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Conn-Rod Mechanism
The connecting rod mechanism.
A connecting rod - conn-rod - mechanism converts rotating motion to reciprocating motion - or vice versa.
The position of the rod can be expressed as
s = r (1 - cos φ) + (λ / 2) r sin^2φ (1)
s = position of the rod (m)
r = radius of crank (m)
φ = ω t
= 2 π n[s] t
= angular position of crank (rad)
ω = crank angular velocity (rad/s)
t = time (s)
n[s] = revolution per second (1/s)
λ = r / l
= crank ratio
l = length of rod (m)
The velocity of the rod can be expressed as
v = ω r sin φ (1 + λ cos φ) (2)
v = velocity of rod (m/s)
The acceleration of the rod can be expressed as
a = ω^2 r (cos φ + λ cos 2φ) (3)
a = acceleration of rod (m/s^2)
Related Topics
• The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
Related Documents | {"url":"https://engineeringtoolbox.com/connecting-rod-mechanism-d_1936.html","timestamp":"2024-11-03T21:44:28Z","content_type":"text/html","content_length":"28788","record_id":"<urn:uuid:d5cea238-9366-4b13-9b50-832dc0a90248>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00356.warc.gz"} |
Shannon’s Information Theory
I never read original papers of the greatest scientists, but I got so intrigued by the information theory that I gave Claude Shannon’s seminal paper a read. This paper is mind-blowing! In this single
paper, Shannon introduced this new fundamental theory. He raised the right questions, which no one else even thought of asking. This would have been enough to make this contribution earthshaking. But
amazingly enough, Shannon also provided most of the right answers with class and elegance. In comparison, it took decades for a dozen of top physicists to define the basics of quantum theory.
Meanwhile, Shannon constructed something equivalent, all by himself, in a single paper.
Shannon’s theory has since transformed the world like no other ever had, from information technologies to telecommunications, from theoretical physics to economical globalization, from everyday life
to philosophy. But instead of taking my words for it, listen to Jim Al-Khalili on BBC Horizon:
I don’t think Shannon has had the credits he deserves. He should be right up there, near Darwin and Einstein, among the few greatest scientists mankind has ever had the chance to have. Yet, he’s
hardly known by the public… More than an explanation of Shannon’s ideas, this article is a tribute to him.
Now, to understand how important his ideas are, let’s go back in time and consider telecommunications in the 1940s. Back then, the telephone network was quickly developing, both in North America and
Europe. The two networks then got connected. But when a message was sent through the Atlantic Ocean, it couldn’t be read at the other end.
Why? What happened?
As the message travelled through the Atlantic Ocean, it got weakened and weakened. Eventually, it was so weak that it was unreadable. Imagine the message was the logo of Science4All. The following
figure displays what happened:
Why not amplifying the message along the way?
This was what was proposed by engineers. However, this led them to face the actual problem of communication.
Which is?
The unpredictable perturbation of the message! This perturbation is called noise. This noise is precisely what prevents a message from getting through.
I don’t see the link between amplification and noise…
When you’re amplifying the message, you’re also amplifying the noise. Thus, even though the noise is small, as you amplify the message over and over, the noise eventually gets bigger than the
message. And if the noise is bigger than the message, then the message cannot be read. This is displayed below:
At that time, it seemed to be impossible to get rid of the noise. There really seemed to be this fundamental limit to communication over long distances. No matter when or how you amplify the message,
the noise will still be much bigger than the message once it arrives in Europe. But then came Claude Shannon…
What did Shannon do?
Wonders! Among these wonders was an amazingly simple solution to communication. This idea comes from the observation that all messages can be converted into binary digits, better known as bits. For
instance, using the PNG format, the logo of Science4All can be digitized into bits as follows:
Bits are not to be confused for bytes. A byte equals 8 bits. Thus, 1,000 bytes equal 8,000 bits.
This digitization of messages has revolutionized our world in a way that we too often forget to be fascinated by.
What do bits change to the communication problem?
Now, instead of simply amplifying the message, we can read it before. Because the digitized message is a sequel of 0s and 1s, it can be read and repeated exactly. By replacing simple amplifiers by
readers and amplifiers (known as regenerative repeaters), we can now easily get messages through the Atlantic Ocean. And all over the world, as displayed below:
This figure is just a representation. The noise rather occurs on the bits. It sort of make bits take values around 0 and 1. The reader then considers that values like 0.1 equal 0, and repeats and
amplify 0 instead of 0.1.
Now, in the first page of his article, Shannon clearly says that the idea of bits is J. W. Tukey’s. But, in a sense, this digitization is just an approximation of Shannon’s more fundamental concept
of bits. This more fundamental concept of bits is the quantification of information, and is sometimes referred to as Shannon’s Bits.
Shannon’s Bits
Obviously, the most important concept of Shannon’s information theory is information. Although we all seem to have an idea of what information is, it’s nearly impossible to define it clearly. And,
surely enough, the definition given by Shannon seems to come out of nowhere. But it works fantastically.
What’s the definition?
According to Shannon’s brilliant theory, the concept of information strongly depends on the context. For instance, my full first name is Lê Nguyên. But in western countries, people simply call me Lê.
Meanwhile, in Vietnam, people rather use my full first name. Somehow, the word Lê is not enough to identify me in Vietnam, as it’s a common name over there. In other words, the word Lê has less
information in Vietnam than in western countries. Similarly, if you talk about “the man with hair”, you are not giving away a lot of information, unless you are surrounded by soldiers who nearly all
have their hair cut.
But what is a context in mathematical terms?
A context corresponds to what messages you expect. More precisely, the context is defined by the probability of the messages. In our example, the probability of calling someone Lê in western
countries is much less likely than in Vietnam. Thus, the context of messages in Vietnam strongly differs from the context of western countries.
OK… So now, what’s information?
Well, we said that the information of Lê is greater in western countries…
So the rarer the message, the more information it has?
Yes! If $p$ is the probability of the message, then its information is related to $1/p$. But this is not how Shannon quantified it, as this quantification would not have nice properties. Shannon’s
great idea was to define information rather as the number of bits required to write the number $1/p$. This number is its logarithm in base 2, which we denote $\log_2(1/p)$.
If you’re uncomfortable with logarithms, read
my article on these mathematical operators
. You don’t need a full understanding of logarithms to read through the rest of the articles though. If you do know about logarithms, you have certainly noticed that, more often than not, Shannon’s
number of bits is not a whole number.
Now, this means that it would require more bits to digitize the word Lê in western countries than in Vietnam, as displayed below:
Why did Shannon use the logarithm?
Because of its nice properties. First, the logarithm enables to bring enormous numbers $1/p$ to more reasonable ones. But mainly, if you consider a half of a text, it is common to say that it has
half the information of the text in its whole. This sentence can only be true if we quantify information as the logarithm of $1/p$. This is due to the property of logarithm to transform
multiplication (which appears in probabilistic reasonings) into addition (which we actually use). Now, this logarithm doesn’t need to be in base 2, but for digitization and interpretation, it is very
useful to do so.
Well, if I read only half of a text, it may contain most of the information of the text rather than the half of it…
This is an awesome remark! Indeed, if the fraction of the text you read is its abstract, then you already kind of know what the information the whole text has. Similarly, Lê Nguyên, even in Vietnam,
doesn’t have twice the information that Lê has.
Does Shannon’s quantification account for that?
It does! And the reason it does is because the first fraction of the message modifies the context of the rest of the message. In other words, the conditional probability of the rest of the message is
sensitive to the first fraction of the message. This updating process leads to counter-intuitive results, but it is an extremely powerful one. Find out more with my article on conditional
Are there applications of this quantification of information?
The whole industry of new technologies and telecommunications! But let me first present you a more surprising application to the understanding of time perception explain in this TedED video by Matt
Danzico. Now that you know about Shannon’s information theory, you should have a new insight into what the video talks about!
Wow! Indeed! But can you explain how Shannon’s theory is applied to telecommuncations?
Yes! As Shannon put it in his seminal paper, telecommunication cannot be thought in terms of information of a particular message. Indeed, a communication device has to be able to work with any
information of the context. This has led Shannon to (re)-define the fundamental concept of entropy, which talks about information of a context.
There’s a funny story about the coining of the term coining of the term “entropy”, which Shannon first wanted to call “uncertainty function”. But John von Neumann gave him the following advice:
You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place,
and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.
Shannon’s Entropy
In 1877, Ludwig Boltzmann shook the world of physics by defining the entropy of gases, which greatly confirmed the atomic theory. He defined the entropy more or less as the logarithm of the number of
microstates which correspond to a macrostate. For instance, a macrostate would say that a set of particles has a certain volume, pressure, mass and temperature. Meanwhile, a microstate defines the
position and velocity of every particle.
The brilliance of Shannon was to focus on the essence of Boltzmann’s idea and to provide the broader framework in which to define entropy.
What’s Shannon’s definition of entropy?
Shannon’s entropy is defined for a context and equals the average amount of information provided by messages of the context. Since each message is given with probability $p$ and has information
$log_2(1/p)$, the average amount of information is the sum for all messages of $p \log_2(1/p)$. This is explained in the following figure, where each color stands for a possible message of the
In the case of a continuous probability with a density function $f$, the entropy can be defined as the integral of $f \log_2(1/f)$. Although it loses a bit of its meaning, it still provides a
powerful understanding of information.
