
Inspired by these works, we introduce fractional order calculation into FuzzDE in this paper, and fractional order FuzzDE ($FuzzDE α$) is proposed. Compared with FuzzDE, $FuzzDE α$ further considers
fractional order information and measures the dynamic changes of time series from multiple fractional orders. In addition, we combine $FuzzDE α$ with other fractional order entropies and propose a
mixed feature bearing fault diagnosis method. Simulated as well as real-world experiments demonstrate the sensitivity of $FuzzDE α$ to the dynamic changes of time series and the excellent performance
on bearing fault diagnosis.
The rest of this paper is organized as follows:
Section 2
presents the theoretical steps of
$FuzzDE α$
and discusses the parameter settings;
Section 3
experiments on the effectiveness of fractional order on FuzzDE through simulated signals;
Section 4
validates the bearing fault diagnosis capability of
$FuzzDE α$
through real-world bearing signals;
Section 5
concludes the whole paper.
2. Fractional Order Fuzzy Dispersion Entropy
2.1. $FuzzDE α$
$FuzzDE α$ is the introduction of the concept of fractional order calculation on the basis of FuzzDE, for a given time series $X = { x 1 , x 2 , ⋯ , x N }$ of length $N$, the specific steps for
$FuzzDE α$ can be expressed as follows:
Step 1: By applying the normal cumulative distribution function (NCDF) to the original time series
$Y = { y 1 , y 2 , ⋯ , y N }$
can be derived with the interval [−1, 1], where the NCDF can be expressed as follows:
$y i = 1 σ 2 π ∫ − ∞ x i e − ( t − γ ) 2 2 σ 2 d t ( i = 1 , 2 , ⋯ , N )$
represent the standard deviation and mean of
, respectively.
Step 2: Normalize the sequence
by converting each element in
to the interval [0, 1]:
$s i = y i M a x − M i n ( i = 1 , 2 , ⋯ , N )$
in which
$S = { s 1 , s 2 , ⋯ , s N }$
is the normalized sequence,
$M a x$
$M i n$
are the maximum and minimum values of the sequence
, respectively.
Step 3: Introduce the class number
to convert the sequence
into a new sequence
$Z c$
$z i c = c y i + 0.5 ( i = 1 , 2 , ⋯ , N )$
where each element
$z i c ( i = 1 , 2 , ⋯ , N )$
$Z c$
is in the interval [0.5,
+ 0.5].
Step 4: Introduce the embedding dimension
and time delay
, reconstruct the sequence
$Z c$
of Step 3 into
$N − ( m + 1 ) τ$
$Z j m , c$
$Z j m , c = { z j c , z j + ( 1 ) τ c , ⋯ , z j + ( m − 1 ) τ c } ( j = 1 , 2 , ⋯ , N − ( m + 1 ) τ )$
determines the number of elements contained in each subsequence
$Z j m , c$
, and
determines the interval between two adjacent elements in the sequence
$Z c$
Step 5: Introduce the fuzzy membership function on the sequence
$Z c$
as follows:
$μ M 1 ( z i c ) = { 0 z i c > 2 2 − z i c 1 ≤ z i c ≤ 2 1 z i c < 1$
$μ M k ( z i c ) = { 0 z i c > k + 1 k + 1 − z i c k ≤ z i c ≤ k + 1 z i c − k + 1 k − 1 ≤ z i c ≤ k 0 z i c < k − 1 ( k = 2 , 3 , ⋯ , c − 1 )$
$μ M c ( z i c ) = { 1 z i c > c z i c − c + 1 c − 1 ≤ z i c ≤ c 0 z i c < c − 1$
stands for the
th class, and
$M k$
is the fuzzy membership function,
$μ M k ( z i c )$
represents the degree of membership of
$z i c$
for the
th class. By the fuzzy membership function, each
$z i c$
will have 1 or 2 different degrees, and the value range is an integer between [1,
], which is the same as the rounding function in the DE [
], but reduces the information loss in the rounding function.
Figure 1
shows the fuzzy membership function.
Step 6: After the processing of the sequence $Z c$ in Step 5, each subsequence $Z j m , c$ can be mapped into a number of new sequences consisting of integers, and these sequences can be represented
by the dispersion patterns $π v 0 v 1 ⋯ v m − 1$, where $v 0$, $v 1$, $v m − 1$ correspond to the integer values of $z j c ,$$z j + ( 1 ) τ c$, and $z j + ( m − 1 ) τ c$ in Equation (1) after fuzzy
processing, respectively.
Step 7: Calculate the degree of membership of each
$Z j m , c$
with respect to the dispersion patterns
$π v 0 v 1 ⋯ v m − 1$
and denote as
$μ π v 0 v 1 ⋯ v m − 1$
$μ π v 0 v 1 ⋯ v m − 1 ( z j m , c ) = ∏ i = 0 m − 1 μ M v i ( z j + ( i ) τ c )$
in this manner, each subsequence
$Z j m , c$
will correspond to multiple dispersion patterns accompanied by different membership degrees. For an example, given a subsequence
$Z 1 2 , 3 = [ 1.149 , 2.306 ]$
, all the member ship degrees can be organized as follows:
$μ M 1 ( z 1 3 ) = 0.851 μ M 2 ( z 1 3 ) = 0.149 μ M 2 ( z 2 3 ) = 0.694 μ M 3 ( z 2 3 ) = 0.306 → μ π 12 ( Z 1 2 , 3 ) = μ M 1 ( z 1 3 ) × μ M 2 ( z 2 3 ) = 0.5906 μ π 13 ( Z 1 2 , 3 ) = μ M 1 ( z 1
3 ) × μ M 3 ( z 2 3 ) = 0.2604 μ π 22 ( Z 1 2 , 3 ) = μ M 2 ( z 1 3 ) × μ M 2 ( z 2 3 ) = 0.1034 μ π 23 ( Z 1 2 , 3 ) = μ M 2 ( z 1 3 ) × μ M 3 ( z 2 3 ) = 0.0456$
Step 8: The frequency of each dispersion pattern
$p ( π v 0 , v 1 , … , v m − 1 )$
can be calculated:
$p π v 0 , v 1 , … , v m − 1 = ∑ j = 1 N − ( m − 1 ) d μ π v 0 v 1 ⋯ v m − 1 ( z j m , c ) N − ( m − 1 ) τ$
Step 9: For writing convenience, we define
$p ( π v 0 , v 1 , … , v m − 1 )$
$P j$
.Then the fractional order calculation is applied, and the
$FuzzDE α$
can be expressed as [
$FuzzDE α ( X , m , c , τ ) = ∑ j P j − P j − α Γ ( α + 1 ) ln P j + ψ ( 1 ) − ψ ( 1 − α )$
is the order of fractional derivative.
$Γ ( ⋅ )$
$ψ ( ⋅ )$
denote the
function and
function respectively.
Step 10: The normalized form
$NFuzzDE α$
$FuzzDE α$
can be computed as:
$NFuzzDE α ( X , m , c , τ ) = FuzzDE α ( X , m , c , τ ) l n ( c m )$
2.2. Parameter Selection
In this subsection, we mainly focus on the discussion of the parameter selection for
$FuzzDE α$
. For the parameter comparison experiments, 50 separate groups of pink noises, white noises and blue noises are selected [
], each with 2048 sample points. Where white noise consists of a homogeneous mixture of signals of different frequencies, with a variety of frequencies in a haphazard manner. Pink noise enhances the
sound intensity of low frequency signals and weakens the intensity of high frequency signals compared to white noise, while blue noise, in contrast, enhances the sound intensity of the high frequency
signal on top of the white noise. Using the control variables method, the effects of three
$FuzzDE α$
parameters, namely the number of classes
, the embedding dimension
and the mapping method, on the mean as well as the standard deviation of the selected noise signals are explored as shown in
Figure 2
Figure 3
Figure 4
, respectively.
To begin with, we conduct comparative experiments on the effects of
, and the interval is set to an integer between 2 and 5 (
= 3, mapping as NCDF),
Figure 2
shows the means and standard deviations of different class number
at different fractional orders.
Comparing the four images, it can be seen that for the average of the entropy values of the three noises, the trend when $m$ equals 3 and $c$ equals 2 is different from the other three in that it has
a slope from large to small, while the others are from small to large. However, the general trend is that it increases with the increase of $α$. For the standard deviation of the entropy values of
the three noise entropy values, the standard deviation of the pink noise is larger, and the others are smaller, and as $α$ increases, the value of the standard deviation also increases, which is
especially evident in the pink noise. In summary, changes in $c$ have an impact on the magnitude of entropy value, but the overall trend in entropy value and the ability to discriminate between
different noises does not change as the fractional order changes.
We next discuss the effect of
, with the interval set to an integer between 3 and 6 (
= 3, mapping as NCDF),
Figure 3
is means and standard deviations of different embedding dimensions
at different fractional orders.
Observing the four subplots, for the average of the $FuzzDE α$ values of the three noises, all four cases of taking values show a similar upward trend. For the standard deviation of the entropy
values of the three noises, there is only a difference between the exact values and the overall trend is almost the same. Thus, it is clear that the effect of $m$ has a greater impact on the
magnitude of the entropy value compared to $c$, but the overall trend and the ability to distinguish between different noises does not change.
Finally, we discuss the effect of the mapping method, which is also an important influencing factor, so we choose different mapping methods for comparison.
Figure 4
shows the means and standard deviations of different mapping approaches at different fractional orders, among which the mapping methods include linear mapping (LM), normal cumulative distribution
function (NCDF), tangent sigmoid (TANSIG), logarithm sigmoid (LOGSIG), and sorting method (SORT) respectively (
= 3,
= 3) [
According to
Figure 4
, the overall trends of the five mapping methods are very similar, but when using the LM mapping method, the standard deviation of the various noise entropy values is significantly larger,
accompanied by the condition that the various noise entropy values overlap each other, which indicates that when the selected mapping method is LM, the stability of
$FuzzDE α$
after mapping is relatively weak, and it is difficult to distinguish the three types of noise. While the standard deviation of other mapping methods is relatively small. Therefore, it is concluded
that NCDF, LOGISG, TANSIG or SORT are the recommended mapping approaches.
In conclusion, $m$ and $c$ have little effect on the experiment, but a large $m$ is more likely to lead to an increase in $FuzzDE α$ values compared to $c$. Among all mapping methods, only LM is not
stable. Therefore, we recommend that $m$ be set to 3–6, $c$ to 2–5 and the mapping method be NCDF, LOGISG, TANSIG or SORT. In the later simulations and the real-world signal experiments, we choose
$m$ = 3, $c$ = 4 and the mapping method to be NCDF.
3. Experiments on Simulated Signals
In this section, we focus on demonstrating the usefulness of fractional order calculations on FuzzDE by simulated signals, mainly including noise signals, chirp signal and MIX signal.
3.1. Noise Signals Experiment
In order to verify the effectiveness of fractional order calculation on FuzzDE, pink noise, white noise and blue noise are selected for comparative experiments, and the fractional orders change from
−0.5 to 0.5 with interval 0.1. 100 independent pink noises, white noises and blue noises are created to prove the discrimination ability of fractional order. The means and standard deviations of
these 100
$FuzzDE α$
. values are calculated respectively as displayed in
Figure 5
As shown in
Figure 5
, the
$FuzzDE α$
value of the three kinds of noise signals has a similar upward trend with the increase of fractional order; when the fractional order is less than −0.1, the mean characteristics of the three noise
signals are mixed together; when the fractional order is greater than −0.1, the difference of mean characteristics of the three noise signals gradually increases, and the
$FuzzDE α$
value of white noise is the largest, with the smallest standard deviation and the most stable
$FuzzDE α$
value. Experiments show that as the fractional order increases (when the fractional order is greater than −0.1),
$FuzzDE α$
has a better distinguishing effect on pink noise, white noise and blue noise.
3.2. Chirp Signal Experiment
Chirp signal is a typical unstable signal, and frequency of chirp signal will change over time [
]. In order to better show the feature extraction effect of
$FuzzDE α$
at different fractional orders, chirp signal is used for simulated experiments. Chirp signal can be expressed as:
$x ( t ) = e ( j 2 π ( f 0 t + 1 2 k t 2 ) )$
$f 0$
is the initiation frequency and is taken as 20 Hz,
is the modulation frequency and is taken as 3, we can understand that the frequency increases from 20 Hz to 80 Hz. The chirp signal lasts 20 s with a sampling frequency of 1000 Hz (20,000 sampling
We take the length of the sliding window as 1000 sampling points, and slide backward from the first sampling point with 90% overlap to obtain 190 samples.
$FuzzDE α$
for chirp signal of each sample are calculated. Chirp signal (top) and the corresponding different entropy curves (bottom) are shown in
Figure 6
It can be observed from
Figure 6
that the waveform of the chirp signal gradually becomes denser as the number of sampling points increases, and the higher the fractional order, the larger the
$FuzzDE α$
value as well as the rate of increase of the curve. In a word, the experimental results show that the higher the fractional order of
$FuzzDE α$
, the better performance of
$FuzzDE α$
in chirp signal feature extraction.
3.3. MIX Signal Experiment
In order to study the influence for fractional order of
$FuzzDE α$
on the effect of feature extraction, we select MIX signal for simulated experiments. MIX signal describes a stochastic sequence that progressively turns into a periodic time series [
], which can be expressed as:
${ M I X ( t ) = ( 1 − Z ) × X ( t ) + Z × Y ( t ) X ( t ) = 2 s i n 2 π t 12$
$X ( t )$
is a periodic signal, the value of
$Y ( t )$
is uniformly distributed from
$− 3$
, and Z is a random number taking 1 or 0 with probabilities P and 1 − P, respectively, and decreasing linearly from 0.99 at the beginning to 0.01 at the end. The sampling frequency of mix signal is
1000 Hz, with a total of 20 s. We take the length of sliding window as 1000 sampling points, and slide backward from the first sampling point with 90% overlap to obtain 190 samples. MIX signal (top)
and the corresponding different entropy curves (bottom) are shown in
Figure 7
As can be seen from
Figure 7
, the MIX signal changes from dense to sparse as the number of sampling points increases; the value of
$FuzzDE α$
decreases as the number of sampling points increases; the higher the fractional order, the larger the
$FuzzDE α$
value and the rate of decline of the curve also increases. Therefore, we can conclude that an increase in fractional order can better reflect the complexity of the MIX signal.
4. Experiments on Bearing Fault Diagnosis
In this section, we focus on the bearing fault diagnosis, and achieve early prevention in order to avoid economic losses and even personal safety due to different bearing faults. Since entropy can be
used to detect changes in the dynamics of weak time series, the higher the entropy value, the more unstable the time series and vice versa. At the same fault size, different faults have similar
amplitude-frequency features, but their bearing fault complexity and dynamics changes are different, and these changes can be reflected in successive subsequences, for which entropy features can be
extracted for fault diagnosis of the bearing signals. The experiments in this section are mainly to verify the effectiveness of
$FuzzDE α$
for bearing fault diagnosis, and the proposed mixed features bearing fault diagnosis method experimental flow chart is shown in
Figure 8
, with the following steps:
Step 1: Input the real-world bearing signals of ten different classes.
Step 2: Segment the input signals into $M$ samples, each with $N$ sample points, by which way, we receive a total of $M$ samples for each class of signal.
Step 3: For each sample, calculate their $FuzzDE α$ values at different fractional orders. For the purpose of contrast, we also introduce fractional order DE ($DE α$), fractional order PE ($PE α$)
and fractional order FDE ($FDE α$) for comparison, with fractional orders of −0.2, −0.1, 0, 0.1, and 0.2 respectively.
Step 4: Mix the 20 features obtained in Step 3 and set the number of selected features to $K$ (initialized to 2), by which way we can acquire a total of $C 20 K$ combinations.
Step 5: Calculate the recognition rate of all $C 20 K$ combinations and select the combination with the highest recognition rate.