I don’t see the link with Boltzmann’s entropy…
In Boltzmann’s setting, all microstates are assumed equally likely. Thus, for all microstate, $1/p$ equals the number of microstates. The average amount of information is therefore the logarithm of
the number of microstates. This shows that Boltzmann’s entropy is nothing more than Shannon’s entropy applied to equiprobable microstates.
Shannon also proved that, given a certain number of states, the entropy of the distribution of states is maximized when all states are equally likely. As a result, when playing the price is right, if
you know that the price is somewhere 1,000$ and 2,000$, then guessing 1,500$ will be what’s providing you the most information in average.
But I’ve heard entropy had to do with disorder…
This is another important interpretation of entropy. For the average information to be high, the context must allow for a large number of unlikely events. Another way of phrasing this is to say that
there is a lot of uncertainties in the context. In other words, entropy is a measure of the spreading of a probability. This spreading is what’s often referred to as disorder. In some sense, the
second law of thermodynamics which states that entropy cannot decrease can be reinterpreted as the increasing impossibility of defining precise contexts on a macroscopic level.
I guess entropy is useful in physics… But in communication?
It is essential! The most important application probably regards data compression. Indeed, the entropy provides the theoretical limit to the average number of bits to code a message of a context. It
also gives an insight into how to do so. Data compression has been applied to image, audio or file compressing, and is now essential on the Web. Youtube videos can now be compressed enough to surf
all over the Internet! But before talking about communication, let’s dig in a major variant of entropy.
Shannon’s Equivocation
By considering a conditional probability, Shannon defined conditional entropy, also known as Shannon’s equivocations. Let’s consider the entropy of a message conditional to its introduction. For any
given introduction, the message can be described with a conditional probability. This defines a entropy conditional to the given introduction. Now, the conditional entropy is the average of this
entropy conditional to the given introduction, when this given introduction follows the probabilistic distribution of introductions. Roughly said, the conditional entropy is the average added
information of the message given its introduction.
It’s getting complicated…
I know! But if you manage to get your head around that, you’ll understand much of the greatest ideas of Shannon.
Does this definition even match common sense?
Yes! Common sense says that the added information of a message to its introduction should not be larger than the information of the message. This translates into saying that the conditional entropy
should be lower than the non-conditional entropy. This is a theorem proven by Shannon! In fact, he went further and quantified this sentence: The entropy of a message is the sum of the entropy of its
introduction and the entropy of the message conditional to its introduction!
I’m lost!
Fortunately, everything can be more easily understood on a figure. The amount of information of the introduction and the message can be drawn as circles. Because they are not independent, they have
some mutual information, which is the intersection of the circles. The conditional entropies correspond to what’s missing from the mutual information to retrieve the entire entropies:
I’m not sure I get it…
Let’see examples. On the left of the following figure is the entropies of two coins thrown independently. On the right is the case where only one coin is thrown, and where the blue corresponds to a
sensor which says which face the coin fell on. The sensor has two positions (heads or tails), but, now, all the information is mutual:
As you can see, in the second case, conditional entropies are nil. Indeed, once we know the result of the sensor, then the coin no longer provides any information. Thus, in average, the conditional
information of the coin is zero. In other words, the conditional entropy is nil.
Waw… This formalism really is powerful to talk about information!
It surely is! In fact, it’s so powerful that some of the weirdest phenomena of quantum mechanics like the mysterious entanglement might be explainable with a generalization of information theory
known as quantum information theory.
I don’t know much about quantum information theory, but I’d love to know more. If you can, please write an article on that topic!
Why are these concepts so important?
They’re essential to understand sequences of symbols. Indeed, if you try to encode a message by encoding each character individually, you will be consuming space to repeat mutual information. In
fact, as Shannon studied the English language, he noticed that the conditional entropy of a letter knowing the previous one is greatly decreased from its non-conditional entropy. Indeed, if a word
starts with a r, then it’s very likely (sure?) that the next letter will be a vowel.
So to optimally compress the information of a text, it’s not enough to encode each character separately like the Morse code does. The structure of information also lies in the concatenation into
longer texts. In fact, Shannon defined the entropy of each character as the limit of the entropy of messages of great size divided by the size.
To study this structure, it’s necessary to use the formalism of Markov chain. To keep it simple here, I won’t. But the role of Markov chains is so essential in plenty of fields that, if you can, you
should write about them!
As it turns out, the decrease of entropy when we consider concatenations of letters and words is a common feature of all human languages… and of dolphin languages too! This has led extraterrestrial
intelligence seekers to search for electromagnetic signals from outer spaces which share this common feature too, as explained in this brilliant video by Art of the Problem:
In some sense, researchers assimilate intelligence to the mere ability to decrease entropy. What an interesting thing to ponder upon!
Shannon’s Capacity
Let’s now talk about communication! A communication consists in a sending of symbols through a channel to some other end. Now, we usually consider that this channel can carry a limited amount of
information every second. Shannon calls this limit the capacity of the channel. It is measured in bits per second, although nowadays we rather use units like megabits per second (Mbit/s) or megabytes
per second (MB/s).
Why would channels have capacities?
The channel is usually using a physical measurable quantity to send a message. This can be the pressure of air in case of oral communication. For longer telecommunications, we use the electromagnetic
field. The message is then encoded by mixing it into a high frequency signal. The frequency of the signal is the limit, as using messages with higher frequencies would profoundly modify the
fundamental frequency of the signal. But don’t bother too much with these details. What’s of concern to us here is that a channel has a capacity.
Can you provide an example?
Sure. Imagine there was a gigantic network of telecommunication spread all over the world to exchange data, like texts and images. Let’s call it the Internet. How fast can we download images from the
servers of the Internet to our computers? Using the basic formatting called Bitmap or BMP, we can encode images pixels per pixels. The encoded images are then decomposed into a certain number of
bits. The average rate of transfer is then deduced from the average size of encoded images and the channel’s capacity:
In the example, using bitmap encoding, the images can be transfered at the rate of 5 images per second. In the webpage you are currently looking at, there are about a dozen images. This means that
more than 2 seconds would be required for the webpage to be downloaded on your computer. That’s not very fast…
Can’t we transfer images faster?
Yes, we can. The capacity cannot be exceed, but the encoding of images can be improved. Now, what Shannon proved is that we can come up with encodings such that the average size of the images nearly
maps Shannon’s entropy! With these nearly optimal encodings, an optimal rate of image file transfer can be reached, as displayed below:
This formula is called Shannon’s fundamental theorem of noiseless channels. It is basically a direct application of the concept of entropy.
Noiseless channels? What do you mean?
I mean that we have here assumed that the received data was identical to what’s sent! This is not the case in actual communication. As opposed to what we have discussed in the first section of this
article, even bits can be badly communicated.
Shannon’s Redundancy
In actual communication, it’s possible that 10% of the bits get wrong.
Does this mean that only 90% of the information gets through?
No! The problem is that we don’t know which are the bits which got wrong. In your case, the information that gets through is thus less than 90%.
So how did Shannon cope with noise?
His amazing insight was to consider that the received deformed message is still described by a probability, which is conditional to the sent message. This is where the language of equivocation or
conditional entropy is essential. In the noiseless case, given a sent message, the received message is certain. In other words, the conditional probability is reduced to a probability 1 that the
received message is the sent message. In Shannon’s powerful language, this all beautifully boils down to saying that the conditional entropy of the received message is nil. Or, even more precisely,
the mutual information equals both the entropies of the received and of the sent message. Just like the sensor detecting the coin in the above example.
What about the general case?
The relevant information received at the other end is the mutual information. This mutual information is precisely the entropy communicated by the channel. Shannon’s revolutionary theorem says that
we can provide the missing information by sending a correction message whose entropy is this conditional entropy of the sent message given the received message. This correction message is known as
Shannon’s redundancy.
This fundamental theorem is described in the following figure, where the word entropy can be replaced by average information:
I’m skipping through a bit of technical details here, as I just want to show you the main idea of redundancy. To be accurate, I should talk in terms of entropies per second with an optimal encoding.
Shannon proved that by adding redundancy with enough entropy, we could reconstruct the information perfectly almost surely (with a probability as close to 1 as possible). This idea is another of
Shannon’s earthshaking idea. Quite often, the redundant message is sent with the message, and guarantees that, almost surely, the message will be readable once received. It’s like having to read
articles again and again to finally retrieve its information.
So redundancy is basically repeating the message?
There are smarter ways to do so, as my students sometimes recall me by asking me to reexplain reasonings differently. Shannon worked on that later, and managed other remarkable breakthroughs.
Similarly to theorems I have mentioned above, Shannon’s theorem for noisy channels provides a limit to the minimum quantity of redundancy required to almost surely retrieve the message. In practice,
this limit is hard to reach though, as it depends on the probabilistic structure of the information.
Does Shannon theorem explain why the English language is so redundant?
Yes! Redundancy is essential in common languages, as we don’t actually catch most of what’s said. But, because English is so redundant, we can guess what’s missing from what we’ve heard. For
instance, whenever you hear I l*v* cake, you can easily fill the blank. What’s particularly surprising is that we actually do most of this reconstitution without even being aware of it! You don’t
believe me? Check the McGurk effect, explained here by Myles Power and Alex Dainis:
It wouldn’t surprise me to find out that languages are nearly optimized for oral communications in Shannon’s sense. Although there definitely are other factors coming in play, which have to explain,
for instance, why the French language is so more redundant than English…
Let’s Conclude
What I’ve presented here are just the few fundamental ideas of Shannon for messages with discrete probabilities. Claude Shannon then moves on generalizing these ideas to discuss communication using
actual electromagnetic signals, whose probabilities now have to be described using probabilistic density functions. Although this doesn’t affect the profound fundamental ideas of information and
communication, it does lead to a much more complex mathematical study. Once again, Shannon’s work is fantastic. But, instead of trusting me, you probably should rather listen to his colleagues who
have inherited his theory in this documentary by UCTV:
The documentary is awesome! You should watch it entirely!