Step 6: Determine the direction of the process by the number of features selected, and if $K$ < 5, skip to Step 7; otherwise, output the combination of 5 features with the highest recognition rate
and the corresponding recognition rate, as a way to avoid the increased computational consumption when the recognition rate has reached the threshold.
Step 7: Determine the direction of the process by the highest recognition rate among $K$ feature combinations. If the recognition rate reaches 100%, then output these feature combinations; otherwise,
let $K$ = $K$ + 1 and back to Step 5.
4.1. Analysis of Experiment Data
This section employs the bearing signal obtained from Case Western Reserve University (CWRU) [
] to verify the effectiveness of the proposed
$FuzzDE α$
. The bearing under test is a deep groove ball bearing type SKF6205 (CWRU, Cleveland, America) with a motor speed set to 1730 r/min and a load of 3 hp. The original bearing signal is acquired by
collecting the acceleration sensor installed at the driving end, and the sampling frequency is 12 kHz. Depending on the states of the bearing and the diameters of the failure, there are 10 different
types of bearing signals marked NORM, IR1, BE1, OR1, IR2, BE2, OR2, IR3, BE3 and OR3, all damage is caused by electro discharge machining as a single point of damage. The details of the selected
bearing signals are shown in
Table 1
. For each class of bearing signal, the length of sample points is 120,000, and
Figure 9
shows the time domain distribution of ten classes of bearing signals.
4.2. Single Feature Extraction and Classification
The ten classes of bearing signals are used as the object of the experiment for single feature extraction. There are 50 samples for each type of bearing signal, and each sample contains 2048 sample
points. While calculating the
$FuzzDE α$
of the bearing signal, the
$DE α$
$PE α$
, and
$FDE α$
are calculated respectively as comparative analysis. The parameter settings are as follows: embedding dimension
is 3, class number
is 4, and the range of fractal order
is from −0.2 to 0.2 with interval 0.1. For other fractional entropies, the parameter settings are the same as
$FuzzDE α$
. The Distribution of fractional entropy features of different classes of bearing signals are exhibited in
Figure 10
Figure 10
, for the four types of fractional entropies, it is difficult to completely distinguish all ten types of bearing signals under different fractional order; for
$FuzzDE α$
$DE α ,$
, and
$FDE α$
, there is always some standard deviation of fractional entropy values close to each other for bearing signals; in addition, the standard deviation of fractional entropy values are significantly
higher than that of
$PE α$
under different fractional order, for
$PE α$
, the standard deviations of fractional entropy values for ten classes of bearing signals are all very close, which is difficult to distinguish. Furthermore, we employ KNN to classify the ten classes
of bearing signals, in which there are 50 samples for each type of bearing signal, the first 25 samples are training samples, and the rest samples are test samples.
Table 2
illustrates the classification recognition rate of different entropies at various fractional orders.
It can be seen from
Table 2
, for four classes of fractional entropies, the recognition rates of bearing signals are all lower than 85% under different fractional orders, and the recognition effect is poor. Therefore, it is
difficult to distinguish ten classes of signals with one feature.
4.3. Double Features Extraction and Classification
In order to improve recognition performance and demonstrate the effectiveness of the mixed feature extraction method proposed in this paper, we choose different entropy-based feature extraction
methods, extract any two fractional orders with the same entropy and choose the best combinations of fractional orders. Since fractional order
has 5 values, a total of
$C 5 2$
combinations can be obtained by each entropy-based feature extraction method. In addition, we use the mixed feature extraction method proposed in this paper to calculate the highest recognition rate,
with a total of
$C 20 2$
Table 3
demonstrate the highest classification recognition rates for each feature extraction method when double features are selected.
Table 3
$FuzzDE α = 0$
$FuzzDE α = 0 . 2$
$FuzzDE α$
when fractional order
is 0 and 0.2 respectively, other combinations of entropy are the same for
$FuzzDE α = 0$
$FuzzDE α = 0 . 2$
. As can be observed in
Table 3
$FuzzDE α$
-based feature extraction method has the best classification effect among the four entropy-based feature extraction methods, but the highest recognition rate is only 91.6%, which cannot fully
recognize the bearing signals. Nevertheless, the mixed feature extraction method proposed in this paper can reach a maximum classification rate of 99.6%, significantly higher than the 91.6% of
$FuzzDE α$
-based feature extraction method, and there are three combinations in total, namely
$FuzzDE α = 0 . 1$
$FDE α = 0 . 1$
$FuzzDE α = - 0 . 1$
$FDE α = 0 . 1$
$FuzzDE α = 0 . 1$
$FDE α = - 0 . 1$
. It is noteworthy that when reaching the highest recognition rate, the three combinations all contain
$FuzzDE α$
, which further proves the importance of
$FuzzDE α$
in bearing fault diagnosis recognition.
Figure 11
shows the distribution of the highest classification recognition rate of mixed double features.
As can be seen from
Figure 11
, under the mixed double features, the distribution of each type of bearing signal is relatively concentrated, and the overlapping part is very small. However, there are few samples that are not
completely distinguishable, for example, a small percentage of IR1 and IR2 samples are mixed. In summary, compared with the entropy-based feature extraction methods, the mixed feature extraction
method proposed in this paper further improves the recognition rate and can better distinguish the ten classes of bearing signals. To sum up, mixed double features extraction method can well
distinguish the ten classes of bearing signals.
4.4. Triple Features Extraction and Classification
In order to further improve the recognition rate of bearing fault diagnosis, we set the number of selected features H to 3. The rest of the steps are the same as
Section 4.3
, and
Table 4
shows the highest classification recognition rates for each feature extraction method when triple features are selected.
Table 4
, it can be seen that as the number of features increases, the recognition rates of the feature extraction methods based on
$FuzzDE α$
$DE α$
$FDE α$
all improved, but the fault diagnosis performance is still much less than that of the mixed double features in
Table 3
, which indicates that different fractional order features with the same entropy still have certain limitations. Furthermore, we can also observe from
Table 4
that the mixed feature extraction method proposed in this paper achieves a recognition rate of 100% for 15 combinations when triple features are selected, further demonstrating the excellent
performance of the mixed feature extraction method for bearing fault diagnosis. To visualize the specific details of these 15 combinations,
Table 5
shows the number of occurrences of each feature in the combinations with 100% recognition rate.
It is clear from
Table 5
$FuzzDE α = - 0 . 1$
has the highest number of occurrences at 11, far more than any other features, which proves the efficiency of
$FuzzDE α$
in bearing fault diagnosis. In addition, among the feature combinations with 100% recognition rate, only
$DE α$
is absent, which is due to the fact that
is an improvement on DE, further validating the conclusion that
is more differentiable than DE. We can also find that although the
$PE α$
has a low recognition rate on bearing fault diagnosis, it can accurately classify some samples that cannot be correctly classified by other entropies. Hence, we can also conclude that different
entropies can distinguish different signal classes, and following the mixed feature extraction method proposed in this paper, while selecting mixed fractional order entropies simultaneously can
effectively improve the performance of bearing fault diagnosis.
Figure 12
depicts the distribution of the triple features at 100% recognition rate with the combination of
$FuzzDE α = - 0 . 1$
$PE α = - 0 . 2$
$FDE α = 0 . 1$
. Compared to
Figure 11
, we can intuitively find that
Figure 12
can perfectly distinguish between IR1 as well as IR2, which are two different sizes of the same fault class, and it is obvious that the mixed double features distribution cannot achieve such results.
5. Conclusions
In this paper, a new non-linear dynamic parameter is proposed, and a mixed features extraction method is put forward based on this new parameter. The main conclusions are as follows.
• Fractional order calculation is introduced on the basis of fuzzy dispersion entropy (FuzzDE), and a new entropy called fractional order FDE ($FuzzDE α$) is proposed. Simulated experiments have
shown that compared with $FuzzDE$, $FuzzDE α$ can provide more features of greater sensitivity to changes in the dynamics of the time series.
• $FuzzDE α$ is combined with $DE α$, $PE α$ as well as $FDE α$ to present a mixed features extraction method. For ten classes of bearing signals, the proposed mixed features fault diagnosis method
achieves 100% recognition rate at only triple features.
• Regardless of how many features are selected, the $FuzzDE α$ proposed in this paper is the most effective in fault diagnosis compared to the other three fractional order entropies, where $FuzzDE
α = - 0 . 1$ appears a total of 11 times in the combination of the triple features with the recognition rate of 100%
Author Contributions
Formal analysis, B.T.; funding acquisition, Y.L.; methodology, B.G.; project administration, B.G.; visualization, S.J.; writing—original draft, B.T.; writing—review and editing, Y.L. All authors have
read and agreed to the published version of the manuscript.
This research was funded by [National Natural Science Foundation of China] grant number [61871318], [Natural Science Foundation of Shaanxi Province] grant number [2022JM-337].
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflict of interests.
FuzzDE Fuzzy Dispersion Entropy
$FuzzDE α$ Fractional order fuzzy dispersion entropy
PE Permutation entropy
$PE α$ Fractional order permutation entropy
DE Dispersion entropy
$DE α$ Fractional order dispersion entropy
FDE Fluctuation-based dispersion entropy
$FDE α$ Fractional order fluctuation-based dispersion entropy
NCDF Normal cumulative distribution function
SE Sample entropy
FRDE Fluctuation-based reverse dispersion entropy
RDE Reverse dispersion entropy
FuzzEn Fuzzy entropy
RCMDE Refined composite multiscale dispersion entropy
$GRCMFDE α$ Generalized refined composite multiscale fluctuation-based fractional dispersion entropy
LM Linear mapping
TANSIG Tangent sigmoid
LOGSIG Logarithm sigmoid
SORT Sorting method
Figure 2. Means and standard deviations of different class number $c$ at different fractional orders. (a) $c$ = 2; (b) $c$ = 3; (c) $c$ = 4; (d) $c$ = 5.
Figure 3. Means and standard deviations of different embedding dimensions $m$ at different fractional orders. (a) $m$ = 3; (b) $m$ = 4; (c) $m$ = 5; (d) $m$ = 6.
Figure 4. Means and standard deviations of different mapping approaches at different fractional orders. (a) LM; (b) NCDF; (c) LOGISG; (d) TANSIG; (e) SORT.
Figure 5. Means and standard deviations of different fractional order entropies under noise signals.
Figure 10. Distribution of fractional entropy features of different classes of bearing signals. (a) $α$ = −0.2; (b) $α$ = −0.1; (c) $α$ = 0; (d) $α$ = 0.1; (e) $α$ = 0.2.
Figure 11. Distribution of the highest classification recognition rate of mixed double features. (a) $FuzzDE α = 0 . 1$. & $FDE α = 0 . 1$; (b)$FuzzDE α = - 0 . 1$ & $FDE α = 0 . 1$; (c) $FuzzDE α =
0 . 1$ & $FDE α = - 0 . 1$.
Figure 12. Distribution of the mixed triple features at 100% recognition rate ($FuzzDE α = - 0 . 1$ , $PE α = - 0 . 2$, $FDE α = 0 . 1$).
Class Label Fault Size (mm) Selected Data
Normal NORM 0 100_normal_3
Inner race fault IR1 0.1778 108_IR007_3
Balling element fault BE1 0.1778 121_B007_3
Outer race fault OR1 0.1778 133_OR007@6_3
Inner race fault IR2 0.3556 172_IR014_3
Balling element fault BE2 0.3556 188_B014_3
Outer race fault OR2 0.3556 200_OR014@6_3
Inner race fault IR3 0.5334 212_IR021_3
Balling element fault BE3 0.5334 225_B021_3
Outer race fault OR3 0.5334 237_OR021@6_3
Entropy Recognition Rates (%)
$α = − 0.2$ $α = − 0.1$ $α = 0$ $α = 0.1$ $α = 0.2$
$FuzzDE α$ 82.8 81.6 74 68.4 67.6
$DE α$ 76.4 79.6 76.4 71.2 66.0
$PE α$ 59.6 56.8 58.4 56.8 54.0
$FDE α$ 79.2 82.8 78.4 77.6 80.4
Table 3. Highest classification recognition rates for each feature extraction method (double features).
Feature Extraction Methods Combinations Recognition Rate (%)
$FuzzDE α$-based $FuzzDE α = 0 & FuzzDE α = 0 . 2$ 91.6
$DE α$-based $DE α = 0 & DE α = 0 . 2$ 88.4
$PE α$-based $PE α = - 0 . 2 & PE α = 0 . 1$ 58.4
$FDE α$-based $FDE α = - 0 . 1 & FDE α = 0$ 90.0
Proposed method $FuzzDE α = 0 . 1$& $FDE α = 0 . 1$ 99.6
(1 of 3)
Table 4. Highest classification recognition rates for each feature extraction method (triple features).
Feature Extraction Methods Combinations Recognition Rate (%)
$FuzzDE α$-based $FuzzDE α = - 0 . 1 & FuzzDE α = 0 & FuzzDE α = 0 . 2$ 92
$DE α$-based $DE α = - 0 . 1 & DE α = 0 & DE α = 0 . 2$ 92
$PE α$-based $PE α = - 0 . 2 & PE α = 0 & PE α = 0 . 1$ 58
$FDE α$-based $FDE α = - 0 . 2 & FDE α = - 0 . 1 & FDE α = 0$ 91.6
Proposed method $FuzzDE α = - 0 . 1 & PE α = - 0 . 2 & FuzzDE α = 0 . 1$ 100
(1 of 15)
Table 5. Number of occurrences of each feature in the combination of the mixed triple features with 100% recognition rate.
Feature Appear Times
$FuzzDE α = - 0 . 1$ $11$
$FuzzDE α = - 0 . 2$ $4$
$PE α = - 0 . 2$ $2$
$PE α = - 0 . 1$ $3$
$PE α = 0$ $3$
$PE α = 0 . 1$ $4$
$PE α = 0 . 2$ $3$
$FDE α = - 0 . 1$ $6$
$FDE α = 0$ $5$
$FDE α = 0 . 1$ $4$
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Li, Y.; Tang, B.; Geng, B.; Jiao, S. Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis. Fractal Fract. 2022, 6, 544. https://doi.org/10.3390/fractalfract6100544
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Li Y, Tang B, Geng B, Jiao S. Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis. Fractal and Fractional. 2022; 6(10):544. https://doi.org/10.3390/
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Li, Yuxing, Bingzhao Tang, Bo Geng, and Shangbin Jiao. 2022. "Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis" Fractal and Fractional 6, no. 10: 544. https://
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An application of the Moore-Penrose inverse to antisymmetric relations
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Proc. Amer. Math. Soc. 78 (1980), 181-186
DOI: https://doi.org/10.1090/S0002-9939-1980-0550489-X
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Let R be a star-ring and let ${R_\dagger }$ denote the set of star-regular elements in R. It is shown that the relation $a\Delta b$, defined by $a{a^\ast }a = a{b^\ast }a$, is antisymmetric on ${R_\
dagger }$ provided that the two-term star-cancellation law and the positive-semidefinite axiom hold in R. This includes the star-regular elements of all ${C^\ast }$-algebras, and in particular those
elements in ${{\mathbf {C}}_{n \times n}}$ and $B(H)$, the bounded linear transformations on Hilbert space H. References
• Adi Ben-Israel and Thomas N. E. Greville, Generalized inverses: theory and applications, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR
• Sterling K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin, 1972. MR 0429975 M. P. Drazin, Natural structures on rings and
semigroups with involution (submitted for publication).