Shannon did not only write the 1948 paper. In fact, the first major breakthrough he did was back when he was a Master’s student at MIT. His thesis is by far the most influential Master’s thesis of
all time, as it shows how exploiting boolean algebra could enable to produce machines that would compute anything. In other words, in his Master’s thesis, Shannon drew the blueprints of computers!
Shannon also made crucial progress in cryptography and artificial intelligence.
I can only invite you to go further and learn more. This is what’s commonly called open your mind. I’m going to conclude with this, but in Shannon’s language… Increase the entropy of your thoughts!
2. I’m confused by the claim that log (1/p) is the number if bits required to write 1/p. It seems to me rhus depends on the number of digits of 1/p. For example what if p = 1/pi – as p can be any
real number between 0 and 1 it could take an infinite number of bits – there are uncountably infinite real numbers between 0 and 1
1. Hi Jeff! Note that p is the probability of a message, not the message itself. So, if you want to find the most efficient way to write pi, the question you should ask is not what pi is, but
how often we mention it. It turns out that in maths, we do often mention pi, so we find a compact way to represent it: We call it “pi”, hence using only 2 letters (and even only one if we use
Greek letters!).
The decimal representation of pi is just another not-very-convenient way to refer to pi.
Regarding other real numbers, well, almost all of them have never been studied, so their probability of being mentioned is literally p=0, which corresponds to an encryption into log(1/p)=∞
bits. This is consistent with the intuition you’re referring to!
3. “… Wonders! Among these wonders was an amazingly simple solution to communication. This idea comes from the observation that all messages can be converted into binary digits, better known as
bits. … ”
Here you repeat yet again the false claim that the conversion of information into digital form was ‘invented’ by Claude Shannon when it had, of course, already been invented, in 1937, by English
engineer Alec Reeves working in Paris for Western Electric. Why do Americans, in particular, have so little respect for Reeves (who invented digital technology in practice) and perhaps rather to
much for Shannon who – belatedy – developed the relevant theory ?
1. Hi David! I have not read enough about Reeves to comment. All I can say is that Shannon’s explanations convinced many others that bits were the way to go. Having said that, I hope you’ll
forgive my ignorance and the many oversimplifications that allow for a better story-telling. I just want to get people excited about information theory.
PS: I’m not American btw… | {"url":"https://www.science4all.org/article/shannons-information-theory/","timestamp":"2024-11-03T22:17:39Z","content_type":"text/html","content_length":"94386","record_id":"<urn:uuid:318fba04-32dd-45b0-9a72-58527e84ec65>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00044.warc.gz"} |
How do you solve trigonometric equations by the quadratic formula? | Socratic
How do you solve trigonometric equations by the quadratic formula?
1 Answer
The quadratic formula can be useful in solving trigonometric (or other) kinds of equation. But strictly speaking, the quadratic formula is used only to solve quadratic equations.
Here is an example:
We can use the quadratic formula to solve
$2 {x}^{2} - 6 x + 3 = 0$.
We get $x = \frac{- \left(- 6\right) \pm \sqrt{{\left(- 6\right)}^{2} - 4 \left(2\right) \left(3\right)}}{2 \left(2\right)} = \frac{3 \pm \sqrt{3}}{2}$
Now what if we needed to solve: ${t}^{6} - 2 {t}^{3} - 2 = 0$?
This is not a quadratic equation.
But, if we substitute $x$ in place of ${t}^{3}$, we get ${x}^{2} - 2 x - 2 = 0$ Which we can solve by the quadratic formula, as above.
But we want $t$, not $x$. That's OK, we have gained this information:
${t}^{3} = \frac{3 \pm \sqrt{3}}{2}$
And (taking 3rd roots on both sides) gives us
$t = \sqrt[3]{\frac{3 \pm \sqrt{3}}{2}}$ . . (No $\pm$ is needed for odd roots)
Now, suppose we need to solve ${\sin}^{2} t - 2 \sin t - 2 = 0$
This is a trigonometric equation, not a quadratic equation. Or is it?
Can't we "turn it into" a quadratic by substituting? (Such equations are sometimes called "quadratic in form".)
Let $x = \sin t$, we get our old friend ${x}^{2} - 2 x - 2 = 0$.
So $x = \sin t = \frac{3 \pm \sqrt{3}}{2}$
Now we need to solve $\sin t = \frac{3 + \sqrt{3}}{2}$ . find $t$.
That's going to be a problem because $\frac{3 + \sqrt{3}}{2}$ is greater than 1.
Solve $\sin t = \frac{3 - \sqrt{3}}{2}$ .
In this case there is a solution, but it is not one of the special angles. Using tables or electronics, we can get the reference angle ${39.3}^{\circ}$ Or the radian angle (the real number) $0.68668$
In degrees the solutions are:
$t = {39.3}^{\circ} + {360}^{\circ} k$ for any integer $k$
$t = {140.7}^{\circ} + {360}^{\circ} k$ for any integer $k$
Impact of this question
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Hitmarkers Modlet
Hey everyone!
here's a modlet I made that add hitmarkers (cross in the center of the screen that dissapear quickly) on succesful shots similar to some famous fps game.
the mod will show White hitmarkers on body shot, Red on headshots and Green on allies/players shot.
(only works with ranged weapon of course)
here's a video showing it ingame with a little surprise in the end (which is for a later mod):
and a quick gif if you don't have time for this:
Haven't tested it thoroughly (and not on multiplayer) let me know if you have any problems.
Download link (A18) : Click Here !
I am sorry I don't see what your trying to show
haha I kinda expected that
the thing in the center appears everytime you hit successfully an enemy with a ranged shot.
it will turn red on headshot, white in the body and green if you're hitting another player.
basically it's an hit indicator that can be really useful when doing things like long-range archery/sniping.
I am sorry I don't see what your trying to show
Really? You cant see the plainly obvious, clear as day thing shown in the gif? The crosshairs turn red when he lands headshots, lets you know when u are landing your headshots.
So weird. This guy that smells like onion in soup form also just released a mod just like this!
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
haha I kinda expected that
the thing in the center appears everytime you hit successfully an enemy with a ranged shot.
it will turn red on headshot, white in the body and green if you're hitting another player.
basically it's an hit indicator that can be really useful when doing things like long-range archery/sniping.
Argh yep I see it now lol old eyes lol first vid I missed it the several times i watched but just then when ya used the sniper I saw the crosshair go red. But yeh than others it still looks white lol
- - - Updated - - -
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
Or asking like I did. :-D
well while it's white on the body it's red in the head.
but whether it's white or red this crosshair in the middle isn't in the game originally I just made it look similar to the one we already have.
well while it's white on the body it's red in the head.
but whether it's white or red this crosshair in the middle isn't in the game originally I just made it look similar to the one we already have.
Sweet thank you great idea and addition :-)
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
Wasnt being sarcastic, I see it clearly in the gif.
When I take my glasses off, ya. But I am pretty much blind then.
I updated the GIF with another one in higher resolution now it's really clear as day
I updated the GIF with another one in higher resolution now it's really clear as day
I see the red cross hairs lol
hey everyone!
Poor Vader:sorrow: Nice job on the modlet though, can't wait for the later one too.
Great modlet..but how can you tease us like that...Darth Vader...Stormtroopers....AAAAAAHHHHHHHHH. Interest piqued.
Quick update, fixed the last hitmarker appearing when leaving/entering a vehicle.
also made it trigger on Rocket Launcher.
probably won't update this one for a while now unless any other bugs get reported or maybe to add different appearance.
Is the Health Bar also your modlet? I'd like to give it a try.
Nope you can find it in here it's called Telric's Health Bar
So weird. This guy that smells like onion in soup form also just released a mod just like this!
Unable to open archive file: 7 Days to Die Dedicated
Sorry gonna need more informations here
• 1 month later...
nice Vader mod, I've been trying to put Jason into the game to no avail.
This topic is now archived and is closed to further replies. | {"url":"https://community.7daystodie.com/topic/15875-hitmarkers-modlet/","timestamp":"2024-11-03T10:20:11Z","content_type":"text/html","content_length":"293034","record_id":"<urn:uuid:09e6e93e-67e8-4d96-b21e-f7d518f9bb0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00278.warc.gz"} |
crystal symmetries
I’ve made a few small gui programs (MATLAB) for finding all unique angles for vectors (or plane normals) in both the cubic and tetragonal crystal spaces. I thought I’d share them here, as I finally
worked out a few kinks. Be aware that the little gui’s use the command window to output their results. Some of my other programs would output latex code so that they could be pasted directly into a
thesis for tables etc., but this one just uses the standard out (ala the command window in Matlab).
Essentially you enter the vectors for the two planes of interest, then hit calculate, and all the unique angle solutions for the crystal space you chose (one gui for each crystal space right now- no
interest in complicating it by combining at this time) gets output in the command window.
I populate some orientation matrices based upon the vectors you give. Then, from matrix calculations, using tetragonal and crystal symmetry operations, we determine all the symmetric orientations.
Then, a simple subspace() command gives us the angles of greatest rise (dihedral angle) between the symmetric orientations. The calculations for these operations can be found in my Thesis (if it ever
gets published), but can also be found in: V. Randle and O. Engler. Texture Analysis: Macrotexture, Microtexture & Orientation Mapping. CRC Press, 2000.