• M. P. Drazin, Pseudo-inverses in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506–514. MR 98762, DOI 10.2307/2308576
• Ivan Erdélyi, On the matrix equation $Ax=\lambda Bx$, J. Math. Anal. Appl. 17 (1967), 119–132. MR 202734, DOI 10.1016/0022-247X(67)90169-2
• Robert E. Hartwig, Block generalized inverses, Arch. Rational Mech. Anal. 61 (1976), no. 3, 197–251. MR 399124, DOI 10.1007/BF00281485
• Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
• C. Radhakrishna Rao, Sujit Kumar Mitra, and P. Bhimasankaram, Determination of a matrix by its subclasses of generalized inverses, Sankhyā Ser. A 34 (1972), 5–8. MR 340272
• N. S. Urquhart, Computation of generalized inverse matrices which satisfy specified conditions, SIAM Rev. 10 (1968), 216–218. MR 227186, DOI 10.1137/1010035
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• © Copyright 1980 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 78 (1980), 181-186
• MSC: Primary 16A28; Secondary 15A09
• DOI: https://doi.org/10.1090/S0002-9939-1980-0550489-X
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[Solved] The average selling price of BlackBerry s | SolutionInn
The average selling price of BlackBerry smartphones purchased by a random sample of 35 customers was $311.
The average selling price of BlackBerry smartphones purchased by a random sample of 35 customers was $311.
Assume the population standard deviation was $35.
a. Construct a 90% confidence interval to estimate the average selling price in the population with this sample.
b. What is the margin of error for this interval?
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relationship between svd and eigendecomposition
What PCA does is transforms the data onto a new set of axes that best account for common data. The columns of \( \mV \) are known as the right-singular vectors of the matrix \( \mA \). \newcommand{\
permutation}[2]{{}_{#1} \mathrm{ P }_{#2}} \newcommand{\mY}{\mat{Y}} Instead of manual calculations, I will use the Python libraries to do the calculations and later give you some examples of using
SVD in data science applications. Now that we are familiar with the transpose and dot product, we can define the length (also called the 2-norm) of the vector u as: To normalize a vector u, we simply
divide it by its length to have the normalized vector n: The normalized vector n is still in the same direction of u, but its length is 1. \right)\,. As a result, we already have enough vi vectors to
form U. Surly Straggler vs. other types of steel frames. Now if we multiply them by a 33 symmetric matrix, Ax becomes a 3-d oval. Why PCA of data by means of SVD of the data? (You can of course put
the sign term with the left singular vectors as well. And \( \mD \in \real^{m \times n} \) is a diagonal matrix containing singular values of the matrix \( \mA \). So if call the independent column
c1 (or it can be any of the other column), the columns have the general form of: where ai is a scalar multiplier. Alternatively, a matrix is singular if and only if it has a determinant of 0. e <-
eigen ( cor (data)) plot (e $ values) The SVD allows us to discover some of the same kind of information as the eigendecomposition. To find the u1-coordinate of x in basis B, we can draw a line
passing from x and parallel to u2 and see where it intersects the u1 axis. Before going into these topics, I will start by discussing some basic Linear Algebra and then will go into these topics in
detail. You can find these by considering how $A$ as a linear transformation morphs a unit sphere $\mathbb S$ in its domain to an ellipse: the principal semi-axes of the ellipse align with the $u_i$
and the $v_i$ are their preimages. \DeclareMathOperator*{\asterisk}{\ast} Thus our SVD allows us to represent the same data with at less than 1/3 1 / 3 the size of the original matrix. u1 shows the
average direction of the column vectors in the first category. So we place the two non-zero singular values in a 22 diagonal matrix and pad it with zero to have a 3 3 matrix. Now assume that we label
them in decreasing order, so: Now we define the singular value of A as the square root of i (the eigenvalue of A^T A), and we denote it with i. \newcommand{\mU}{\mat{U}} So to find each coordinate
ai, we just need to draw a line perpendicular to an axis of ui through point x and see where it intersects it (refer to Figure 8). The vectors can be represented either by a 1-d array or a 2-d array
with a shape of (1,n) which is a row vector or (n,1) which is a column vector. it doubles the number of digits that you lose to roundoff errors. So we can flatten each image and place the pixel
values into a column vector f with 4096 elements as shown in Figure 28: So each image with label k will be stored in the vector fk, and we need 400 fk vectors to keep all the images. The singular
value decomposition is similar to Eigen Decomposition except this time we will write A as a product of three matrices: U and V are orthogonal matrices. M is factorized into three matrices, U, and V,
it can be expended as linear combination of orthonormal basis diections (u and v) with coefficient . U and V are both orthonormal matrices which means UU = VV = I , I is the identity matrix. What is
the relationship between SVD and eigendecomposition? In this specific case, $u_i$ give us a scaled projection of the data $X$ onto the direction of the $i$-th principal component. \newcommand{\
hadamard}{\circ} Then we approximate matrix C with the first term in its eigendecomposition equation which is: and plot the transformation of s by that. The value of the elements of these vectors can
be greater than 1 or less than zero, and when reshaped they should not be interpreted as a grayscale image. Higher the rank, more the information. First come the dimen-sions of the four subspaces in
Figure 7.3. $$, where $\{ u_i \}$ and $\{ v_i \}$ are orthonormal sets of vectors.A comparison with the eigenvalue decomposition of $S$ reveals that the "right singular vectors" $v_i$ are equal to
the PCs, the "right singular vectors" are, $$ Then we pad it with zero to make it an m n matrix. Then we reconstruct the image using the first 20, 55 and 200 singular values. These vectors have the
general form of. Their entire premise is that our data matrix A can be expressed as a sum of two low rank data signals: Here the fundamental assumption is that: That is noise has a Normal
distribution with mean 0 and variance 1. \newcommand{\set}[1]{\lbrace #1 \rbrace} The only difference is that each element in C is now a vector itself and should be transposed too. If is an
eigenvalue of A, then there exist non-zero x, y Rn such that Ax = x and yTA = yT. Relationship between eigendecomposition and singular value decomposition, We've added a "Necessary cookies only"
option to the cookie consent popup, Visualization of Singular Value decomposition of a Symmetric Matrix. The values along the diagonal of D are the singular values of A. But that similarity ends
there. \newcommand{\dash}[1]{#1^{'}} single family homes for sale milwaukee, wi; 5 facts about tulsa, oklahoma in the 1960s; minuet mountain laurel for sale; kevin costner daughter singer An
eigenvector of a square matrix A is a nonzero vector v such that multiplication by A alters only the scale of v and not the direction: The scalar is known as the eigenvalue corresponding to this
eigenvector. Each matrix iui vi ^T has a rank of 1 and has the same number of rows and columns as the original matrix. So A^T A is equal to its transpose, and it is a symmetric matrix. Graph neural
network (GNN), a popular deep learning framework for graph data is achieving remarkable performances in a variety of such application domains. First, we calculate the eigenvalues and eigenvectors of
A^T A. What is the relationship between SVD and eigendecomposition? \newcommand{\sC}{\setsymb{C}} \newcommand{\infnorm}[1]{\norm{#1}{\infty}} In fact, for each matrix A, only some of the vectors have
this property. Each of the matrices. In linear algebra, the Singular Value Decomposition (SVD) of a matrix is a factorization of that matrix into three matrices. An important property of the
symmetric matrices is that an nn symmetric matrix has n linearly independent and orthogonal eigenvectors, and it has n real eigenvalues corresponding to those eigenvectors. Let the real values data
matrix $\mathbf X$ be of $n \times p$ size, where $n$ is the number of samples and $p$ is the number of variables. However, it can also be performed via singular value decomposition (SVD) of the data
matrix $\mathbf X$. Also conder that there a Continue Reading 16 Sean Owen $$A^2 = AA^T = U\Sigma V^T V \Sigma U^T = U\Sigma^2 U^T$$ Every real matrix \( \mA \in \real^{m \times n} \) can be
factorized as follows. That is because the columns of F are not linear independent. Making sense of principal component analysis, eigenvectors & eigenvalues -- my answer giving a non-technical
explanation of PCA. The Sigma diagonal matrix is returned as a vector of singular values. If a matrix can be eigendecomposed, then finding its inverse is quite easy. \newcommand{\integer}{\mathbb{Z}}
Eigendecomposition and SVD can be also used for the Principal Component Analysis (PCA). now we can calculate ui: So ui is the eigenvector of A corresponding to i (and i). The result is a matrix that
is only an approximation of the noiseless matrix that we are looking for. The only way to change the magnitude of a vector without changing its direction is by multiplying it with a scalar. We form
an approximation to A by truncating, hence this is called as Truncated SVD. Please answer ALL parts Part 1: Discuss at least 1 affliction Please answer ALL parts . In addition, it returns V^T, not V,
so I have printed the transpose of the array VT that it returns. Ok, lets look at the above plot, the two axis X (yellow arrow) and Y (green arrow) with directions are orthogonal with each other. We
know that the singular values are the square root of the eigenvalues (i=i) as shown in (Figure 172). And this is where SVD helps. \newcommand{\mH}{\mat{H}} Connect and share knowledge within a single
location that is structured and easy to search. A symmetric matrix transforms a vector by stretching or shrinking it along its eigenvectors, and the amount of stretching or shrinking along each
eigenvector is proportional to the corresponding eigenvalue. Hence, $A = U \Sigma V^T = W \Lambda W^T$, and $$A^2 = U \Sigma^2 U^T = V \Sigma^2 V^T = W \Lambda^2 W^T$$. Singular value decomposition
(SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. )The singular values $\
sigma_i$ are the magnitude of the eigen values $\lambda_i$. In the first 5 columns, only the first element is not zero, and in the last 10 columns, only the first element is zero. Depends on the
original data structure quality. So x is a 3-d column vector, but Ax is a not 3-dimensional vector, and x and Ax exist in different vector spaces. On the right side, the vectors Av1 and Av2 have been
plotted, and it is clear that these vectors show the directions of stretching for Ax. So what does the eigenvectors and the eigenvalues mean ? How to handle a hobby that makes income in US. Figure 22
shows the result. relationship between svd and eigendecomposition. Now let A be an mn matrix. \renewcommand{\BigO}[1]{\mathcal{O}(#1)} That will entail corresponding adjustments to the \( \mU \) and
\( \mV \) matrices by getting rid of the rows or columns that correspond to lower singular values. For example in Figure 26, we have the image of the national monument of Scotland which has 6 pillars
(in the image), and the matrix corresponding to the first singular value can capture the number of pillars in the original image. u2-coordinate can be found similarly as shown in Figure 8. X = \left(
It also has some important applications in data science. That is because B is a symmetric matrix. PCA is a special case of SVD. \newcommand{\sX}{\setsymb{X}} They correspond to a new set of features
(that are a linear combination of the original features) with the first feature explaining most of the variance. If we multiply both sides of the SVD equation by x we get: We know that the set {u1,
u2, , ur} is an orthonormal basis for Ax. \newcommand{\vr}{\vec{r}} The main idea is that the sign of the derivative of the function at a specific value of x tells you if you need to increase or
decrease x to reach the minimum. You can easily construct the matrix and check that multiplying these matrices gives A. This decomposition comes from a general theorem in linear algebra, and some
work does have to be done to motivate the relatino to PCA. relationship between svd and eigendecomposition. \newcommand{\unlabeledset}{\mathbb{U}} S = \frac{1}{n-1} \sum_{i=1}^n (x_i-\mu)(x_i-\mu)^T
= \frac{1}{n-1} X^T X The problem is that I see formulas where $\lambda_i = s_i^2$ and try to understand, how to use them? Machine learning is all about working with the generalizable and dominant
patterns in data. The image background is white and the noisy pixels are black. Now we can write the singular value decomposition of A as: where V is an nn matrix that its columns are vi. This data
set contains 400 images. george smith north funeral home In Figure 19, you see a plot of x which is the vectors in a unit sphere and Ax which is the set of 2-d vectors produced by A. This vector is
the transformation of the vector v1 by A. \newcommand{\setsymb}[1]{#1} \newcommand{\expe}[1]{\mathrm{e}^{#1}} So now we have an orthonormal basis {u1, u2, ,um}. In linear algebra, the singular value
decomposition (SVD) is a factorization of a real or complex matrix.It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. So the singular values
of A are the length of vectors Avi. TRANSFORMED LOW-RANK PARAMETERIZATION CAN HELP ROBUST GENERALIZATION in (Kilmer et al., 2013), a 3-way tensor of size d 1 cis also called a t-vector and denoted by
underlined lowercase, e.g., x, whereas a 3-way tensor of size m n cis also called a t-matrix and denoted by underlined uppercase, e.g., X.We use a t-vector x Rd1c to represent a multi- Please note
that unlike the original grayscale image, the value of the elements of these rank-1 matrices can be greater than 1 or less than zero, and they should not be interpreted as a grayscale image. In any
case, for the data matrix $X$ above (really, just set $A = X$), SVD lets us write, $$ data are centered), then it's simply the average value of $x_i^2$. A matrix whose columns are an orthonormal set
is called an orthogonal matrix, and V is an orthogonal matrix. Similarly, we can have a stretching matrix in y-direction: then y=Ax is the vector which results after rotation of x by , and Bx is a
vector which is the result of stretching x in the x-direction by a constant factor k. Listing 1 shows how these matrices can be applied to a vector x and visualized in Python. In fact, in the
reconstructed vector, the second element (which did not contain noise) has now a lower value compared to the original vector (Figure 36). So every vector s in V can be written as: A vector space V
can have many different vector bases, but each basis always has the same number of basis vectors. \newcommand{\mSigma}{\mat{\Sigma}} This result indicates that the first SVD mode captures the most
important relationship between the CGT and SEALLH SSR in winter. Imagine that we have a vector x and a unit vector v. The inner product of v and x which is equal to v.x=v^T x gives the scalar
projection of x onto v (which is the length of the vector projection of x into v), and if we multiply it by v again, it gives a vector which is called the orthogonal projection of x onto v. This is
shown in Figure 9. by x, will give the orthogonal projection of x onto v, and that is why it is called the projection matrix. So it acts as a projection matrix and projects all the vectors in x on
the line y=2x. I downoaded articles from libgen (didn't know was illegal) and it seems that advisor used them to publish his work.
Havant Tip Booking System, Parker High School Football Roster, Princess Diana Ghost Prince William Wedding, Shawn Mendes Tour 2022, Articles R | {"url":"https://unser-altona.de/7pjvtp6/iazia/archive.php?id=relationship-between-svd-and-eigendecomposition","timestamp":"2024-11-13T17:47:46Z","content_type":"text/html","content_length":"65899","record_id":"<urn:uuid:94920295-bdf0-4035-87a4-da9e1fb0990b>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00748.warc.gz"} |
Vampire 2019: Author Index
[ ]
Vampire 2019: Author Index
Author Papers
Biere, Armin SAT, Computer Algebra, Multipliers
Gonzalez-Dios, Itziar Towards Word Sense Disambiguation by Reasoning
Holden, Sean Bayesian Optimisation for Heuristic Configuration in Automated Theorem Proving
Jamnik, Mateja Bayesian Optimisation for Heuristic Configuration in Automated Theorem Proving
Kauers, Manuel SAT, Computer Algebra, Multipliers
Mangla, Chaitanya Bayesian Optimisation for Heuristic Configuration in Automated Theorem Proving
Paulson, Lawrence C. Bayesian Optimisation for Heuristic Configuration in Automated Theorem Proving
Riener, Martin Experimenting with Theory Instantiation in Vampire
Rigau, German Towards Word Sense Disambiguation by Reasoning
Ritirc, Daniela SAT, Computer Algebra, Multipliers
Suda, Martin Aiming for the Goal with SInE
Słowik, Agnieszka Bayesian Optimisation for Heuristic Configuration in Automated Theorem Proving
Álvez, Javier Towards Word Sense Disambiguation by Reasoning | {"url":"https://easychair.org/publications/volume/Vampire_2019/authors","timestamp":"2024-11-11T08:14:23Z","content_type":"text/html","content_length":"5756","record_id":"<urn:uuid:ca1af3a1-207d-4ba4-afb8-4a5b5cb00f57>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00215.warc.gz"} |
Using the LIMIT Clause to Retrieve Random Data with Restrictions in SQL | IT trip
Structured Query Language (SQL) is a powerful tool for managing databases, and the LIMIT clause is particularly useful for controlling the amount of data retrieved from a database. When combined with
random data fetching, this can lead to potent queries that serve a wide array of applications. This article delves into how to use the LIMIT clause to retrieve random data in a restricted manner,
effectively managing both performance and specificity of the data.