Please note- it’s your job to check if these are correct, I make no warranties about this stuff.
These are very simple, but hopefully they’ll help a bit for those working in cubic and tetragonal spaces.
Here’s the cubic symmetry angle calculator:
Vector Angle Calculator Cubic Symmetries
Here’s the tetragonal symmetry angle calculator:
Vector Angle Calculator Tetragonal Symmetries
I hope they work for you- please let me know if you have problems, if I have time I’ll try and help.
I'm a graduate student (PhD Candidate) at the University of Illinois at Urbana-Champaign.
I've studied and researched in two fields of Materials Science and Engineering (Polymers and Semiconductors). My interests are as diverse as my musical tastes and I usually have my hand in some crazy
project during my free time.
I'm available for consulting and have access to a world-renown materials research user-facility supported by the D.O.E. If you would like to know more, please contact me.
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Measurement of the B+/B0 production ratio in e+e- collisions at the ϒ (4S) resonance using B →J/ψ (ℓℓ)K decays at Belle
We measure the ratio of branching fractions for the ϒ(4S) decays to B+B- and B0B¯0 using B+→J/ψ(ℓℓ)K+ and B0→J/ψ(ℓℓ)K0 samples, where J/ψ(ℓℓ) stands for J/ψ→ℓ+ℓ- (ℓ=e or μ), with 711 fb-1 of data
collected at the ϒ(4S) resonance with the Belle detector. We find the decay rate ratio of ϒ(4S)→B+B- over ϒ(4S)→B0B¯0 to be 1.065±0.012±0.019±0.047, which is the most precise measurement to date. The
first and second uncertainties are statistical and systematic, respectively, and the third uncertainty is due to the assumption of isospin symmetry in B→J/ψ(ℓℓ)K.
Bibliographical note
Publisher Copyright:
© 2023 authors. Published by the American Physical Society.
ASJC Scopus subject areas
• Nuclear and High Energy Physics
Dive into the research topics of 'Measurement of the B+/B0 production ratio in e+e- collisions at the ϒ (4S) resonance using B →J/ψ (ℓℓ)K decays at Belle'. Together they form a unique fingerprint. | {"url":"https://pure.korea.ac.kr/en/publications/measurement-of-the-bb0-production-ratio-in-ee-collisions-at-the-%CF%92","timestamp":"2024-11-07T14:02:28Z","content_type":"text/html","content_length":"53873","record_id":"<urn:uuid:3913ff8b-1ce7-4693-9eec-6b4ce9a4cacb>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00821.warc.gz"} |
Universal bayes consistency in metric spaces
We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes consistent in all metric spaces admitting any such
learner, making it an “optimistically universal” Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, the k-NN classifier and its variants are
not generally universally Bayes consistent, except under additional structural assumptions, such as an inner product, a norm, finite dimension or a Besicovitch-type property. The metric spaces in
which universal Bayes consistency is possible are the “essentially separable” ones-a notion that we define, which is more general than standard separability. The existence of metric spaces that are
not essentially separable is widely believed to be independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal
Bayes-consistent learner for the given metric space. In particular, this yields the first impossibility result for universal Bayes consistency. Taken together, our results completely characterize
strong and weak universal Bayes consistency in metric spaces.
• Bayes consistency
• Classification
• Metric space
• Nearest neighbor
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Order of Magnitude Calculations
Next: Mathematical Notation Up: Introduction Previous: Conversion of Units
Idea: An order of magnitude calculation is an estimate to determine if a more precise calculation is necessary. We round off or guess at various inputs to obtain a result that is usually reliable to
within a factor of 10. Specifically, to get the order of magnitude of a given quantity, we round off to the closest power of 10 (example: 75 kg ^2 kg). | {"url":"http://theory.uwinnipeg.ca/physics/intro/node7.html","timestamp":"2024-11-12T23:47:10Z","content_type":"text/html","content_length":"3271","record_id":"<urn:uuid:8c16f1c1-31cd-43db-aba6-38db45f3be68>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00378.warc.gz"} |
Theta is an options Greek that measures the time sensitivity of an option i.e. an option’s time decay. It represents how the price of an option declines relative to time hence is denoted as a
negative number. Theta is not linear or constant; it increases as time to expiry reduces given that at expiry, an option no longer has any sensitivity to time.
Related terms | {"url":"https://ondemand.euromoney.com/discover/glossary/theta","timestamp":"2024-11-12T06:05:54Z","content_type":"text/html","content_length":"97885","record_id":"<urn:uuid:e8f02b96-4a49-4631-b1e1-16f0254664ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00258.warc.gz"} |
Home - Dot Product Calculator
Dot Product Calculator
Enter values to find the online calculator dot product of two vectors with the dot product calculator.
Define & calculating each vector
Dot product calculator calculates the dot product of two vectors a and b in Euclidean space. Enter i, j, and k for both vectors to get scalar number.
A Dot Product Calculator is a tool that computes the dot product (also known as scalar product or inner product) of two vectors in Euclidean space. The dot product is a scalar value that represents
the extent to which two vectors are aligned. It has numerous applications in geometry, physics, and engineering.
To use the dot product calculator, you need to enter the components (component) i, j, and k for both vectors, which are generally represented as a = (a1, a2, a3) and b = (b1, b2, b3). These
components correspond to the x, y, and z dimensions in a three-dimensional Euclidean space ( second vector).
Scalar Product of Two Vectors
The dot product calculator computes this scalar value for you, given the components i, j, and k for both vectors. This scalar number can then be used for various purposes, such as determining the
angle between two vectors, testing if vectors are orthogonal (dot product equals zero), or finding the projection of one vector onto another.
In summary, a Dot Product Calculator simplifies the process of finding the dot product of two vectors in Euclidean space by requiring only the i, j, and k components of both vectors to calculate the
scalar number
a . b
Vector dot product calculator shows step by step scalar multiplication.
What is dot product?
Dot product is an algebraic operation that takes two equal-length sequences of numbers usually coordinate vectors, and returns a single number.
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
a . b usually read as a dot b.
Dot product formula (two vectors)
Use this equation to calculate dot product of two vectors if magnitude (length) is given. Vector Multiplication calculation for vector components
a ∙ b = |a| × |b| × cos(θ)
|a| is length of vector a
|b| is length of vector b
θ is the angle between a and b
Vector Directions
We can also find dot product by using the direction of both vectors.
(a[i ]a[j] a[k]) ∙ (b[i] b[j] b[k]) = (a[i] ∙ b[i] + a[j] ∙ b[j] + a[k] ∙ b[k])
i, j, and k refers to x, y, and z coordinates on Cartesian plane.
calculator for vector components/coordinates & other calculations including scalar quantity
Multiplying the dot product. A calculation of the second vector
How to find dot product of two vectors?
The dot product of two vectors can be calculated by using the dot product formula.
Dot Product Example Method 1 – Vector Direction
Vector a = (2i, 6j, 4k)
Vector b = (5i, 3j, 7k)
Place the values in the formula.
a ∙ b = (2, 6, 4) ∙ (5, 3, 7)
(ai aj ak) ∙ (bi bj bk) = (ai ∙ bi + aj ∙ bj + ak ∙ bk)
(2 6 4) ∙ (5 3 7) = (2 ∙ 5 + 6 ∙ 3 + 4 ∙ 7)
(2 6 4) ∙ (5 3 7) = (10 + 18 + 28)
Solution – a ∙ b = 56
Dot Product Example Method 2 – Vector Magnitude
|a| = 15, |b| = 10, θ = 30°
Place the values in the formula.
a · b = |a| × |b| × cos(θ)
a · b = 15 × 10 × cos(30°)
Solution – a · b = 23.14
Dot Product Formula from tutorial.math.lamar.edu.
Two non-zero vectors are perpendicular if and only if their scalar product equals to zero
Dot product of two vectors a and b is a scalar quantity equal to the sum of pairwise products of coordinate vectors a and b
Dot Product Cross Product Magnitude Angle Unit Projection Scalar Projection Orthogonal Projection | {"url":"https://dotproductcalculator.com/","timestamp":"2024-11-10T09:49:10Z","content_type":"text/html","content_length":"45627","record_id":"<urn:uuid:c1bc1a3f-940c-4633-8b1f-639d879acf6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00648.warc.gz"} |
Matlab Based Projects | Matlab based Projects TopicsMatlab Based Projects
Matlab based Projects is used to enhance images quality where input and Output results are stored in m-files. We offer projects on Matlab based applications for students of engineering. Many latest
and advanced features are required for doing projects on Matlab, which is provided by us. These features have three types of files namely mat-files, command line and m-files. Storage data and loading
process is done by mat-files. Commands such as Dos command window are executed through command line.
Image types in Matlab:
Intensity and binary are the two types of images. Data type such as unit 2 unit 16 and double are present in intensity. It also has data matrix. For binary just as its name it has logical of binary
numbers 1’s and 0’s.
Linear Algebra:
Problems and errors of linear equation, Eigen value and linear computation are solved by Matlab simulation tool. There are two types of Linear algebra namely vector algebra and matrix algebra.
Vector is defined as the size of matrix in one dimension. The vector column and low should be differentiated. One of the prominent tasks of vector algebra is vector manipulation. Results of scalar
vector should be multiplied with the direction. Many rows and columns of matrix are taken into account for matrix operations.
Commands used in MATLAB:
Numeric display format commands used in MATLAB is a follows:
• Format + – it can be defined by negative, positive or zero.
• Format bank – 2 decimal digits.
• Format short – default 4 decimal digits.
• Format loose – less compact display modes are to be reset.
• Format compact – line feeds are suppressed.
• Format long – 16 decimal digits.
• Format long e- 16 digits plus exponent.
• Format short e – 5 digits plus exponent.
• Format rat – rational Approximation.
Input /output in MATLAB programming:
Ways such as built-in functions, externally defined and usage of explicit files are need to perform input and output function through Matlab variables and matrices. Input variables from keyboard and
formatted output variables are the Matlab output and input variables. Inputs for Matlab operation are usually movies and 3D objects.
Image compression:
The process of compressing data or an image for the purpose of transmission is called data compression. The size of input data is reduced in this process. This technique is further divided into two
namely loss and lossless compression. Lossless compression is more secured way of transmitting data than loss compression.
Noise reduction:
Input image is processed and removed of any noise. Thus the image is ready for next level of image processing.
Future Enhancement:
Machine listening concepts like intelligent instrument recognition and creating sense of sounds is the future of Matlab tools. We offer more projects on machine listening concepts. | {"url":"https://academiccollegeprojects.com/matlab-based-projects/","timestamp":"2024-11-09T09:22:06Z","content_type":"text/html","content_length":"237245","record_id":"<urn:uuid:79ab8dfc-8fb0-488e-b04f-2197912ac68e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00785.warc.gz"} |
On December 8, 1865, French mathematician Jacques Salomon Hadamard was born. Hadamard made major contributions in number theory, complex function theory, differential geometry and partial
differential equations. Moreover, he is also known for his description of the mathematical though process in his book Psychology of Invention in the Mathematical Field. “It is important for him who
wants to discover not to confine himself to one chapter of science, but to keep in…
Read more
Siméon Denis Poisson’s Contributions to Mathematics
On June 21, 1781, French mathematician, geometer, and physicist Siméon Denis Poisson was born. He is known known for his work on definite integrals, advances in Fourier series, electromagnetic
theory, and probability, esp. the eponymous Poisson series, the Poisson integral and the Poisson equation from potential theory. His works also included applications to electricity and magnetism, and
astronomy. Simeon Denis Poisson – The Youth of a Mathematician Poisson was born in Pithiviers, Loiret,…
Read more
Emil Artin and Algebraic Number Theory
On March 3, 1898, Austrian mathematician Emil Artin was born. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory,
contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Early Years Emil Artin was born in Vienna to
parents Emma Maria Artin, a soubrette on…
Read more
Bernhard Riemann’s innovative approaches to Geometry
On September 17, 1826, influential German mathematician Bernhard Riemann was born. Riemann‘s profound and novel approaches to the study of geometry laid the mathematical foundation for Albert
Einstein’s theory of relativity. He also made important contributions to the theory of functions, complex analysis, and number theory. “Nevertheless, it remains conceivable that the measure relations
of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry],…
Read more
James Joseph Sylvester – Lawyer and Mathematician
On September 3, 1815, English mathematician James Joseph Sylvester was born. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics.
He also was the founder of the American Journal of Mathematics. “It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his
journey sit down and invent a definite integral or two…
Read more
Hermann Minkowski and the four-dimensional Space-Time
On June 22, 1864, German mathematician Hermann Minkowski was born. Minkowski developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics,
and the theory of relativity. But he is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein’s special theory of relativity can be
understood geometrically as a theory of four-dimensional space–time, since known as the “Minkowski…
Read more
Number Theory, Topology, and Fractals with Wacław Sierpiński
On March 14, 1882, Polish mathematician Wacław Franciszek Sierpiński was born. Sierpiński is known for contributions to set theory, research on the axiom of choice and the continuum hypothesis,
number theory, theory of functions and topology. Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet and the Sierpiński curve), as are Sierpiński numbers and
the associated Sierpiński problem. Wacław Sierpiński – Early Years in Russian occupied Poland Wacław…
Read more
Charles Hermite’s admiration for simple beauty in Mathematics
On December 24, 1821, French mathematician Charles Hermite was born. He was the first to prove that e, the base of natural logarithms, is a transcendental number. Furthermore, he is famous for his
work in the theory of functions including the application of elliptic functions and his provision of the first solution to the general equation of the fifth degree, the quintic equation. “There
exists, if I am not mistaken, an entire…
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Carl Jacobi and the Elliptic Functions
On December 10, 1804, German mathematician Carl Gustav Jacob Jacobi was born. He made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. “Any
progress in the theory of partial differential equations must also bring about a progress in Mechanics.” – Carl Jacobi, Vorlesungen über Dynamik [Lectures on Dynamics] (1842/3) Carl Jacobi – A Child
Prodigy Carl Jacobi was the son of a banker and grew up in a rather…
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God made the integers, all the rest is the work of man – Leopold Kronecker
On December 7, 1823, German mathematician Leopold Kronecker was born, who worked on number theory and algebra. He criticized Cantor’s work on set theory, and his most cited quote says, “Die ganzen
Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk” (traditionally rendered: “God made natural numbers; all else is the work of man“.) Leopold Kronecker – Early Life Leopold Kronecker
was born in Liegnitz, Prussia (now Legnica, Poland) in a wealthy Jewish…
Read more | {"url":"http://scihi.org/tag/number-theory/","timestamp":"2024-11-12T02:02:53Z","content_type":"text/html","content_length":"595190","record_id":"<urn:uuid:778a41a7-4e70-4303-be63-35f652581b8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00071.warc.gz"} |
Learn Computer Science
I have a confession to make. I, (and most of us) have been stuck on fad diets for CS. It’s so nice to read easy books, like learn C++ in 21 days, or doing some leetcode problems. You learn a bit,
feel that warm glow of getting smarter, and trick yourself into feeling productive.
You’re not really getting any better, without learning the fundamentals.
I decided to cobble up a set of resources to go through, as a fusion of Teach Yourself Computer Science and Steve Yegge’s recommendations.
Teach Yourself CS has a detailed list of good resources for the more practical aspects of CS, whereas Steve Yegge’s recommendations focus more on the mathematical side – the first four courses he
recommends are: discrete math, linear algebra, statistics, and theory of computation.
Steve Yegge’s list omits Computer Architecture, which is a glaring omission – an Operating Systems course doesn’t have enough time to cover all the interesting parts of concurrency, parallelism, and
optimization that a computer architecture course would.
Teach Yourself CS doesn’t mention Theory of Computation, and is lighter on the math background, giving one resource for math. Theory of Computation is a bit more dated (swallowed up by all the other
fields), but is still useful for its applications.
To that end, I’ve fused them both, and skimmed some resources to put on this list.
This list is incomplete and changing all the time, but hey, isn’t that what agile development is all about?
1. Structure and Interpretation of Computer Programs
Discrete Math
1. Discrete Mathematics, an open introduction
Linear Algebra
1. No bullshit guide to linear Algebra, Savov
1. Statistics fourth ed, Friedman et. al
2. Think stats, Downey
3. Think Bayes, Downey
Theory of Computation
1. Theory of Computation, Hefferon
2. Computational Complexity, Arora and Barak
Computer Architecture
1. Computer Architecture, a Programmers Perspective
2. Computer Architecture, Patterson and Hennessey
1. Some Assembly Required
Algorithms and Data Structures
1. The Algorithm Design Manual, Skiena
2. The Art of Multiprocessor programming
Operating Systems
1. Operating Systems, Three Easy Pieces
1. Computer Networks, a Systems approach
1. Database Internals
1. Crafting Interpreters
Distributed Systems
1. Designing Data Intensive Applications | {"url":"https://takashiidobe.com/gen/learn-computer-science","timestamp":"2024-11-03T16:26:44Z","content_type":"text/html","content_length":"8691","record_id":"<urn:uuid:cb4a66f7-b4a9-4d97-9376-d749fb45d5dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00312.warc.gz"} |
Simo Sort: A Sorting Algorithm for Elements with Low Variance
I was writing an article on determining the fastest algorithm for sorting and while I was writing I got influenced by the many sorting algorithms and I devised an algorithm for sorting. It's called
Simo because that's the name of someone who is very special to my heart :)
How it works
It is a recursive sorting algorithm that repeats the following steps:
1. Get the average value of numbers in the array
2. Divide the array into two arrays (array 1 and array 2) and the average value will be chosen as the pivot.
3. Any value smaller than the pivot will be in first array and any value larger than the pivot will be in the second array.
4. Repeat the same algorithm on array 1 and array 2 till you reach the ending condition
For optimization purposes:
1. The ending condition is not a normal ending for a recursive function but it uses some conditions that I devised through try and error, these conditions raise the performance alot.
2. If the array size is less than 15 an Insertion Sort is used to decrease overhead, 15 has been found empirically as the optimal cutoff value in 1996.
The algorithm
This is how the algorithm works:
low is the index of the first element and high is the index of last element.
template <class T>
void simoSort(T a[],int low,int high)
//Optimization Condition
double average=0;
int n = (high - low + 1);
for (int i = low; i <= high; i++)
int k=low;
for(int i = low;i<=high; i++)
int tempValue = a[i];
a[i] = a[k];
a[k] = tempValue;
//Optimized Stop Condition
if(low<k-1 && k-1!=high)
if(high>k && low!=k)
//Insertion Sort to reduce Overhead when recursion is not deep enough
for (int i = low + 1; i <= high; i++) {
int value = a[i];
int j;
for (j = i - 1; j >= low && a[j] > value; j--) {
a[j + 1] = a[j];
a[j + 1] = value;
"Example 1" shows how the algorithm works in a normal case.