Introduction to LIMIT and Random Data Fetching in SQL
When dealing with large datasets in SQL, sometimes you don’t need to fetch all the records. You might need just a few records for sampling or for some quick analysis. This is where the LIMIT clause
comes into play. Furthermore, sometimes it’s beneficial to fetch records in a random order, not just the first or last few based on certain conditions.
What is the LIMIT Clause?
The LIMIT clause is used in SQL to specify the maximum number of records to be returned by a query. This can be particularly useful when you are dealing with large tables and only need a subset of
Why Fetch Random Data?
Fetching data in a random sequence is essential for various statistical analyses, machine learning model training, A/B testing, and even for load balancing in certain applications. By doing so, you
eliminate the bias that can occur from the order in which the data is stored in the database.
Using LIMIT for Random Data Retrieval
Using the LIMIT clause to fetch random data in SQL is not directly straightforward, but it’s entirely possible. You generally use a combination of sorting and limiting functions to achieve this.
Fetching Random Rows
Here is a simple example that demonstrates how to fetch five random rows from a table named `employees`.
SELECT * FROM employees
ORDER BY RAND()
LIMIT 5;
In this query, `ORDER BY RAND()` sorts the rows in a random order, and then `LIMIT 5` restricts the output to only the first five randomly sorted rows.
Random Sampling with Conditions
If you need to fetch random data but with certain conditions, you can include a WHERE clause. Here’s an example:
SELECT * FROM employees
WHERE department = 'Engineering'
ORDER BY RAND()
LIMIT 5;
This query fetches five random rows where the department is Engineering.
Performance Considerations
While this method is simple, it may not be efficient for large tables as it sorts the entire table before applying the LIMIT clause. A more performance-efficient way could be to use subqueries or to
index your tables appropriately.
Advanced Techniques
As you become more comfortable using the LIMIT clause for random data fetching, you might want to explore more advanced techniques, such as weighted random sampling or clustering.
Weighted Random Sampling
In some applications, you might need to sample data based on certain weights or probabilities. SQL allows you to do this through calculated fields and temporary tables.
Sometimes, your data may have inherent clusters that you want to take into account when sampling. SQL allows for this through its various aggregate functions and JOIN operations.
The LIMIT clause in SQL is a versatile tool that becomes even more powerful when used in conjunction with functions for random sorting. Whether you’re conducting statistical analyses, training
machine learning models, or simply managing large databases, understanding how to retrieve random, restricted data can save both time and computational resources. As always, make sure to consider the
specific needs of your project and the limitations of your database system when implementing these techniques. | {"url":"https://en.ittrip.xyz/sql/sql-limit-random-data","timestamp":"2024-11-04T02:28:23Z","content_type":"text/html","content_length":"185440","record_id":"<urn:uuid:57068a1a-8174-40db-b5fe-a64fb2e8aef9>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00511.warc.gz"} |
Modeling the Physics of Selective Laser Sintering Using the Discrete Element Method
In the preceding discussion, the potential of the discrete element method, often known as DEM, in replicating powder-based stacking was brought to light. In order to make the most of the capabilities
of this technology, a thorough algorithm has been devised. This algorithm incorporates the appropriate DEM formulation in order to simulate the SLS process in a manner that is both precise and
This multi-particle DEM framework addresses several major physical events in the fusion process, including:
•Particle dynamics: This aspect of the simulation focuses on the motion of particles caused by contact between particles and surfaces, as well as the forces that arise between particles as a result
of these interactions. The algorithm accounts for the complex behavior of particles as they collide and deform, providing a physically realistic simulation.
•Laser input: The algorithm also considers the absorption of the supplied laser energy by particles. The laser input is modeled in a way that accurately reflects the physical behavior of the
particles and their interactions with the laser.
•Thermodynamics of particles: The algorithm also models the heat transfer between particles that occurs through conductive contact. As a result of this heat transfer, particles may experience thermal
softening, which can significantly affect the particles' dynamics and behavior. The algorithm accounts for these effects, providing a more realistic simulation of the thermodynamics of particles.
The simulation of the selective laser sintering process through the discrete element method incorporates several essential tasks designed to accurately replicate the associated physical phenomena.
These operations have been methodically organized and are depicted in Figure 2, which outlines the details of our DEM simulation framework. The subsequent sections provide a comprehensive examination
of the mathematical formulations implemented in our Python code, specifically tailored for each DEM modeling task. This detailed approach ensures that each step of the simulation process is grounded
in rigorous mathematical principles, allowing for a more accurate and effective representation of the SLS process in our computational model.
Figure 2. The tasks required for the simulation of the SLS process by the DEM
2.1 Particle system definition
A simulation that utilizes the discrete element method (DEM) involves many particles of varying shapes, such as spheres, cylinders, and polygons, that interact with each other or with a bounding
plane. These interactions can include collisions, rolling, and sliding. These interactions result in reaction forces, such as normal and tangential forces, that affect the position and velocity of
the particles. These forces can cause the particles to change direction, speed up, or slow down.
As shown in Figure 3, these interactions and forces are represented by various variables, such as position $\left(p_i\right)$, velocity $\left(v_i\right)$, tangential force $\left(f_t\right)$, and
normal force $\left(f_n\right)$, which are used to track the movement and behavior of the particles in the simulation. The normal and tangential directions are denoted by $\vec{n}$ and $\vec{\tau}$,
respectively, and $\delta_{i j}$ represents the overlap between particles $i$ and $j$.
It's important to note that while Figure 3 and Figure 4 depict spherical particles, this representation may not always be accurate. In many cases, the particles in a simulation may not be spherical
and can be irregular shapes such as cylinders, polygons, etc. [17]. Contact stiffness, which is the measure of the resistance of an object to deformation when a force is applied, is dependent on
local curvature, and this can vary significantly between spherical and non-spherical particles. Additionally, interactions between non-spherical particles, such as friction, rolling, and sliding,
generate tangential forces and moments that can significantly affect the particles' motion and behavior. Therefore, it should be used with caution when assuming all particles to be spherical, as this
approximation may not be universally applicable in all cases.
Figure 4 illustrates a structural visualization of the particulate system in the powder bed within our DEM simulation. This visualization explains how the particles are arranged and organized in the
bed, where the grains are seen to be evenly distributed.
The simulation time variable is $\tau$, the simulation time step is $\Delta \tau$, and $n_\tau$ is the total number of iterations in the simulation. The time step selection must consider various
factors, including the duration of the simulation. Longer time steps are necessary for more extended simulations, while smaller ones are required for more detailed and precise solutions.
The available range of time steps is determined by the selected integration method. In explicit methods, which are the most commonly used, the critical time step imposes a limit on the time step
An exciting work by Rougier et al. [18] contains a comparative study of the explicit time integration schemes commonly used in the discrete element method.
Figure 3. Parameters and mechanism of contact between particles in a powder bed
Figure 4. Representation of the particulate system within the powder bed in our DEM simulation
2.2 Contact search
The discrete element method (DEM) differs from continuous methods in that there are no connections between nodes through elements. This means that properties such as forces and accelerations,
velocities, and displacement can be transferred through contact criteria between pairs of elements.
Contact search is a crucial component of the method in terms of computational cost, often taking up 60-80% of simulation time, and can be particularly challenging when dealing with non-spherical/
circular particles. To reduce simulation time, an approach that limits contact search is necessary.
Due to the poly-dispersity of the particle system, where the difference between the maximum and minimum size is greater than ten, collision detection can be divided into two steps. Additionally, it
is assumed that all particles are spherical. Various algorithms can be employed for contact search to minimize the contact search required to lower the simulation duration. Literature on collision
detection methods can be found in references [19-23].
The most frequently employed techniques for collision detection are grid-based and tree-based algorithms [24], both of which have multiple forms and variations. The structure of these algorithms is
outlined in Figure 5.
(A) Grid-based techniques: This method divides the simulation space into a grid of cells, and particles are assigned to the appropriate cell based on their position. When a particle moves, it is
reassigned to the appropriate cell. This method is efficient but can lead to errors if the cell size is not chosen correctly.
(B) Tree-based techniques: These algorithms divide the simulation space into smaller regions, such as octrees or bounding volume hierarchies, and particles are assigned to the appropriate region
based on their position. This method is more accurate than grid-based techniques but can be computationally expensive.
In our DEM simulations, collision detection plays a critical role in identifying particle interactions. Each particle, denoted as particle 'i', is associated with a unique list, $L_i$, which contains
the indices of other particles that come within a pre-defined proximity, indicating a collision.
A collision is considered to have occurred when the distance between two particles is less than the collision radius. The distance between the particles is computed using the following equation:
$L_i=\left\{j:\left\|p_i-p_j\right\| \leq d_i\right\} ; d_i=r_i+r_j$ (1)
where, $\left\|p_i-p_j\right\|$ represents the normalized Euclidean distance between particles $i$ and $j$, and $d_i$ is the collision radius, calculated as the sum of the radii of the tested
This collision detection framework underpins the subsequent application of contact laws, ensuring that physical interactions between particles are accurately captured and analyzed within our
Figure 5. Collision detection algorithms in DEM simulations
2.3 Contact modeling
In this section, we present the laws that govern the physical interactions between particles in the powder bed, based on the lists of contacts identified in the previous section.
We differentiate two types of contact laws in the powder bed: the contact law be-tween the particles and the contact law with the boundary plane. These laws describe the mechanical and thermal
interactions within the medium.
2.3.1 Inter-particle contact law
Mechanical contact law:
When two discrete particles come into contact, it transmits forces to each of these particles. These forces oppose the relative motion of each particle to its neighbor in both normal and tangential
The connection is defined geometrically by a line passing through the centers of the two particles $i$ and $j$. A unit vector $\vec{n}$ is defined in the same direction as this line. The direction of
the parameters involved in the contact mechanism is depicted in Figure 3.
Building upon the principles established by researchers [14, 25], we integrate advanced formulations into our DEM simulations to characterize the interactions between particles. Specifically, the
mechanical contact laws for calculating the normal and shear forces between two particles, i and j, have been refined to include damping effects, enhancing the simulation's accuracy:
• The normal force ($f_{n_{i j}}$) considers both the elastic (Hookean) response and the damping effect due to the relative velocities of the particles:
$f_{n_{i j}}=-K_n . \delta_{i j} . \vec{n}-c_n . v_{i j} . m^*$ (2)
• The shear force ($f_{t_{i j}}$), influenced by the normal force, models frictional interactions and is defined as:
$f_{t_{i j}}=-\mu_c\left\|f_{n_{i j}}\right\| \frac{\left(v_i-v_j\right)}{\left\|v_i-v_j\right\|}$ (3)
In this context, $c_n$ refers to the damping coefficient, and $v_{i j}$ denotes the relative normal velocities. The effective mass is represented as $m^*=m_i m_j /\left(m_i+m_j\right)$, where $m_k$,
for $k$ in $\{i, j\}$, signifies the mass of the k-th particle. Additionally, $\delta_{i j}$ represents the penetration distance between particles $i$ and $j$, computed using the following formula:
$\delta_{i j}=d_i-\left\|p_i-p_j\right\|$ (4)
where, $d_i$ is the sum of the radii of the particles in contact, already utilized in the collision detection process (refer to section 2.2), $\left\|p_i-p_j\right\|$ represents the Euclidean
distance between the centroids of particles $i$ and $j . K_n$ is the normal stiffness coefficient, $\mu_c$ represents the Coulomb friction coefficient, and $v_i$ and $v_j$ are the velocity vectors of
particles $i$ and $j$.
Therefore, the total force $\left(f_i\right)$ exerted on particle $i$ is the sum of all the normal and tangential forces exerted by the neighboring particles.
$f_i=\sum_j\left(f_{n_{i j}}+f_{t_{i j}}\right)$ (5)
Thermal contact law:
In our DEM simulation, particle interactions lead not only to mechanical impacts but also to significant thermal exchanges, as modeled based on methodology of the study [26], focusing solely on
conduction for heat transfer between particles. This specific thermal interaction, separate from mechanical forces, is vital for a precise depiction of powder bed dynamics in SLS processes. Governed
by Fourier's law, the heat transfer is quantified as:
$\mathrm{q}_{\mathrm{ij}}=\lambda_{\mathrm{c}}\left(\mathrm{T}_{\mathrm{j}}-\mathrm{T}_{\mathrm{i}}\right)$ (6)
where, $\lambda_c$ represents the heat conduction between the particles and $\mathrm{T}_{\mathrm{i}}$ and $\mathrm{T}_{\mathrm{j}}$ are the temperatures of particles $i$ and $j$. The heat absorbed by
particle $i$ as a result of contact with another particle can be succinctly represented as:
$q_i=\sum_{j=1}^{n_c} q_{i j}$ (7)
where, $n_c$ indicates the number of contacts formed by particle $i$.
2.3.2 Contact law with the boundary planes
Mechanical contact law:
In our DEM framework, the interaction dynamics between a particle and a boundary plane (wall) are modeled with the same principles as those between individual particles. This consistency ensures a
unified approach to force calculations across the simulation. Specifically, we focus on the normal force exerted by the wall on particles, which is the force acting perpendicular to the surface of
the wall, denoted as (k).
This normal force is added to the force of the particles to update their position and velocity. To do this, the distance between the particles and the wall must be calculated $\left(x_i\right)$, and
the force with the walls is determined using the following equation:
$f_{n_{i, k}}=k_b \cdot \beta \cdot \overrightarrow{n_k} ; \beta=\left\{\begin{array}{c}0 \text { if } x_i \geq r_i \\ r_i-x_i \text { if } x_i<r_i\end{array}\right.$ (8)
The distance between particle $i$ and the wall is represented by $x_i, \overrightarrow{n_k}$ is the normal unit vector of the boundary surface, and $k_b$ is the boundary stiffness.
Thermal contact law:
In our DEM simulations, we approach heat transfer between particles and walls identically, treating it as strictly conductive to maintain consistency and simplicity. However, for particles at the
upper surface, the scenario differs; they experience heat exchange with the chamber's internal air through both convection and radiation, acknowledging the distinct thermal environments experienced
by these particles compared to those fully embedded within the powder bed.
In the case of wall transfer, a conduction heat flow is added to particle i, which is described by the following equation:
$q_i+=\lambda_{c b}\left(T_w-T_i\right)$ (9)
For heat transfer on the top surface, particle i receives a heat flow through both convection and radiation, represented by the following equation:
$q_i+=h_c\left(T_a-T_i\right)+\zeta \cdot \sigma_{S B} \cdot\left(T_a{ }^4-T_i^4\right)$ (10)
where, $T_w$ and $T_a$ represent the temperatures of the wall and fabrication chamber respectively. $\lambda_{c b}$ is the conduction coefficient with the wall, $h_c$ is the convection coefficient
with the chamber, $\zeta$ is the emissivity of the material, and $\sigma_{S B}$ is the Stefan-Boltzmann constant.
2.4 Energy deposition
Powder-based additive manufacturing processes involve using a laser source to transfer energy in order to create a physical object. The position of the laser source can be changed depending on the
desired scanning strategy.
This means that the laser beam can be moved to different locations and angles in order to achieve the desired final product. The motion of the laser beam is described mathematically by a function,
denoted by $\varphi(\tau)$, which specifies the exact coordinates of the laser at each time step.
$\varphi(\tau)=\left\{\begin{array}{cc}\varphi_0(\tau), & 0 \leq \tau<\tau_0 \\ \varphi_1(\tau), & \tau_0 \leq \tau<\tau_1 \\ \cdot \\ \cdot \\ \varphi_n, & \tau_{n-1} \leq \tau<\tau_n\end{array}\
right.$ (11)
The laser beam scans a group of particles in a powder bed at each time step. The absorption of the light, or flow, by these particles is studied and modeled in academic literature [27, 28]. One
common model used is the cylindrical and Gaussian thermal flow model, which considers the shape and distribution of the flow within the powder bed.