"Example 2" shows how the algorithm works in its best case scenario where elements of the array have little variance between them. In the case where there where only 2 numbers "1" and "0" the
algorithm had a complexity of O(n)
Note: When the algorithm stops in the shown figures it means a leaf has reached an ending condition.
The Search is Stable and has an upper bound space Complexity of O(1)
After doing the time complexity analysis calculations.
• The Average and Worst case scenarios are of (5/6)*nlogn complexity which maps to O(nlogn)
• The Best Case is O(n) as shown in Example 2.
If any one can prove these values wrong please tell me and I will modify them, thanks.
The algorithm is currently faster than quick sort and bucket sort when the variance of the elements is low(not more than 5 different elements) according to the I benchmark that I made, and I'm
currently writing an article that will compare all sorting algorithms including this one.
The current version of the algorithm only works on integers. it does not work on chars or double values.
1.0 (10 March 2012)
I hope that this article would at least slightly help those who are interested in this issue. Feel free to tell me any comments on the algorithm :) | {"url":"https://www.codeproject.com/Articles/344046/Simo-Sort-A-Sorting-Algorithm-for-Elements-with-Lo","timestamp":"2024-11-07T08:45:39Z","content_type":"text/html","content_length":"26756","record_id":"<urn:uuid:a65bd0d5-dbe6-4923-93cc-b0906e01cde8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00138.warc.gz"} |
geometry Archives - Mark Proffitt
All four of these solids are Round Squares. I thought I had a general rule for calculating the combinations when I found the first two but then I found another and another. Is there a general formula
for all the possible combinations. What are the parameters? Numerically there could be 6 but I do not think geometrically 6 are possible. This gets more confusing for Round Triangles or Square
Cylinder is the obvious round square.
Cones are round triangle. Two triangles for a square. Two cones form a round square.
Combinations of 3D shapes are either unions or intersections. The intersection of a cylinder and a wedge makes a Round Square Triangle. The union of two half height Round Square Triangles form a
Round Square.
Intersection of two cylinders rotates 90 degrees from each other leaves a square and two rounded sides for the Fourth Round Square.
I found the first two using pencil and paper. The third with FreeCAD thinking in a subtractive then additive fashion. The fourth I found using OpenSCAD. The different approaches enabled finding the
alternatives. I would not have found the fourth type using pencil and paper, the CAD software showed me what I described.
How would you go about finding a proof for a general formula?
FutureMaps created using the Predictive Innovation^® are truly maps of the innovation space. Each innovation has a specific address and any desired set of innovations can be located by using the
proper combination of parameters.
The Predictive Innovation^® constructs an n dimensional taxonomy (hypercube) to describe the innovation space. This is in essence a fractal model. By increasing dimensions, greater resolution can be
achieved at predictable levels. This improves upon hierarchical taxonomies by allowing for multiple classifications and sparsely populated hypercubes.
Similar approaches have been used in computer graphics to highly accurately represent natural systems. This shares some similarity to the work of Stephen Wolfram on cellular automata models.
Extra Credit
If all that was a bunch of geeky gobly gook to you then you can at least enjoy the pretty picture and this fun song by Jonathan Coulton’s called “Mandelbrot Set”. You might want to visit his website.
He has some great music and he has done some cool things with Creative Commons.
FutureMaps created using the Predictive Innovation Method are truly maps of the innovation space. Each innovation has a specific address and any desired set of innovations can be located by using the
proper combination of parameters.
The Predictive Innovation Method constructs an n dimensional taxonomy (hypercube) to describe the innovation space. This is in essence a fractal model. By increasing dimensions, greater resolution
can be achieved at predictable levels. This improves upon hierarchical taxonomies by allowing for multiple classifications and sparsely populated hypercubes.
Similar approaches have been used in computer graphics to highly accurately represent natural systems. This shares some similarity to the work of Stephen Wolfram on cellular automata models.
Extra Credit
If all that was a bunch of geeky gobly gook to you then you can at least enjoy the pretty picture and this fun song by Jonathan Coulton’s called “Mandelbrot Set”. You might want to visit his website.
He has some great music and he has done some cool things with Creative Commons. | {"url":"https://markproffitt.com/tag/geometry/","timestamp":"2024-11-05T15:43:52Z","content_type":"application/xhtml+xml","content_length":"78445","record_id":"<urn:uuid:953a242a-6fa1-4fe1-879d-4ae2a2ce7ae4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00349.warc.gz"} |
How many integers less than 1 billion are very round numbers? - Answers
If the previous digit is 5 or more then round up but if the previous digit is less than 5 then round down.
By the uniqueness of numbers, it is equal to 18 billion: no more, no less.
it is below zero * * * * * They are whole numbers that are less than zero.
129,999 , all smaller integers, and all negative numbers are.
This is the 'null' or 'empty' set.There are no numbers greater than '-3' and less than '-9'.
If both numbers are positive....yes If either or both numbers are negative ....no
The set of positive integers less than 50 is finite (there are 49).The set of all integers less than 50 is infinite, because it includes an infinite number of negative numbers.
Positive odd integers less than 8 are: 1 3 5 and 7 | {"url":"https://math.answers.com/math-and-arithmetic/How_many_integers_less_than_1_billion_are_very_round_numbers","timestamp":"2024-11-05T17:06:17Z","content_type":"text/html","content_length":"159220","record_id":"<urn:uuid:16fb9ab5-40b1-42cb-8316-900e93353b94>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00019.warc.gz"} |
Latest Posts - Page 4
This comprehensive guide will help you understand the unit circle and its various components.
Learn how to apply the product and quotient rules in calculus to solve complex equations.
This article explains the concept of standard deviation in sampling distribution and provides an in-depth understanding of how it is calculated.
This article provides an in-depth explanation of the factored form of a quadratic equation and how to use it to solve various types of equations.
Learn how to use the unit circle to explore mathematical concepts like trigonometry, angles, and radians.
Learn how to perform matrix multiplication in MATLAB with this step-by-step guide.
Learn how to find the greatest common factor of 8 and 12 with this simple guide.
Learn how to calculate and interpret standard deviation in psychology.
Learn how to find the greatest common factor of 8 and 12.
Learn what the standard deviation abbreviation is and how it is used in statistics. Understand the importance of standard deviation and how it is calculated. | {"url":"https://mathemista.com/page/4/","timestamp":"2024-11-07T14:12:51Z","content_type":"text/html","content_length":"79521","record_id":"<urn:uuid:bb10a3b4-238a-4aa8-8c31-caf00d5e09a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00138.warc.gz"} |
Week 3 discussion - Course Help Online
• Simplify each expression using the rules of exponents and examine the steps you are taking.
• Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in
sentences describing the thought behind your math work.Principal rootProduct ruleQuotient ruleReciprocalnth rootRefer to Inserting Math Symbols for guidance with formatting. Be aware with regards
to the square root symbol, you will notice that it only shows the front part of a radical and not the top bar. Thus, it is impossible to tell how much of an expression is included in the radical
itself unless you use parenthesis. For example, if we have √12 + 9 it is not enough for us to know if the 9 is under the radical with the 12 or not. Therefore, we must specify whether we mean it
to say √(12) + 9 or √(12 + 9), as there is a big difference between the two. This distinction is important in your notation.Another solution is to type the letters “sqrt” in place of the radical
and use parenthesis to indicate how much is included in the radical as described in the second method above. The example above would appear as either “sqrt(12) + 9” or “sqrt(12 + 9)” depending on
what we needed it to say.
Your initial post should be at least 250 words in length.
(3) Simplifying Expressions involving Variables
Simplify each expression. Assume the variables represent any
real numbers and use absolute value as necessary. See
Example 8
55. (x4)1/4
56. ()1/6
57. (*)/2
58. (110)1/2
59. (*)/3
60. (W)1/3
61. (9x4y2)1/2
62. (164/41/4 | {"url":"https://coursehelponline.com/week-3-discussion/","timestamp":"2024-11-13T21:45:09Z","content_type":"text/html","content_length":"40973","record_id":"<urn:uuid:dee321a6-cd56-4392-ba74-c440d151f6bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00785.warc.gz"} |
Symmetric Molecules
For molecules with non-trivial point group symmetry, the Fourier projection theorem implies that each 2D projection image corresponds to multiple slices, where the number of slices equals the size of
the symmetry group. As a consequence, determining the 3D structure requires fewer images for molecules with larger point-group symmetries. Our algorithms for orientation assignment currently support
cyclically-symmetric molecules.
Further reading:
G. Pragier, Y. Shkolnisky, A common lines approach for ab-initio modeling of cyclically-symmetric molecules, arxiv preprint. | {"url":"https://spr.math.princeton.edu/symmetric","timestamp":"2024-11-14T20:42:35Z","content_type":"text/html","content_length":"17862","record_id":"<urn:uuid:ff4a38f7-11bc-4f01-ba5d-f190647f68b5>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00705.warc.gz"} |
Area Calculator - webtodoweb.com
Area calculation is the process of determining the amount of space enclosed by a two-dimensional shape. It’s a measure of the extent of a surface. The method of calculating the area depends on the
shape in question.
Area Calculator 📐
Calculating areas is not just a skill learned in school; it’s an essential tool used across various industries. From construction and agriculture to interior design and even art, the ability to
accurately calculate area is indispensable. To make this task easier, our website offers a free and user-friendly Area Calculator that can handle a wide range of geometric shapes.