In our DEM simulation, we will use the Gaussian model to represent the heat distribution in the laser beam. This is an improvement over previous work that has been published [14], as the Gaussian
model provides a more accurate representation of how the flow spreads in the powder bed. This is because the Gaussian model accounts for the natural propagation and decay of the flow as it moves
through the powder. The following equation characterizes the Gaussian model:
$q_0(r)=\frac{2 P}{\pi r_l^2} \cdot e^{-2 r^2 / r_l^2}$ (12)
where, $r_l$ is the laser beam radius, and $P$ is the laser power.
As a result of the laser beam scanning, the particles in the powder bed absorb an additional amount of heat flow. The amount of heat flow absorbed by each particle is not uniform, it varies depending
on the particle's distance from the center of the laser beam and if the particle is wholly or partially scanned, as illustrated in Figure 6.
$q_i+=\left\{\begin{array}{cc}q_0 * \frac{S_i}{S_l}, & 0 \leq\left\|p_i-\beta_l\right\| \leq r_l-r_i \\ q_0 * \frac{S_{\text {inter }}}{S_l}, & r_l-r_i \leq\left\|p_i-\beta_l\right\| \leq r_l+r_i \\
0 & \text { otherwise }\end{array}\right.$ (13)
The distance between particle $i$ and the center of the laser beam is represented by $\left\|p_i-\beta_l\right\|$, while $S_i$ represents the area of particle $i, S_l$ represents the area of laser
sintering, $S_{\text {inter}}$ represents the interface between the particle and the heat input area. The total energy absorbed by all sintered particles is described by $q_0$, which is distributed
among these particles according to their sintering area ratio.
Figure 6. Visualization of the particles in the powder bed as the laser beam scans them
2.5 Bond formation
After the energy is applied, the particles form bonds with their neighbors. These bonds are determined by specific formation criteria and are stored in a list $B_i$, which lists the indices of
particles that are bonded to particle $i$. To collect these connections, we use the contact list $\left(L_i\right)$ as outlined in Section 2.2, which defines the contacts of particle $i$.
The criteria for the formation of links or bonds between particles include the following:
• Temperature exceeding the sintering temperature (T[s]): $T_i, T_j \geq T_s$
• Sufficient normal contact force: $f_{n_{i j}} \leq f_{c r i t}$
• Low particle speeds: $v_i \leq v_{\text {crit }}$ and $v_j \leq v_{\text {crit }}$
$f_{\text {crit }}, v_{\text {crit }}$ and $T_s$ are critical criteria that vary for each material used in the SLS process.
2.6 Time step integration
Upon computing the physical interactions affecting each particle within our DEM framework, it is essential to extend these impacts temporally. For this purpose, we employ explicit time integration
schemes tailored for DEM simulations, as detailed in reference [18].
The same update scheme used in the work of Lakraimi et al. [14] is applied as follows:
$\begin{gathered}v_i(\tau+\Delta \tau)=v_i(\tau)+\frac{f_i(\tau)}{m_i} \Delta \tau \\ p_i(\tau+\Delta \tau)=p_i(\tau)+v_i(\tau) \Delta \tau \\ T_i(\tau+\Delta \tau)=T_i(\tau)+\frac{q_i(\tau)}{m_i *
c_p} \Delta \tau\end{gathered}$ (14)
where, $m_i$ and $c_p$ represent the mass and the mass heat capacity of particle $i$, respectively.
2.7 Implementation
The methodology elaborated in previous sections, which outlines the conceptual framework of our DEM simulation, is systematically organized and depicted in Figure 7. This figure illustrates the
comprehensive algorithm developed for simulating the SLS process via the discrete element method. This algorithm, as showcased in Figure 7, has been effectively implemented and visualized utilizing
the Python programming language.
Figure 7. The general framework for simulating the SLS process under the projection of a moving laser source | {"url":"https://iieta.org/journals/ijcmem/paper/10.18280/ijcmem.120103","timestamp":"2024-11-03T16:11:35Z","content_type":"text/html","content_length":"114925","record_id":"<urn:uuid:786cb946-fe8c-4b65-8c97-7110dc76d527>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00828.warc.gz"} |
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We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π
-statements in the structure 〈
ω [2]
), ∈, NS
[ω 1]
〉 for some set
of reals in
(ℝ), for the following statements: (1) The cofinality of the null ideal is ℵ
. (2) There exists a good basis of the strong measure zero ideal. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson
demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time.
The supplier therefore functions as an
/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution
of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer.
We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder
levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We
conclude by recommending an approach which is intuitive and computationally straightforward.
Conductance data for sodium nitrite, chloride, and acetate in water and
-dimethylformamide (DMF)-water mixtures (74.82
42.48) for the concentration range 0.001–0.04
, as well as the densities, viscosities, and dielectric constants of the solvent mixtures at 35°C, are reported. The data have been analyzed by the Fuoss (1975) equation. The existence of a maximum
in the viscosity at a 13 mole ratio of DMF and water is indicated. The Walden products for all the three salts pass through a maximum while the equivalent conductances show a minimum with increasing
DMF content. The maxima in the Walden product are attributed to the dehydration of ions by the cosolvent (DMF).Part I:
Indian J. Chem. 14
A, 1015 (1976).Deceased.
In this article, a novel method is proposed for investigating the set controllability of Markov jump switching Boolean control networks (MJSBCNs). Specifically, the switching signal is described as a
discrete-time homogeneous Markov chain. By resorting to the expectation and switching indicator function, an expectation system is constructed. Based on the expectation system, a novel verifiable
condition is established for solving the set reachability of MJSBCNs. With the newly obtained results on set reachability, a necessary and sufficient condition is further derived for the set
controllability of MJSBCNs. The obtained results are applied to Boolean control networks with Markov jump time delays. Examples are demonstrated to justify the theoretical results.
With the rapid development of wireless sensor technology, recent progress in wireless sensor and actuator networks (WSANs) with energy harvesting provide the possibility for various real-time
applications. Meanwhile, extensive research activities are carried out in the fields of efficient energy allocation and control strategy design. However, the joint design considering physical plant
control, energy harvesting, and consumption is rarely concerned in existing works. In this paper, in order to enhance system control stability and promote quality of service for the WSAN energy
efficiency, a novel three-step joint optimization algorithm is proposed through control strategy and energy management analysis. First, the optimal sampling interval can be obtained based on energy
harvesting, consumption, and remaining conditions. Then, the control gain for each sampling interval is derived by using a backward iteration. Finally, the optimal control strategy is determined as a
linear function of the current plant states and previous control strategies. The application of UAV formation flight system demonstrates that better system performance and control stability can be
achieved by the proposed joint optimization design for all poor, sufficient, and general energy harvesting scenarios.
To take into account the temporal dimension of uncertainty in stock markets, this paper introduces a cross-sectional estimation of stock market volatility based on the intrinsic entropy model. The
proposed cross-sectional intrinsic entropy (
) is defined and computed as a daily volatility estimate for the entire market, grounded on the daily traded prices—open, high, low, and close prices (OHLC)—along with the daily traded volume for all
symbols listed on The New York Stock Exchange (NYSE) and The National Association of Securities Dealers Automated Quotations (NASDAQ). We perform a comparative analysis between the time series
obtained from the
and the historical volatility as provided by the estimators: close-to-close, Parkinson, Garman–Klass, Rogers–Satchell, Yang–Zhang, and intrinsic entropy (
), defined and computed from historical OHLC daily prices of the Standard & Poor’s 500 index (S&P500), Dow Jones Industrial Average (DJIA), and the NASDAQ Composite index, respectively, for various
time intervals. Our study uses an approximate 6000-day reference point, starting 1 January 2001, until 23 January 2022, for both the NYSE and the NASDAQ. We found that the
market volatility estimator is consistently at least 10 times more sensitive to market changes, compared to the volatility estimate captured through the market indices. Furthermore, beta values
confirm a consistently lower volatility risk for market indices overall, between 50% and 90% lower, compared to the volatility risk of the entire market in various time intervals and rolling windows.
Financial and economic time series forecasting has never been an easy task due to its sensibility to political, economic and social factors. For this reason, people who invest in financial markets
and currency exchange are usually looking for robust models that can ensure them to maximize their profile and minimize their losses as much as possible. Fortunately, recently, various studies have
speculated that a special type of Artificial Neural Networks (ANNs) called Recurrent Neural Networks (RNNs) could improve the predictive accuracy of the behavior of the financial data over time. This
paper aims to forecast: (i) the closing price of eight stock market indexes; and (ii) the closing price of six currency exchange rates related to the USD, using the RNNs model and its variants: the
Long Short-Term Memory (LSTM) and the Gated Recurrent Unit (GRU). The results show that the GRU gives the overall best results, especially for the univariate out-of-sample forecasting for the
currency exchange rates and multivariate out-of-sample forecasting for the stock market indexes.
Domain adaptation aims to learn a classifier for a target domain task by using related labeled data from the source domain. Because source domain data and target domain task may be mismatched, there
is an uncertainty of source domain data with respect to the target domain task. Ignoring the uncertainty may lead to models with unreliable and suboptimal classification results for the target domain
task. However, most previous works focus on reducing the gap in data distribution between the source and target domains. They do not consider the uncertainty of source domain data about the target
domain task and cannot apply the uncertainty to learn an adaptive classifier. Aimed at this problem, we revisit the domain adaptation from source domain data uncertainty based on evidence theory and
thereby devise an adaptive classifier with the uncertainty measure. Based on evidence theory, we first design an evidence net to estimate the uncertainty of source domain data about the target domain
task. Second, we design a general loss function with the uncertainty measure for the adaptive classifier and extend the loss function to support vector machine. Finally, numerical experiments on
simulation datasets and real-world applications are given to comprehensively demonstrate the effectiveness of the adaptive classifier with the uncertainty measure. | {"url":"https://slh.alljournals.cn/search.aspx?subject=mathematical_chemical&major=sx&orderby=referenced&field=key_word&q=stock+prices%2C+delay%2C+better-of+options%2C+equivalent+martingale+measure.&encoding=utf8","timestamp":"2024-11-12T03:32:00Z","content_type":"application/xhtml+xml","content_length":"66231","record_id":"<urn:uuid:b1dca465-685c-44fe-acdb-dd19f32ed183>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00185.warc.gz"} |
Short-term prediction of horizontal winds in the mesosphere and lower thermosphere over coastal Peru using a hybrid model
The mesosphere and lower thermosphere (MLT) are transitional regions between the lower and upper atmosphere. The MLT dynamics can be investigated using wind measurements conducted with meteor radars.
Predicting MLT winds could help forecast ionospheric parameters, which has many implications for global communications and geo-location applications. Several literature sources have developed and
compared predictive models for wind speed estimation. However, in recent years, hybrid models have been developed that significantly improve the accuracy of the estimates. These integrate time series
decomposition and machine learning techniques to achieve more accurate short-term predictions. This research evaluates a hybrid model that is capable of making a short-term prediction of the
horizontal winds between 80 and 95 km altitudes on the coast of Peru at two locations: Lima (12°S, 77°W) and Piura (5°S, 80°W). The model takes a window of 56 data points as input (corresponding to 7
days) and predicts 16 data points as output (corresponding to 2 days). First, the missing data problem was analyzed using the Expectation Maximization algorithm (EM). Then, variational mode
decomposition (VMD) separates the components that dominate the winds. Each resulting component is processed separately in a Long short-term memory (LSTM) neural network whose hyperparameters were
optimized using the Optuna tool. Then, the final prediction is the sum of the predicted components. The efficiency of the hybrid model is evaluated at different altitudes using the root mean square
error (RMSE) and Spearman’s correlation (r). The hybrid model performed better compared to two other models: the persistence model and the dominant harmonics model. The RMSE ranged from 10.79 to
27.04 (Formula presented.), and the correlation ranged from 0.55 to 0.94. In addition, it is observed that the prediction quality decreases as the prediction time increases. The RMSE at the first
step reached 6.04 (Formula presented.) with a correlation of 0.99, while at the sixteenth step, the RMSE increased up to 30.84 (Formula presented.) with a correlation of 0.5.
Profundice en los temas de investigación de 'Short-term prediction of horizontal winds in the mesosphere and lower thermosphere over coastal Peru using a hybrid model'. En conjunto forman una huella | {"url":"https://cris.pucp.edu.pe/es/publications/short-term-prediction-of-horizontal-winds-in-the-mesosphere-and-l","timestamp":"2024-11-15T04:39:33Z","content_type":"text/html","content_length":"56127","record_id":"<urn:uuid:1abf73a3-7c50-493e-96e5-b8d0ac8215c6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00669.warc.gz"} |
Hexadecimal to Binary
Use this Hexadecimal to Binary converter to convert any base-16 hex number into the zeros and ones. Enter a hex number value into the box and convert that hex number into the correct binary number
with ease. Hex numbers contain 0-9 in numbers and A-z in letters. For example, enter "4444" into the text area and click on the convert button. You will get the "100010001000100" answer in the
conversion box. | {"url":"https://binarytotext.net/hexadecimal-to-binary-converter","timestamp":"2024-11-02T01:51:55Z","content_type":"text/html","content_length":"7594","record_id":"<urn:uuid:b603d2e7-0ce0-42e4-8077-d64a0e0eb410>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00673.warc.gz"} |
Arterial Stiffness: Different Metrics, Different Meanings
Findings from basic science and clinical studies agree that arterial stiffness is fundamental to both the mechanobiology and the biomechanics that dictate vascular health and disease. There is,
therefore, an appropriately growing literature on arterial stiffness. Perusal of the literature reveals, however, that many different methods and metrics are used to quantify arterial stiffness, and
reported values often differ by orders of magnitude and have different meanings. Without clear definitions and an understanding of possible inter-relations therein, it is increasingly difficult to
integrate results from the literature to glean true understanding. In this paper, we briefly review methods that are used to infer values of arterial stiffness that span studies on isolated cells,
excised intact vessels, and clinical assessments. We highlight similarities and differences and identify a single theoretical approach that can be used across scales and applications and thus could
help to unify future results. We conclude by emphasizing the need to move toward a synthesis of many disparate reports, for only in this way will we be able to move from our current fragmented
understanding to a true appreciation of how vascular cells maintain, remodel, or repair the arteries that are fundamental to cardiovascular properties and function.
Issue Section:
Research Papers
Mechanical factors have long been known to play vital roles in arterial health, disease, and treatment. Studies dating back to the late 19th century showed that altered blood flow and pressure
typically result in changes in luminal radius and wall thickness [1,2], thus foreshadowing the need to quantify the mechanoregulation of arterial structure and function. It was not until the 1970s,
however, that it was discovered how such changes occur. Experiments on cultured endothelial cells and vascular smooth muscle cells revealed flow- and stretch-induced changes in secreted proteins that
resulted from altered gene expression [3,4]. Hence was born the field of vascular mechanobiology, reviews of the first three decades of which can be found elsewhere [5,6]. Importantly, careful
studies of the aorta during this same defining period revealed that circumferential lamellar tension (∼2N/m), and by inference circumferential lamellar stress (∼150kPa), is similar within the
medial layer across multiple mammalian species [7]. This finding, coupled with subsequent observations that aortic material stiffness at mean arterial pressure is also similar (∼500kPa) across
invertebrates and vertebrates [8], strongly suggests the existence of a “mechanical homeostasis”—that is, vascular cells seek to establish and then maintain particular mechanical quantities near
target (homeostatic) values. Indeed, findings at subcellular, cellular, tissue, and organ levels suggest that such homeostatic processes exist across many spatial and temporal scales [9].
Two seminal contributions by Fung and his colleagues advanced our ability to quantify arterial wall stress and the associated stiffness, which are important both for assessing the mechanics of the
wall and understanding its mechanobiology. First, Fung observed via uniaxial experiments on soft tissues that material stiffness (in this case, a change in the first Piola-Kirchhoff stress with
respect to stretch) relates nearly linearly to the stress itself [10]. Importantly, this finding suggests directly that the first Piola-Kirchhoff stress increases nearly exponentially with stretch.
Indeed, a similar observation had previously been reported for the overall structural stiffness of the pressurized eye based on its pressure–volume relation [11]. Motivated by these findings, Fung
later hypothesized the existence of an exponential stored energy function $W$ that yields an exponential relationship between the second Piola-Kirchhoff stress $S$ and Green strain $E$, where $S=∂W/
∂E$. Now known as Fung elasticity, this hyperelastic function can be written as $W=c(exp(Q)−1)$, where $c$ is a material parameter and the scalar function $Q$ depends quadratically on $E$ [12]. An
associated metric of material stiffness, in referential form, is thus $C=∂S/∂E=∂2W/∂E∂E$. Notwithstanding the importance of Fung's constitutive hypothesis in quantifying the mechanical behavior of
many soft tissues and solving associated initial-boundary value problems, the observation that material stiffness relates linearly to stress implies that it should be challenging to delineate whether
mechanobiological responses correlate better with stress or stiffness, consistent with the aforementioned observations that homeostatic values of aortic stress and material stiffness are similar
across species [7,8]. Second, Fung showed that the existence of residual stress in arteries dramatically affects the calculated distribution of Cauchy stress $t$ across the arterial wall [13], which
in combination with basal smooth muscle cell tone results in a nearly homogenized transmural distribution [12]. This finding supports the concept that vascular cells seek to establish and then
maintain mechanical stress or stiffness near a homeostatic target, regardless of location within the wall. Importantly, it now seems that different homeostatic targets may exist for different cell
types that populate different layers of the wall [14], thus reinforcing the concept of a cell-specific mechanical homeostasis that can manifest at the vessel level as well.
The importance of arterial stiffening in human disease was anticipated as early as the turn of the 20th century by Sir W^m Osler, one of four founding fathers of Johns Hopkins medicine, but
confirmation had to await a seminal clinical study at Paris, France that revealed that an increased pulse wave velocity ($PWV$) is an initiator and indicator of diverse cardiovascular,
neurovascular, and renovascular disease and thus all-cause mortality [15]. Carotid-to-femoral $PWV$ is now considered to be the gold standard for clinical assessment of arterial stiffness [16], which
increases with diabetes, hypertension, natural aging, particular connective tissue disorders, and so forth [17–19]. This metric of stiffness is best measured by dividing the vascular centerline
distance between two recording sites by the time that it takes the (foot of the) pressure pulse wave to travel between these sites (“foot-to-foot” $PWV$). Intuitive understanding of the $PWV$ often
comes from the Moens–Korteweg equation, derived in the 1870s and commonly written as $PWV=Eh/2ρa$, where $E$ is an intrinsic (linear) material stiffness, $h$ is the wall thickness, $a$ is the
luminal radius, and $ρ$ is the mass density of the fluid. Understood in this way, we see that $PWV$ depends on both the intrinsic material stiffness and the geometry of the wall, hence rendering it
an integrated (along the centerline distance) measure of structural, not material, stiffness.
Not surprisingly, there continues to be increasingly greater interest in measuring and understanding arterial stiffness in basic science studies and clinical assessments. Quantifying stiffness helps
us to understand critical questions related to, among other issues, fundamental aspects of vascular mechanobiology, effects of a distensible wall on the hemodynamics, disease progression, and even
the efficacy of particular clinical treatments. In this paper, we briefly review and contrast different methods for measuring and interpreting arterial stiffness both to emphasize that which has been
learned and that which remains unclear. In particular, we emphasize that many different metrics of arterial stiffness reported in the literature have different theoretical underpinnings and thus
different meanings. Caution is thus warranted when comparing values of stiffness across studies as well as when interpreting fundamental implications of a particular metric on either the cell biology
or the systems physiology.
We first review the theoretical basis for some of the experimental methods that are commonly used to infer material properties of the arterial wall and its constituent parts. The presentation is
organized by physical scale, not chronological introduction within the field. Regardless of scale, it is critical to delineate concepts of material versus structural stiffness, and similarly, it is
essential to note that values of stiffness depend on the type (tensile, compressive, or shear) and magnitude (small or large) of the loading as well as the mechanical state in which the experimental
loads are imposed, especially if the sample is otherwise unloaded during testing. Unfortunately, many papers often simply report a value for stiffness without emphasizing the precise definition or
experimental conditions, hence obscuring the range of applicability. Here, we attempt to delineate some of these issues.
Atomic Force Microscopy.
Invented in the early 1980s at IBM, atomic force microscopy (AFM) uses a laser to detect small deflections of a cantilever probe as it interacts with the surface of a specimen. The atomic force
microscope can be used in various modes to assess the topography of a surface or to probe mechanical properties. With regard to the latter, one can perform precise indentation (compression) tests or
functionalize the probe to enable tractions (tension) to be applied at the surface. As implied by its name, the most precise atomic force microscopes measure nanoscale forces and motions, thus it is
not surprising that AFM has been used extensively in biology, biophysics, and bioengineering, often to assess aspects of cell stiffness, cell-matrix interactions, or local matrix stiffness. With
larger probes, one can also assess bulk tissue stiffness. Associated data consist primarily of the applied force (inferred from knowledge of the cantilever properties and deflections) and the imposed
motion of the probe, often the depth of penetration into the surface. Quantities such as stress and stiffness can be determined by solving the associated initial-boundary value problem.
Data on indentation force (
) and depth (
) are often interpreted using a classical 19th century solution by Hertz for the localized indentation of a semi-infinite half-space that exhibits a linearly elastic and isotropic material behavior
under small strains and rotations [
]. In this way, one can infer an associated Young's modulus
, a measure of the intrinsic compressive material stiffness for a specialized class of material behaviors. Albeit often not noted, this classical Hertz solution also assumes that the half-space is
unloaded prior to the indentation by the probe. Of course, all arteries and their attendant cells are under significant loads in vivo and often are loaded in vitro in cell or organ culture, hence any
Young's modulus inferred from the Hertz equation must be viewed cautiously. In particular, it has been shown both analytically [
] and numerically [
] that the indentation force-depth relationship, and thus inferred stiffness, depends strongly on the pre-existing state of stress/strain in the material. Because analytical findings typically
provide more intuitive insight than numerical results, note that an exact solution relevant to AFM can be obtained using a theory of small deformations superimposed on large (Appendix
) for the special case of an initially isotropic planar specimen subjected to finite equibiaxial in-plane stretching prior to a superimposed small indentation in the out-of-plane direction [
]. In this case, the indentation force
and (small) depth of penetration
are related linearly via
where the “transverse structural stiffness”
depends nonlinearly on the finite in-plane stretch experienced by the specimen as well as on the intrinsic material stiffness of the material and the geometry of the rigid probe. For example, for a
neo-Hookean material behavior defined by a stored energy function
, where
is the shear modulus and
is the right Cauchy–Green tensor, with
being the deformation gradient tensor, it can be shown that the transverse stiffness, having units of N/m, can be written as [
where 2$ro$ is the diameter (d[o]) of a flat-ended cylindrical probe and $λ$ is the equibiaxial in-plane stretch. A Hertz-type solution can be recovered as $λ→1$, for which $α→16roμ = 4doE/3$, with
Young's modulus $E(=6μ)$ having units of N/m^2. Hence, $E$ can be computed easily given the diameter of the probe and the slope of the force-depth data ($f,δ$) for an otherwise unloaded specimen.
Of course, because of inherent uncertainty in all experimental data, the stiffness parameter(s) should be estimated using least-squares regressions of data via an over-determined system of equations.
Moreover, in the case of a nonlinear behavior, the intrinsic material stiffness (e.g., shear modulus $μ$ for a neo-Hookean response) should be inferred from force-depth data over a range of in-plane
stretches that are relevant to the biological or physiological regime of interest in order to characterize well the overall material behavior. Noting that adherent cells spread out and develop
significant cytoskeletal stress when cultured on any substrate, which is to say that they develop a finite nearly in-plane state of stress or stretch, it has been shown that a small-on-large
framework can be used to assess cell stiffness from AFM for more general exponential stored energy functions [24].
Indeed, although a neo-Hookean relation admits a simple analytical solution that allows one to intuit effects of finite in-plane stretches on AFM measured stiffness (Eq. (1)), the Fung exponential
model is more appropriate for most soft tissues and even cells. Appendix B summarizes a general Fung model and Appendix C provides results for a small-on-large solution for AFM based on an
isotropic Fung exponential model. As expected, the numerical implementation is straightforward. Nevertheless, results from AFM for assessing vascular cell stiffness [25,26] and arterial properties [
27–30] typically have been interpreted in terms of a single Young's modulus inferred from the Hertz solution, which tacitly ignores the important nonlinear dependence on in-plane prestretch and the
nonlinear (often exponential) material behavior. Hence, the inferred values of stiffness are typically well below in vivo values and could be misleading regarding in vivo mechanobiology.
Although not discussed in detail here, additional methods are also used to assess cell stiffness, including magnetic twisting cytometry (MTC) and optical tweezers [31]. MTC is similar to AFM except
that one either affixes onto or embeds within a living cell a ferromagnetic bead that can be cyclically twisted using a magnetic field. Associated data are often interpreted in terms of the so-called
storage ($G′$) and loss ($G″$) shear moduli, which are basic descriptors of a one-dimensional (1D) linearly viscoelastic behavior over small strains and rotations [32]. For example, MTC was used
to show that the stiffness of vascular smooth muscle cells (i.e., elastic storage and dissipative loss moduli) increases in aging [33]. Similar to the Hertz solution, however, this inference of
material properties does not explicitly account for the underlying nonlinear material behaviors or the finite deformations that adherent cells experience in vitro when spreading on a surface or to
which they are likely exposed in vivo. Again, a small-on-large approach for data analysis could be more appropriate.
Biaxial Biomechanical Testing.
Arteries are subjected in vivo to complex multiaxial loads due to blood pressure and flow as well as axial prestresses that develop largely due to somatic growth [
]. Flow-induced shear stresses are important mechanobiologically and can be determined by solving equations that govern the hemodynamics (e.g., Navier–Stokes solutions within the context of
fluid–solid interactions), yet they are typically five orders of magnitude smaller (1.5Pa) than the in-plane intramural stresses (∼150kPa) and thus are neglected in most analyses of the wall stress
field. The in-plane circumferential (
) and axial (
) components of Cauchy stress are tensile, whereas the out-of-plane Cauchy radial (
) stress is compressive, on the order of −15kPa, and typically dictated largely by the traction boundary conditions. For these reasons, biaxial loading has long been preferred for studying the
biomechanical properties and function of blood vessels, particularly arteries and veins [
]. Associated pressure-diameter and axial force-length data provide direct insight into the structural stiffness of these vessels. Material stiffness can be inferred from a global equilibrium
solution that relates the loads that are measured during standard biaxial tests on excised cylindrical samples to components of the Cauchy stress as [
$P=∫rirotθθ−trrrdr, L=π∫riro(2tzz−trr−tθθ)rdr$
where $r∈[ri,ro]$, with $ri$ and $ro$ being the inner and the outer radius of the sample, respectively, $P$ is the distending pressure, and $L$ is the reduced axial load. If only the mean (i.e.,
radially averaged, $〈…〉$) values of stress are of interest, then these two integral relations can be replaced with algebraic ones, $〈tθθ〉≡σθθ=Pri/h$ and $〈tzz〉≡σzz=(L+πri2P)/πh(2ri+h)$, where
$h=ro−ri$ is the wall thickness. Mean values of stress are surprisingly useful because of the aforementioned effect of residual stress in homogenizing the transmural distribution of stress.
Regardless, some investigators infer stiffness from plots of stress versus stretch or strain (yielding so-called tangent moduli), but stiffness depends on the full multiaxial deformation and is best
computed from an appropriate constitutive relation as noted above for the referential stiffness ($∂S/∂E$). Given the complex microstructure of the vascular wall, most investigators now prefer the
so-called fiber-family constitutive models [35–37] to characterize multiaxial data, though the Fung exponential provides good fits to data in many cases (recall that Appendix B reviews a general
orthotropic Fung relation). Best-fit values of material parameters for any appropriate constitutive relation can be determined via nonlinear regression of pressure-diameter and axial force-length
data, with data from multiple biaxial protocols typically combined to improve the parameter estimation, again via (nonlinear) least squares regression. In this regard, it is important to note that
many studies nevertheless report only pressure-diameter data at a single value of axial stretch and often do not measure the associated axial force. Such data are essentially one-dimensional and not
useful for calculating in vivo relevant biaxial stress or stiffness [38].
There are, in addition, a few other issues regarding the in vivo applicability of constitutive relations that are inferred in vitro. First, values of arterial stiffness change with pressure over a
cardiac cycle, hence the theory of small deformations superimposed on large has been used to compute “single” values of the spatial material stiffness
that are often representative over a cardiac cycle and thereby render computations of fluid-solid-interactions more efficient [
]. In component form, we have (see Appendix
where $δij$ is the Kronecker delta and a superscript $o$ denotes a deformation quantity that is associated with an original finite deformation about which the small superimposed deformation occurs.
For example, the large deformation could be for an artery from a traction-free reference configuration to a finitely distended and extended in vivo configuration near mean arterial pressure about
which relatively small motions occur over a cardiac cycle. Second, there is a pressing need to understand better how central arterial stiffness and resistance vessel function interact to affect both
local and global hemodynamics and cardiovascular function [19,39]. Third, the effects of perivascular tethering can be as important as the structural stiffness (i.e., material stiffness and geometry)
in affecting the hemodynamics though there has not been much quantification of this effect [40,41]. Fourth, constitutive relations that are inferred in vitro are generally established under
quasi-static conditions, whereas in vivo loading is pulsatile and hence intrinsically dynamic. Associated differences in strain-rate may influence calculated values of measured stiffness [42,43].
Finally, although well suited for nearly cylindrical samples, standard distension–extension tests are not sufficient for more complex arterial geometries, particularly those manifesting in disease.
Fortunately, new methods are emerging that enable local material properties to be inferred using full-field strain measurements and inverse methods for material characterization [44,45].
Although not discussed in detail here, there are also many reports of vascular function and properties based on either uniaxial loading tests (e.g., Ref. [46]) or ring myography (e.g., Ref. [25]).
Originally conceived to study isometric contraction, ring tests are performed by placing a short ring-like sample of the vascular wall on two mounting fixtures that are separated by a finite
distance. The sample thus stretches primarily in the circumferential direction as the fixtures are separated and the sample deforms from a circular to an oval to a more uniaxial geometry within the
central region of measurement. Measurement of the force acting on (passive) or generated by (active) the sample thus allows one to estimate 1D stress-stretch information or to construct dose-response
curves while holding a sample at a fixed separation distance (isometric). This approach is particularly valuable for high-throughput comparisons of different drugs or their doses, but the associated
contraction of the smooth muscle is under nonphysiological loading [47]. Inference of passive or active stiffness from these ring tests is further compromised by the lack of an exact solution to the
full boundary value problem (finite bending and uniaxial extension of an annulus) and the lack of biaxial loading [12,34]. In particular, both the radial and axial stretches reduce below one as
circumferential stretch increases with the separation of the mounting fixtures. Axial stretch is typically much greater than unity in vivo and is important to both the mechanics and the
mechanobiology [48]. Finally, in-plane biaxial tests can be performed on excised samples (e.g., Ref. [49]) and are generally very informative for they can largely mimic the in vivo state of stress,
with the exception of radial compression due to a distending blood pressure. The primary caveat with in-plane biaxial tests is that large specimens can retain some of their natural cylindrical
curvature in the unloaded state, thus requiring combined bending and extension to load the sample biaxially. Uniaxial tests performed on different samples with different gross orientations can avoid
some issues of residual curvature, but associated protocols are limited and cannot explore directly the inherent coupling that is important in dictating the multiaxial mechanical behavior.
Pulse Wave Velocity and Distensibility.