Calculators, Converters, and Randomizers
Our website is a comprehensive platform that offers a variety of tools designed to assist you in your daily tasks. Among these tools, you’ll find calculators, converters, and randomizers that can
help you with everything from unit conversions to generating random numbers. However, the focus of this blog is our Area Calculator, a tool designed to simplify the process of calculating areas for
various shapes.
Using the Area Calculator
Navigating and using the Area Calculator is a straightforward process. Here’s a step-by-step guide to help you get started:
1. Length: Enter the value of the length in the designated field.
2. Width: Enter the value of the width in the designated fields.
3. Calculate: Click the ‘Calculate’ button to get your results.
The Area Calculator provides not only the area but also a detailed explanation of the calculations, making it an educational tool as well.
Supported Shapes
Our Area Calculator is versatile, supporting a wide range of shapes. Here’s a list to give you an idea:
• Square
• Rectangle
• Triangle
• Circle
• Parallelogram
• Trapezoid
• Ellipse
• Octagon
• Sector of a circle
For each shape, it’s crucial to know the required measurements and units to ensure accurate calculations.
Formulas and Explanations
For those who like to understand the mechanics behind the calculations, our Area Calculator uses universally accepted geometric formulas for each shape. We also provide diagrams and visual aids to
help you better understand the process.
Area of a Square
Calculating the area of a square is one of the simplest tasks you can perform with our online calculator. The formula used is ( A = a^^2 ), where ( A ) is the area and ( a ) is the length of one
side. This straightforward formula makes it easy to quickly find the area of any square.
Area of a Rectangle
The area of a rectangle is calculated by multiplying its length by its width. The formula is ( A = l \times w ). Our online Calculator makes this process effortless. If you’re dealing with irregular
shapes, you can approximate their area by breaking them down into smaller rectangles.
Area of a Triangle
Triangles come in various forms, and our Calculator is equipped to handle them all. Whether you’re dealing with an equilateral, isosceles, or scalene triangle, the calculator has you covered.
The most commonly used formula is
\[( A = \frac{1}{2} \times b \times h )\]
where ( b ) is the base and ( h ) is the height.
Area of a Circle
The formula for calculating the area of a circle is
\[( A = \pi \times r^2 ), \]
where ( r ) is the radius. The calculator simplifies this process, requiring only the radius or diameter to provide an accurate area measurement.
Area of a Parallelogram
Adding the base and vertical height together yields the area of a parallelogram.
The formula is:
\[ ( A = b \times h ) \]
Our online Calculator also allows you to transform a parallelogram into a rectangle to find its area, offering more flexibility in your calculations.
Area of a Trapezoid
The formula for the area of a trapezoid is
\[ ( A = \frac{1}{2} \times (a + b) \times h ), \]
where ( a ) and ( b ) are the lengths of the parallel bases and ( h ) is the height. The Area Calculator provides a step-by-step breakdown of this calculation.
Area of an Ellipse (Oval)
For an ellipse, the formula is
\[ ( A = \pi \times a \times b ),\]
where ( a ) and ( b ) are the major and minor radii, respectively. The calculator makes it easy to input these values and get an accurate area measurement.
Area of a Sector
The area of a sector is calculated using ( A = \frac{1}{2} \times r^2 \times \theta ), where ( \theta ) is the angle in radians. Accurate angle measurement is essential, and our Area Calculator
ensures you get it right.
Area of an Octagon
For a regular octagon, the area can be calculated using a specific formula. Our Area Calculator guides you through this, ensuring you confirm the regularity of the shape for accurate calculations.
The utility of an Area Calculator extends beyond academic exercises. It’s a practical tool that can be used in real-world applications such as engineering projects, crafts, and even calculating the
amount of paint or material needed for various tasks.
We’ve walked you through the functionalities and capabilities of our Area Calculator, covering the various shapes it supports and the formulas it uses. We encourage you to utilize this tool to make
your area calculations more efficient and accurate.
The information provided in this blog is based on standard geometric principles. For more details and to use the Area Calculator, please visit our webpage.
Additional Resources
For those interested in further exploring the world of geometry, our website offers a range of calculators and tools designed to assist you in various calculations.
Feel free to explore and make your calculations more efficient and accurate.
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Yes 😃No ☹️ | {"url":"https://webtodoweb.com/area-calculator/","timestamp":"2024-11-06T02:30:43Z","content_type":"text/html","content_length":"132587","record_id":"<urn:uuid:291d9902-15ff-4260-aebb-c6cbd15825b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00819.warc.gz"} |
CInt Function
किसी भी स्ट्रिंग या न्यूमेरिक एक्सप्रेशन को पूर्णांक में बदलता है.
CInt (Expression As Variant) As Integer
If the argument is string, the function trims the leading white space; then it tries to recognize a number in following characters. The syntax below are recognized:
• Decimal numbers (with optional leading sign) using decimal and group separators of locale configured in LibreOffice (group separators are accepted in any position), with optional exponential
notation like "-12e+1" (where an optionally signed whole decimal number after e or E or d or D defines power of 10);
• Octal numbers like "&Onnn...", where "nnn..." after "&O" or "&o" is sequence no longer than 11 digits, from 0 to 7, up to the next non-alphanumeric character;
• Hexadecimal numbers like "&Hnnn...", where "nnn..." after "&H" or "&h" is sequence of characters up to the next non-alphanumeric character, and must be no longer than 8 digits, from 0 to 9, A to
F, or a to f.
The rest of the string is ignored. If the string is not recognized, e.g. when after trimming leading whitespace it doesn't start with plus, minus, a decimal digit, or "&", or when the sequence after
"&O" is longer than 11 characters or contains an alphabetic character, the numeric value of expression is 0.
If the argument is an error, the error number is used as numeric value of the expression.
If the argument is a date, number of days since 1899-12-30 (serial date) is used as numeric value of the expression. Time is represented as fraction of a day.
After calculating the numeric value of the expression, it is rounded to the nearest integer (if needed), and if the result is not between -32768 and 32767, LibreOffice Basic reports an overflow
error. Otherwise, the result is returned. | {"url":"https://help.libreoffice.org/latest/hi/text/sbasic/shared/03100500.html","timestamp":"2024-11-11T20:39:54Z","content_type":"text/html","content_length":"14533","record_id":"<urn:uuid:0a7136a0-2887-45bb-adc4-573f055c5270>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00806.warc.gz"} |
ArrayTester - Part A
15.10. ArrayTester - Part A¶
The following is a free response question from 2018. It was question 4 on the exam. You can see all the free response questions from past exams at https://apstudents.collegeboard.org/courses/
Question 4. This question involves reasoning about arrays of integers. You will write two static methods, both of which are in a class named ArrayTester.
public class ArrayTester
* Returns an array containing the elements of column c of arr2D in the same
* order as they appear in arr2D. Precondition: c is a valid column index in
* arr2D. Postcondition: arr2D is unchanged.
public static int[] getColumn(int[][] arr2D, int c)
/* to be implemented in part (a) */
* Returns true if and only if every value in arr1 appears in arr2.
* Precondition: arr1 and arr2 have the same length. Postcondition: arr1 and
* arr2 are unchanged.
public static boolean hasAllValues(int[] arr1, int[] arr2)
/* implementation not shown */
/** Returns true if arr contains any duplicate values; false otherwise. */
public static boolean containsDuplicates(int[] arr)
/* implementation not shown) */
* Returns true if square is a Latin square as described in part (b); false
* otherwise. Precondition: square has an equal number of rows and columns.
* Precondition: square has at least one row.
public static boolean isLatin(int[][] square)
/* to be implemented in part (b) */
Part a. Write a static method getColumn, which returns a one-dimensional array containing the elements of a single column in a two-dimensional array. The elements in the returned array should be in
the same order as they appear in the given column. The notation arr2D [r][c] represents the array at row r and column c.
The following code segment initializes an array and calls the getColumn method.
int [] [] arr2D = { { 0, 1, 2 }, { 3, 4, 5 }, { 6, 7, 8 }, { 9, 5, 3 } }; int[] result = ArrayTester.getColumn (arr2D, 1);
When the code segment has completed execution, the variable result result will have the following contents. result: {1, 4, 7, 5}
15.10.1. Try and Solve It¶
Complete the method getColumn below.
You have attempted of activities on this page | {"url":"https://runestone.academy/ns/books/published/csawesome/FreeResponse/ArrayTesterA.html","timestamp":"2024-11-14T10:15:21Z","content_type":"text/html","content_length":"31270","record_id":"<urn:uuid:1620a576-a9f5-4cd4-bc91-a3fdbd7518e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00244.warc.gz"} |
CATMI 2023
Home Programme Venue Excursion
The registration is now open. The registration is free, but the daily costs for lunches at the conference hotel is NOK 300. Invited speakers will have lunches covered.
To register, fill out the registration form.
Registration deadline: 1st of June.
The meeting is an activity organised by the Lie-Størmer Center, a newly founded Norwegian research center for fundamental structures in computational and pure mathematics, and aims at bringing
together a mix of people from mathematics and informatics to exchange ideas on how we apply concepts and tools from category theory, type theory, and homotopy theory to structure complex problems and
research in mathematics, computations and theoretical computer science.
Category theory, as a mathematical theory, is less a collection of theorems than a language to organise our thinking. The last decades has seen its growing impact: Applied category theory has emerged
modelling diverse areas, for instance data bases and networks. In computer science, it provides appropriate conceptual tools to structure complex problems, organise research areas and ask the right
questions. In mathematics, formulating results using categorical notions, gives a better understanding of which fundamental structures are universally at work, and gives guidance on natural problem
As soon as they are more complex than mere numbers, mathematical structures are only specified up to isomorphism, corresponding to the fact that computations are only determined up to any specific
implementation. Concrete implementations are necessary to get anything specific done. Theoretical informatics (computing science) is still fragmented into different disciplines and schools. It is,
therefore, highly desirable to provide abstraction and unification to guide our thinking.