Whereas AFM, MTC, biaxial testing, ring myography, and allied methods are employed in vitro on excised samples, which generally enables significant experimental control, clinical studies necessarily
require less invasive in vivo methods, measurements, and metrics. As noted earlier, the current clinical gold standard metric of arterial stiffness is the carotid-to-femoral pulse wave velocity,
which is a measured quantity that represents a structural response that is integrated over a particular vascular path. There are, in addition, more local metrics of structural stiffness that are used
clinically. One such metric is the so-called distensibility
is the luminal diameter, with
denoting values at systole and diastole, respectively. The parameter
=1 or 2, yielding values of
that differ numerically by approximately a factor of 2. Taking
implies using cross-sectional area rather than diameter, which potentially avoids problems with low spatial imaging resolution. Note further that the Bramwell–Hill form for pulse wave velocity can be
written as
, hence localizing the value of
. Another local measure of structural stiffness (the pressure–strain or Peterson modulus) can be computed as
Clinical findings confirm that local measures such as
can correlate well with the more global
], hence suggesting that these metrics are complementary. We end this overview by noting that
, and
all intrinsically depend on the operating point (i.e., blood pressure) about which they are measured/calculated [
]. In a research setting, such dependence can sometimes be controlled post hoc using regression methods, but this is less feasible in clinical practice. Regression may be especially problematic in
studies of hypertension where (a change in) blood pressure may have both direct (through nonlinear mechanics) and indirect (causing arterial remodeling) effects on the measured metric of stiffness, a
distinction that cannot be made with regression analyses [
]. This issue motivated considerable research for “pressure-corrected” metrics of stiffness. As should be clear from the preceding text, however, arterial mechanics is highly nonlinear and fully
capturing it requires complex constitutive relations with many parameters, with value(s) of stiffness always dependent on the level of stress. Nevertheless, simple metrics have been proposed that,
within individual limitations, work surprisingly well [
]. Examples include the cardio-ankle vascular indices (
), which are metrics of global structural stiffness that essentially correct a measured foot-to-foot
for its pressure dependence via another metric of local structural stiffness (
), namely
$CAVI=lnPsysPdiasPWV2·2ρPsys−Pdias, CAVI0=PWV2·2ρPdias−lnPdiasPref$
$β=lnPsysPdiasddiasdsys−ddias, and β0=β−lnPdiasPref$
being an arbitrary but constant reference value of pressure [
]. These so-called pressure-corrected metrics are motivated by the observation that blood pressure relates approximately exponentially with diameter [
represent modifications to the original metrics (
) to correspond better with an actual exponential relationship; additionally,
to scale with diastolic instead of mean blood pressure, which seems more in line with experimental observations on the pressure dependency of
Of course, $PWV$ can also be estimated using computational methods given much more information: the spatially distributed geometry, material properties, and boundary conditions. Such calculations
avoid implicit assumptions inherent in the Moens–Korteweg and related equations and they enable informative parametric studies, as, for example, how vascular taper or distal resistances affect the
pulse wave. Unsteady 1D and three-dimensional models have been used to compute $PWV$ [57,58], but again additional intuitive insight can be gleaned from analytical solutions. Toward this end, the
theory of small deformations superimposed on large has also been used to obtain analytical results for a simple cylindrical geometry [59]. This solution shows explicitly that both the diastolic
distension and the axial prestretch are important contributors to the computed values of $PWV$; in the limit as the strains become small, this relation recovers the Moens–Korteweg equation. As in
the case of the AFM, however, this analysis shows clearly that complexities due to nonlinear material behaviors and finite deformations play important roles in determining the precise value of
stiffness that is computed.
Fung and colleagues used nonlinear regression to determine best-fit values of the seven material parameters in an exponential stored energy function $W$ (Eqs. (A1) and (A2) in Appendix B) from both
uniaxial (radial compression [60]) and biaxial (pressure-diameter-axial force-stretch [13]) data for a rabbit thoracic artery. Not surprisingly, the parameter values differed for the predominantly
compressive versus tensile behaviors. Figure 1 compares these results together; note the anisotropy and strong nonlinearity in the tensile (circumferential and axial) behaviors and the asymmetry
between the tensile and compressive behaviors even at modest strains. Results are not shown for greater compressive strains since the original tests used modest levels of compression and the
associated analytical solution reveals potential bifurcations in the equilibrium solutions at higher compressive strains. Importantly, given that the slope of the stress-stretch curves reflects (but
does not define) the stretch-dependent spatial material stiffness, note the expected tremendous differences between the low values of material stiffness near the unloaded configuration (low stress)
and high values near the in vivo configuration (physiologic stress). Figure 2 shows calculated circumferential behaviors for different degrees of fixed axial stretch from one to the in vivo value
(1.691 for this particular rabbit artery). This result reveals the strong biaxial coupling, with circumferential stress and stiffness affected dramatically by the value of axial stretch. The
aforementioned ring myography and uniaxial tests disregard such coupling and thus can underestimate the actual stiffness dramatically.
Figure 3 shows calculated results for a simulated AFM indentation test with the material modeled with the same Fung-exponential type of constitutive relation and the indentation performed at
different levels of fixed equibiaxial in-plane stretch. Since the analytical small-on-large solution holds for isotropic material behaviors [21], we first estimated new values of the material
parameters in the Fung-exponential that yield an isotropic response similar to the mean of the anisotropic response shown in Fig. 1. Again, it can be seen that there is a strongly coupled response
between different directions of loading, here out-of-plane versus in-plane. Most published works on arterial wall and vascular cell stiffness disregard this coupling effect and use AFM to test
samples, in the absence of a pre-existing in-plane stress, to compute Young's modulus that strictly holds only for small strains. Such a situation is not physiological and again is expected to
underestimate the actual in vivo stiffness dramatically.
To emphasize further both the multiaxial nature of stress and stiffness, and couplings therein, we show three-dimensional plots in Fig. 4 for simple cases of combined circumferential and axial
extensions of Fung elastic materials. That is, assuming principal homogeneous deformations and incompressibility, the stored energy function and thus associated wall stress and stiffness depend on
the in-plane principal stretches alone. The primary in-plane components of the mean Cauchy stress can thus be thought of conceptually in terms of components of the left stretch tensor $V=FFT$,
namely $σθθ=σ̂θθ(λθ,λz)$ and $σzz=σ̂zz(λθ,λz)$. As it can be seen in the figure, for both isotropic (left columns; cf. Fig. 3) and anisotropic (right columns; cf. Figs. 1 and 2) behaviors,
circumferential and axial deformations contribute similarly to the overall elastic energy storage and thus stress and material stiffness, here in a spatial rather than referential description since
spatial quantities (defined relative to the current configuration) are most relevant in vivo.
It is now widely accepted that both cell- and matrix-level stiffness are fundamental to mechanobiological responses within the vasculature, including modulation of cell phenotype. Similarly, it is
widely accepted that tissue-level stiffness is fundamental to the hemodynamics, particularly propagation of the arterial pressure wave that dictates many aspects of end organ health or disease. There
is, therefore, an appropriately growing literature on arterial stiffness. Yet, the many different reported metrics have different meanings because of different types of loads that are imposed
relative to different biomechanical states. We summarize in Table 1 some of these methods and metrics and, for purposes of illustration, we list in Table 2 some values of stiffness resulting from
these different methods. As it can be seen, reported values of stiffness (some compressive, some shear, most tensile) differ by orders of magnitude, as should be expected for highly nonlinear
material behaviors when assessed relative to different configurations, ranging from otherwise unloaded to in vivo relevant. Assuming that the associated calculations were performed correctly, each of
these different values of stiffness should be viewed as reliable. The key question, which we do not attempt to answer here, is therefore: How can we extract from these disparate metrics, having
different meanings, a unified understanding of vascular cell mechanobiology and biomechanics and their roles in dictating vascular health or disease progression? We submit that focused effort should
be directed toward answering this question, for without such our ability to use basic science findings to inform clinical decisions will remain largely wanting since understanding will remain largely
fragmented and incomplete.
Table 1
Modality Application Configuration for stiffness measurement Vascular axes of stiffness measurement Mode of stiffness measurement Loading rate Typical output metric (s)
Atomic force microscopy In vitro Unloaded^1 Radial or axial Compressive Quasi-static E (M)
Wire myography In vitro Uniaxially loaded Circumferential Tensile Quasi-static^2a E (M), C (S)
Pressure myography In vitro Uniaxially or biaxially loaded Circumferential and axial Tensile Quasi-static^2b E (M), W (M), C (S)
Biomechanical biaxial testing In vitro Biaxially loaded Circumferential and axial Tensile Quasi-static W (M), C (S)
Surface transit time PWV In vivo Loaded — Tensile Dynamic PWV (S)
Magnetic Resonance Imaging In vivo Loaded Circumferential Tensile Dynamic C (S), D (S), PWV (S)^3^,^4
Ultrasound echotracking In vivo Loaded Circumferential Tensile Dynamic E (M), C (S),D (S), PWV (S)^3
Modality Application Configuration for stiffness measurement Vascular axes of stiffness measurement Mode of stiffness measurement Loading rate Typical output metric (s)
Atomic force microscopy In vitro Unloaded^1 Radial or axial Compressive Quasi-static E (M)
Wire myography In vitro Uniaxially loaded Circumferential Tensile Quasi-static^2a E (M), C (S)
Pressure myography In vitro Uniaxially or biaxially loaded Circumferential and axial Tensile Quasi-static^2b E (M), W (M), C (S)
Biomechanical biaxial testing In vitro Biaxially loaded Circumferential and axial Tensile Quasi-static W (M), C (S)
Surface transit time PWV In vivo Loaded — Tensile Dynamic PWV (S)
Magnetic Resonance Imaging In vivo Loaded Circumferential Tensile Dynamic C (S), D (S), PWV (S)^3^,^4
Ultrasound echotracking In vivo Loaded Circumferential Tensile Dynamic E (M), C (S),D (S), PWV (S)^3
C, compliance coefficient; $D$, distensibility; E, Young's modulus; W, strain energy; PWV, pulse wave velocity. M, material stiffness metric; S, structural stiffness metric.
Typically unloaded, but has been performed on tissue maintained in a biaxially loaded state [61];
typically quasi-static, but has been performed under dynamic loading (^a[62], ^b[63]);
calculated using the Bramwell–Hill relationship;
measured by estimating transit time.
Table 2
Vessel/mouse Modality Definition/state Value, units, age Reference
Thoracic aorta, C57BL/6J E=3.1kPa (2
In Vitro, unloaded, cut open, radially indented from luminal side (endothelium months)
E=3.6kPa (6 months) AFM intact)/compressive [29]
(age 2–18 months) E=16.9kPa (12
E=21.8kPa (18 months)
Suprarenal abdominal aorta, C57/Sv129 AFM In Vitro, unloaded or pressurized to 100mmHg and elongated to In Vivo axial E=18.7kPa [61]
stretch, ring, axially indented/compressive (unloaded)
(age 10–13 months) E=12.3–76.4kPa (loaded;
bimodal distribution)
Aorta, C57BL/6J AFM In Vitro, unloaded, cut open, radially indented from luminal side (endothelium E=24kPa [28]
(age 11 months)
Ascending thoracic aorta, C57BL/6J E=2.8–12.7kPa (0.5
AFM In Vitro, unloaded, cut open, radially indented from luminal side (endothelium months)^1 [30]
(age 0.5–3.5 months) intact)/compressive E=5.0–38.8kPa (2
E=4.4-36.7kPa (3.5 months)^1
Ascending thoracic aorta, C57BL/6J Biaxial testing In Vitro, loaded, intact, pressurized to 128mmHg, elongated to In Vivo axial $Cθθθθ$=2.76MPa [64]
$Czzzz$=2.26MPa stretch/tensile
(age 15.2±0.1 weeks)
Suprarenal abdominal aorta, C57BL/6J In Vitro, loaded, intact, pressurized to 100mmHg, elongated to In Vivo axial ∂S[θθ]/∂E[θθ]=
Biaxial testing stretch/tensile 1.33MPa [65]
(age 5–6 months)
Carotid-to-femoral arterial bed, C57BL/6J Applanation tonometry In Vivo, 4–5% sevoflurane or 75mg/kg sodium pentobarbital anesthesia, noninvasive PWV=3.96 m/s [66]
/PWV (sevoflurane)^2
(age 5.6±0.2 months) PWV=2.89 m/s (sodium
Abdominal aorta, C57BL/6 Ultrasound echotracking In Vivo, 125mg/kg tribromoethanol anesthesia, noninvasive/PWV PWV=2.70 m/s^3 [67]
(age 3–4 months)
Aorta (regionally dependent), C57BL/6J Ultrasound echotracking/pressure In Vivo, 1.5% PWV=5.2 m/s^4
Isoflurane anesthesia, noninvasive (ultrasound), catheter PWV=3.0 m/s^5 [68]
invasive (catheter)/PWV (age 3 months)
PWV=3.5 m/s^6
Vessel/mouse Modality Definition/state Value, units, age Reference
Thoracic aorta, C57BL/6J E=3.1kPa (2
In Vitro, unloaded, cut open, radially indented from luminal side (endothelium months)
E=3.6kPa (6 months) AFM intact)/compressive [29]
(age 2–18 months) E=16.9kPa (12
E=21.8kPa (18 months)
Suprarenal abdominal aorta, C57/Sv129 AFM In Vitro, unloaded or pressurized to 100mmHg and elongated to In Vivo axial E=18.7kPa [61]
stretch, ring, axially indented/compressive (unloaded)
(age 10–13 months) E=12.3–76.4kPa (loaded;
bimodal distribution)
Aorta, C57BL/6J AFM In Vitro, unloaded, cut open, radially indented from luminal side (endothelium E=24kPa [28]
(age 11 months)
Ascending thoracic aorta, C57BL/6J E=2.8–12.7kPa (0.5
AFM In Vitro, unloaded, cut open, radially indented from luminal side (endothelium months)^1 [30]
(age 0.5–3.5 months) intact)/compressive E=5.0–38.8kPa (2
E=4.4-36.7kPa (3.5 months)^1
Ascending thoracic aorta, C57BL/6J Biaxial testing In Vitro, loaded, intact, pressurized to 128mmHg, elongated to In Vivo axial $Cθθθθ$=2.76MPa [64]
$Czzzz$=2.26MPa stretch/tensile
(age 15.2±0.1 weeks)
Suprarenal abdominal aorta, C57BL/6J In Vitro, loaded, intact, pressurized to 100mmHg, elongated to In Vivo axial ∂S[θθ]/∂E[θθ]=
Biaxial testing stretch/tensile 1.33MPa [65]
(age 5–6 months)
Carotid-to-femoral arterial bed, C57BL/6J Applanation tonometry In Vivo, 4–5% sevoflurane or 75mg/kg sodium pentobarbital anesthesia, noninvasive PWV=3.96 m/s [66]
/PWV (sevoflurane)^2
(age 5.6±0.2 months) PWV=2.89 m/s (sodium
Abdominal aorta, C57BL/6 Ultrasound echotracking In Vivo, 125mg/kg tribromoethanol anesthesia, noninvasive/PWV PWV=2.70 m/s^3 [67]
(age 3–4 months)
Aorta (regionally dependent), C57BL/6J Ultrasound echotracking/pressure In Vivo, 1.5% PWV=5.2 m/s^4
Isoflurane anesthesia, noninvasive (ultrasound), catheter PWV=3.0 m/s^5 [68]
invasive (catheter)/PWV (age 3 months)
PWV=3.5 m/s^6
Computationally separated, numbers denote intimal/medial moduli. AFM, atomic force microscopy; PWV, pulse wave velocity; $Cθθθθ$ and $Czzzz$, linearized circumferential and axial spatial material
stiffness obtained using the theory of small-on-large. ∂S[θθ]/∂E[θθ] and ∂S[zz]/∂E[zz], referential material stiffness defined as the derivative of second Piola-Kirchhoff stress with respect to Green
carotid-to-femoral transit-time PWV;
PWV in the window of an ultrasound probe;
aortic arch-to-femoral bifurcation transit-time PWV (ultrasound);
abdominal transit-time PWV (blood pressure catheter, 2cm path length);
distensibility-based local abdominal PWV obtained from Bramwell–Hill equation. In studies where interventions were performed, control groups are displayed here.
In some ways, we have emphasized the obvious—the value of stiffness depends on its definition (material versus structural), the configuration to which it refers (referential-unloaded, versus
spatial-current), and the conditions under which it is evaluated (compression versus tension versus shear in an otherwise unloaded state or not). Yet, we are not aware of a prior consistent
discussion of methods used in vascular mechanics across scales from atomic force microscopy to biaxial tests on cylindrical segments to in vivo measurements. As we have noted, a number of metrics
reported in the literature are based on solutions from classical elasticity because of the associated simplicity, not the theoretical relevance. Strictly speaking, therefore, most of these results
are applicable only for linearly elastic isotropic responses under small strains and rotations when measured about an unloaded state—conditions not applicable to mechanobiologically, physiologically,
or clinically relevant situations. Although we noted two analytical examples wherein a material stiffness applicable to a nonlinear behavior can be compared directly in the limit to a small strain
Young's modulus [21,59], it is generally problematic and uninformative to compare findings based on Hertz, Moens–Korteweg, or similar equations with those based on nonlinear constitutive relations or
appropriate linearizations thereof, including the theory of small deformations superimposed on large. Yet, values of Young's modulus continue to be reported and, in some cases, results are similarly
presented based on concepts from linearized viscoelasticity, including storage and loss moduli. Of course, one could argue that such results can be insightful when one consistently compares values
across studies using the same methods and metrics whether they strictly hold theoretically or not in configurations relevant to in vivo conditions or not. For example, a compressive Young's modulus
inferred from AFM has been reported by multiple groups to increase for isolated smooth muscle cells in aging and hypertension relative to that in normalcy [25,69], hence these data are trying to tell
us something—the question is, What? That is, although results may be reproducible and reliable, the key question should focus on their possible relevance in vivo. Similar methods have been used to
show that the nuclear protein lamin-A scales with tissue stiffness [70], with tissues ranging from compliant (adipose or liver) to stiff (ligaments and bones). Again, the results are reproducible,
reliable, and provocative, though not evaluated at in vivo values of stiffness. How such results should be interpreted or compared to values that are relevant to the in vivo condition remains an open
We submit that in vivo relevant conditions and metrics should be used when possible. Toward this end, the theory of small deformations superimposed on large deformations [71] can serve as a
theoretically appropriate method to compute linearized values of stiffness while accounting for underlying nonlinear material behaviors and in vivo relevant finite deformations, with applicability
including interpretations of AFM data [21], defining biaxial material stiffness in intact excised vessels for simulations of hemodynamics [35], and computing $PWV$ either analytically [59] or
numerically [58]. With regard to the need to measure and report metrics that are relevant to the in vivo conditions, it would be prudent to remember the words of Y. C. Fung written ∼50years ago.
First, “The main difficulty [problem] lies in the customary use of infinitesimal theory of elasticity to the media which normally exhibit finite deformations” [10] and “the greatest need lies in the
direction of collecting data in multiaxial loading conditions and formulating a [constitutive] theory for the general rheological behavior of living tissues…” [72].
In conclusion, it is critical to quantify arterial stiffness—material and structural—because deviations from normal values associate with both the phenotypic modulation of vascular cells and the
clinical severity of disease or disease risk. That said, there is also a fundamental conceptual issue that must be considered carefully as we seek to advance our understanding of the underlying
mechanobiology. Although vascular cells clearly attempt to establish and then maintain certain mechanical quantities near homeostatic values [9,73], the continuum quantities of stress and strain, and
metrics such as material stiffness that are derived from them, are actually mathematical concepts, not physical realities [74]. Hence, even though it appears that mean wall stress and stiffness are
normally regulated near homeostatic targets across mammalian species [7,8,75], we should not expect a cell to necessarily respond to a stress (i.e., a linear transformation, or tensor, that
transforms an outward unit vector into a traction vector at a point). Rather, it is more likely that forces acting at the molecular level change the conformation of important biomolecules and thereby
stimulate cell signaling and downstream gene products. There is, therefore, a pressing need to understand better the micromechanics of mechanosensing by cells and the associated mechanoregulation of
matrix [76] and to associate such phenomena with convenient continuum metrics such as stress and stiffness. Nevertheless, until, and possibly after, such multiscale understanding is achieved, direct
correlations of mechanobiological and (patho)physiological responses with wall stress and stiffness should continue to be identified. Toward this end, an increased use of concepts of
nondimensionalization and allometric scaling [77] should also become a priority. There is, therefore, a need for continued development of new concepts and techniques in vascular mechanics and
mechanobiology and we conclude with words of Fung in his foreword to the inaugural issue of the journal Biomechanics and Modeling in Mechanobiology in 2002 [78]—“let us enjoy the work.”
JDH also gratefully acknowledges the Y.C. Fung Young Investigator Award from ASME that he received in 1990.
Funding Data
• US NIH (Funder ID: 10.13039/100000002; Grant Nos. R01 HL105297, P01 HL134605).
• Netherlands Organization for Scientific Research (NWO) (Funder ID: 10.13039/501100003246; Rubicon Grant 452172006).
Small on Large
The polymath A. Cauchy knew equations of nonlinear elasticity in the 1820s, but analytical solutions to this class of problems had to await the semi-inverse approach of R. Rivlin in the late 1940s.
Because of the inherent complexities, analytical solutions to problems in nonlinear elasticity were yet possible for only a relatively small class of motions [79] and many employed the advances in
finite element methods that soon arrived [80]. It is often difficult to develop intuition from numerical solutions of highly nonlinear problems, however, thus the concept of “small deformations
superimposed on large deformations” became useful in extending the range of possible analytical solutions. As noted herein, small-on-large solutions have been found useful in interpreting
experimental results associated with atomic force microscopy [21], quantifying mechanical properties from biaxial tests on excised arteries [35], and relating the structural stiffness that can be
inferred in vivo from measurements of pulse wave velocity to the material stiffness inferred from in vitro tests [59]. It seems appropriate, therefore, to briefly outline steps of this approach; the
interested reader is referred elsewhere for more details [71].
Briefly, let the location of a material particle in an original configuration be denoted by $X$ and in a finitely deformed configuration by $x$. In addition, let the location of this particle in a
configuration that is close to the finitely deformed one be denoted by $y$. The total deformation gradient is thus $F=(∂y/∂x)(∂x/∂X)$, which can be written as $F=FsFo$, where subscript $o$ denotes
the original finite deformation and subscript $s$ denotes the superimposed small deformation. It proves convenient to write $y=x+u$, where u is a small displacement vector. Hence, $Fs=∂x/∂x+∂u/∂x$
which we write as $Fs=I+h$, where $h$ is a displacement gradient with $h=tr(hhT)≪1$. The right Cauchy–Green tensor can then be written as $C=FTF=[I+hFo]TI+hFo=
FoTFo+FoThTFo+FoThFo+FoThThFo≅Co+FoThT+hFo=Co+FoT2εFo$ if we neglect terms that are higher order in the (small) displacement gradient and we recognize the small strain tensor $ε=(h+hT)/2$. Hence,
the right Cauchy–Green tensor in the final configuration simply equals that in an intermediate (finitely deformed) configuration, denoted by subscript o, plus a part that is linear in the small
displacement gradient.
Just as the total deformation gradient $F=I+hFo= Fo+hFo$ consists of a contribution that is due to a finite deformation to an intermediate configuration plus an addition, one can also assume that the
total second Piola-Kirchhoff stress can be computed as $S=So+S*$, where the first contribution is associated with the original finite deformation and the second with a superimposed small
deformation, namely $S*≅(∂S/∂C)o:C*$ with $S=∂W/∂Co$ and $C*=C−Co=FoT2εFo$ from above. Importantly, it can be shown that we can write the total Cauchy stress $t$, for an incompressible response and
to order $h$, as [35] $t=to+ht̂o+t̂ohT−p*I+FoŜ*FoT$, where $t̂o$ is the extra (i.e., deformation-dependent) part of the Cauchy stress in the finitely deformed intermediate configuration and $p*$ is a
Lagrange multiplier that enforces isochoric motions during the small superimposed deformation. Finally, using the result above for the second Piola-Kirchhoff stress in terms of the stored energy
function $W$, and recognizing that a small displacement gradient can be written in terms of the infinitesimal strain and rotation (i.e., $h=h+hT/2+(h−hT)/2$, or, $h=ε+Ω$), we arrive at our final
result: $t+p*I=to+Cε+DΩ$, where the spatial material stiffness (fourth order) tensor $C$ can be written for an artery as given in component form in the third equation in the main text. Noting that
the infinitesimal rotation tensor $Ω$ vanishes with assumed principal deformations, this derivation reveals clearly that the spatial material stiffness associated with a superimposed small
deformation depends directly and strongly on the prestress (or prestretch) and the nonlinear properties represented by $W$, which are amplified nonlinearly by the initial finite deformation when
evaluated in the current (spatial) configuration [35].
Fung Elasticity
The illustrative solutions in Figs.
were obtained using the same Fung exponential strain energy function to facilitate comparisons across methods. This function is defined by [
where, in terms of principal Green strains,
) being material parameters that need to be determined from nonlinear regressions of data. For principal deformations, as of interest herein, the Green strains
/2 for (no sum on)
Uniaxial Compression.
Uniaxial radial compression testing comparable to that reported by Chuong and Fung [
] was simulated by prescribing radial stretches
from 1 to 0.85 (Fig.
, negative abscissa). Incompressibility was assumed, and circumferential and axial directions were assumed to be traction-free consistent with the reported experiment, though it would have been
better, albeit difficult, to have induced a finite in-plane deformation prior to the radial compression. Nonetheless, given the assumption of a homogeneous deformation, the in-plane Cauchy stresses
in the actual experiment were
. Constitutively, Cauchy stress was calculated from
$tii=−p+λi2∂W∂Eii,with no sum oni=θ,z,r or Θ,Z,R$
being a Lagrange multiplier enforcing incompressibility. From the traction-free conditions and homogeneous state of deformation, the Lagrange multiplier
for compression tests (
can be determined from Eq.
using either
$pcomp=λθ2∂W∂EΘΘ or pcomp=λz2∂W∂EZZ$
after which either
was determined as
determined iteratively to satisfy
or vice versa for the axial stretch. Numerical simulations revealed equivalent outcomes using these two constraints.
Simulations of the compression experiment were then performed using reported values of the constitutive parameters for a rabbit thoracic artery (sample 11, incompressible case) given in Ref. [60]: $c
=43.12 kPa$, $b1=0.8230$, $b2=0.9125$, $b3=1×10−7$, $b4=1.1237$, $b5=0.4125$, and $b6=0.3768$.
Biaxial Stretching.
Equibiaxial in-plane stretch testing similar to that performed by Vande Geest et al. [
] was simulated by prescribing
from 1.0 to 1.8 (Fig.
, positive abscissa), with
. Cauchy stress was then obtained using Eq.
, with
) determined from
, because of the traction-free condition on the top and bottom surfaces and the assumed homogeneous deformation, as
Simulations of tensile loading were performed using values of the constitutive parameters, again for a rabbit thoracic artery [13]: $c=22.40 kPa$,$b1=1.0672$, $b2=0.4775$, $b3=0.0499$, $b4=
0.0903$, $b5=0.0585$, and $b6=0.0042$. Note the differences between the values for compression and tension despite the same functional form of $W$ and the same sample tested by the same
Additional in-plane biaxial tension experiments (tension in
directions), with
, were simulated with the same tensile constitutive parameters (
) that were used in the prior section. Such loading simulates possible in-plane biaxial testing [
] but also mimics mean values of stress in possible distension–extension tests [
]. Indeed, corresponding in vivo loading states were determined by simulating pressure-diameter testing as follows. Unloaded inner and outer radii were
, with
$li=8.75 mm$
$lo=12.5 mm$
the inner and outer unloaded circumferences given in Ref. [
], with unloaded thickness
. Loaded inner radius
was varied iteratively and loaded outer radius
was computed as
with loaded thickness
. The mean circumferential stretch
is then
Prescribing axial stretch
allows the radial stretch
to be determined from incompressibility (
). Components of Cauchy stress again follow from Eq.
from Eq.
. Luminal distending pressure
is now obtained from the mean value of Cauchy stress and geometry as
Increasing values of $ri$ thus yield desired increases in pressure (e.g., up to 120mmHg).
Simulations of Atomic Force Microscopy for Fung Elasticity
The effect of in-plane prestretch on out-of-plane indentation testing, as in AFM, was simulated using the solution in Ref. [
]. For these derivations to hold, the in-plane stretch must be equibiaxial and the material must be isotropic. For consistency with our other simulations (Appendix
), we again used a Fung exponential strain energy function, though one that is parameterized isotropically, namely
$b1=b2=b3 and b4=b5=b6$
To obtain parameter values that model tension similar to the mean anisotropic response shown in Fig.
, we simulated equibiaxial testing of an isotropic Fung elastic material (described by unique parameters
, and
), and fitted these three parameters to the mean of the anisotropic case. Specifically, we calculated the homogeneous (mean) equibiaxial Cauchy stresses
, and minimized
being the anisotropic Cauchy stresses shown in Fig.
(positive abscissa), with
a data point index and
the total number of data points (i.e., equilibrium configurations assessed during testing). This procedure yielded
$c=26.97 kPa$
, and
, here for a “virtual” rabbit thoracic artery.
To use the formulation in Ref. [
], we reformulated the isotropic form of
in terms of invariants of the Green strain tensor
, namely
, and then wrote
in terms of invariants of the right Cauchy–Green tensor
. Toward this end, we used the full formulation for
] where
which for isotropy (Eq.
), with
, can be written as
or more compactly
is a symmetric tensor. Note, too, that the two (exponential) material parameters for isotropy are
). Next, recognize that the right Cauchy–Green tensor
, with invariants
Re-arranging Eqs.
and inserting into Eq.
Given $W(Q)$ written in this form, the equations in Ref. [21], written as functions of $∂W/∂IC$ and $∂W/∂IIC$, can be used directly to simulate AFM indentations for an isotropic Fung elastic
material, similar to the calculation discussed in the text for the simpler neo-Hookean material.
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If cp and cd are semi conjugate diameters of an ellipse (x^-Turito
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If CP and CD are semi – conjugate diameters of an ellipse ^2 + CD^2 =
A. a + b
B. a^2 + b^2
C. a^2 – b^2
Properties of conjugate diameters
i)Sum of squares of any two conjugate semi diameters of ellipse is equal to sum of square of semi axes of ellipse.
ii) Square of semi diameter which is conjugate to the diameter through a point is equal to the product of focal distances of a point on an ellipse.
The correct answer is: a^2 + b^2
Given : CP and CD are semi – conjugate diameters of an ellipse
Let coordinates of P(a cos
Coordinates of D( a cos
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Discovering nonlinear resonances through physics-informed machine learning
For an ensemble of nonlinear systems that model, for instance, molecules or photonic systems, we propose a method that finds efficiently the configuration that has prescribed transfer properties.
Specifically, we use physics-informed machine learning (PIML) techniques to find the parameters for the efficient transfer of an electron (or photon) to a targeted state in a nonlinear dimer. We
create a machine learning model containing two variables, χ[D] and χ[A, ]representing the nonlinear terms in the donor and acceptor target system states. We then introduce a data-free
physics-informed loss function as 1.0 − P[j], where P[j] is the probability, the electron being in the targeted state, j. By minimizing the loss function, we maximize the occupation probability to
the targeted state. The method recovers known results in the targeted energy transfer (TET) model, and it is then applied to a more complex system with an additional intermediate state. In this
trimer configuration, the PIML approach discovers desired resonant paths from the donor to acceptor units. The proposed PIML method is general and may be used in the chemical design of molecular
complexes or engineering design of quantum or photonic systems.
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