The workshop is a forum for exchanging experiences in how we apply concepts and tools from category theory, type, and homotopy theory to structure complex problems and research in mathematics,
computations and theoretical computer science. | {"url":"http://catmi.no/","timestamp":"2024-11-05T10:24:06Z","content_type":"application/xhtml+xml","content_length":"4980","record_id":"<urn:uuid:67307e4a-d566-4aa1-ae59-6a85bc2de800>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00733.warc.gz"} |
Multiplication By 12 Worksheets
Mathematics, specifically multiplication, forms the keystone of countless scholastic disciplines and real-world applications. Yet, for several students, mastering multiplication can present a
difficulty. To address this hurdle, educators and moms and dads have actually accepted an effective tool: Multiplication By 12 Worksheets.
Introduction to Multiplication By 12 Worksheets
Multiplication By 12 Worksheets
Multiplication By 12 Worksheets -
This page has lots of games worksheets flashcards and activities for teaching all basic multiplication facts between 0 and 10 Basic Multiplication 0 through 12 On this page you ll find all of the
resources you need for teaching basic facts through 12 Includes multiplication games mystery pictures quizzes worksheets and more
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
Importance of Multiplication Method Comprehending multiplication is pivotal, laying a strong structure for innovative mathematical ideas. Multiplication By 12 Worksheets provide structured and
targeted method, fostering a deeper comprehension of this fundamental arithmetic operation.
Development of Multiplication By 12 Worksheets
1 12 multiplication Worksheet For Kids Learning Printable
1 12 multiplication Worksheet For Kids Learning Printable
These free 12 times table worksheets provide you with an excellent tool to practice and memorise the tables The 12 times table is probably the hardest multiplication table to memorise However there
are several tips to help you learn this table quicker Let s take a look at some of the sums 1 x 12 12 alternatively this is 1 x 10 1 x 2
Multiplication by 12 worksheets gives different methods to solve types of multiplication problems and it will help to solve the equations problems in future Multiplication by 12 worksheets are very
useful for kids to grow their math skills They also give methods for getting kids to practice multiplication and other important concepts
From conventional pen-and-paper exercises to digitized interactive styles, Multiplication By 12 Worksheets have actually advanced, accommodating varied knowing styles and preferences.
Sorts Of Multiplication By 12 Worksheets
Fundamental Multiplication Sheets Easy exercises concentrating on multiplication tables, aiding students develop a solid math base.
Word Problem Worksheets
Real-life situations integrated into problems, enhancing vital reasoning and application skills.
Timed Multiplication Drills Tests developed to improve speed and accuracy, assisting in rapid psychological mathematics.
Benefits of Using Multiplication By 12 Worksheets
Times Table Grid To 12x12
Times Table Grid To 12x12
This basic Multiplication worksheet is designed to help kids practice multiplying by 12 with multiplication questions that change each time you visit This math worksheet is printable and displays a
full page math sheet with Horizontal Multiplication questions With this math sheet generator you can easily create Multiplication worksheets
Learn to Multiply by 12s Print this worksheet for your class so they can learn to multiply by 12s Find the missing factors complete the multiplication wheel and skip counting by 12s are just a few of
the activities on this worksheet 3rd and 4th Grades View PDF
Enhanced Mathematical Skills
Consistent practice sharpens multiplication efficiency, improving total math capacities.
Enhanced Problem-Solving Abilities
Word troubles in worksheets establish logical thinking and technique application.
Self-Paced Learning Advantages
Worksheets fit specific understanding rates, fostering a comfortable and adaptable discovering setting.
Just How to Develop Engaging Multiplication By 12 Worksheets
Integrating Visuals and Colors Lively visuals and shades catch focus, making worksheets aesthetically appealing and engaging.
Including Real-Life Situations
Relating multiplication to daily situations adds relevance and usefulness to exercises.
Customizing Worksheets to Various Skill Levels Tailoring worksheets based on differing proficiency degrees makes sure inclusive learning. Interactive and Online Multiplication Resources Digital
Multiplication Devices and Gamings Technology-based resources offer interactive understanding experiences, making multiplication appealing and enjoyable. Interactive Web Sites and Applications On the
internet platforms give diverse and obtainable multiplication method, supplementing traditional worksheets. Customizing Worksheets for Different Knowing Styles Aesthetic Learners Visual aids and
layouts help understanding for learners inclined toward aesthetic understanding. Auditory Learners Spoken multiplication troubles or mnemonics satisfy students that understand concepts with acoustic
means. Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic students in recognizing multiplication. Tips for Effective Execution in Knowing Consistency in Practice Routine
technique enhances multiplication skills, advertising retention and fluency. Balancing Repeating and Variety A mix of repeated exercises and varied problem styles preserves rate of interest and
understanding. Providing Useful Responses Feedback aids in recognizing areas of improvement, urging continued progression. Difficulties in Multiplication Practice and Solutions Motivation and
Engagement Difficulties Dull drills can result in uninterest; cutting-edge approaches can reignite inspiration. Getting Rid Of Anxiety of Math Unfavorable perceptions around mathematics can prevent
development; creating a favorable understanding atmosphere is necessary. Impact of Multiplication By 12 Worksheets on Academic Efficiency Studies and Study Findings Research indicates a favorable
correlation in between consistent worksheet use and boosted mathematics performance.
Multiplication By 12 Worksheets emerge as flexible devices, fostering mathematical proficiency in students while fitting varied discovering designs. From fundamental drills to interactive on-line
sources, these worksheets not only enhance multiplication skills yet also promote important reasoning and analytical capabilities.
Printable Timed Multiplication Quiz PrintableMultiplication
Worksheet On 12 Times Table Printable Multiplication Table 12 Times Table
Check more of Multiplication By 12 Worksheets below
Third Grade Multiplication Practice
Printable Multiplication Table 1 12 Pdf PrintableMultiplication
Multiplication Time 1 Worksheet
16 MULTIPLICATION WORKSHEETS 1 TO 12 Worksheets
Multiplication Worksheets X3 PrintableMultiplication
Multiplication Worksheets Numbers 1 Through 12 Mamas Learning Corner
Multiplying 1 to 12 by 12 100 Questions A Math Drills
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
Multiplication Facts Worksheets Math Drills
It is quite likely that there are students who have mastered all of the multiplication facts up to the 12 times tables In case they want need an extra challenge this sections includes multiplication
facts worksheets above 12 with the expectation that students will use mental math or recall to calculate the answers
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
It is quite likely that there are students who have mastered all of the multiplication facts up to the 12 times tables In case they want need an extra challenge this sections includes multiplication
facts worksheets above 12 with the expectation that students will use mental math or recall to calculate the answers
16 MULTIPLICATION WORKSHEETS 1 TO 12 Worksheets
Printable Multiplication Table 1 12 Pdf PrintableMultiplication
Multiplication Worksheets X3 PrintableMultiplication
Multiplication Worksheets Numbers 1 Through 12 Mamas Learning Corner
4Th Grade Multiplication Worksheets Free 4th Grade Multiplication Worksheets Best Coloring
Multiplication By Twelves Worksheet
Multiplication By Twelves Worksheet
Multiplication Drills 1 12 Free Printable
FAQs (Frequently Asked Questions).
Are Multiplication By 12 Worksheets ideal for all age teams?
Yes, worksheets can be customized to various age and ability levels, making them adaptable for numerous students.
How commonly should students practice utilizing Multiplication By 12 Worksheets?
Constant method is essential. Routine sessions, ideally a couple of times a week, can generate considerable enhancement.
Can worksheets alone improve mathematics abilities?
Worksheets are an important tool however ought to be supplemented with varied knowing methods for thorough ability development.
Exist on the internet systems offering free Multiplication By 12 Worksheets?
Yes, numerous academic web sites offer free access to a variety of Multiplication By 12 Worksheets.
Just how can parents sustain their youngsters's multiplication practice at home?
Encouraging regular practice, giving support, and developing a positive learning atmosphere are advantageous actions. | {"url":"https://crown-darts.com/en/multiplication-by-12-worksheets.html","timestamp":"2024-11-06T11:53:39Z","content_type":"text/html","content_length":"28782","record_id":"<urn:uuid:0e62c75d-739c-4ab8-8f62-d6e0a6d5c6c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00275.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
"nongravitational forces"
"نیروهای ِناگرانشی"
"niruhâ-ye nâgerâneši" (#)
Fr.: "forces non-gravitationnelles"
The forces of jets from a comet's nucleus that can cause a rocket-like effect and alter a comet's direction of motion slightly.
→ non-; → gravitational; → force.
Einstein's gravitational constant
پایای ِگرانشی ِاینشتین
pâyâ-ye gerâneši-ye Einstein (#)
Fr.: constante gravitationnelle d'Einstein
The coupling constant appearing in → Einstein's field equations, expressed by: κ = 8πG/c^4, where G is the Newtonian → gravitational constant and c the → speed of light.
→ einstein; → gravitational; → constant.
Gaussian gravitational constant
پایای ِگرانشی ِگاؤس
pâyâ-ye gerâneši-ye Gauss
Fr.: constante gravitationnelle de Gauss
The constant, denoted k, defining the astronomical system of units of length (→ astronomical unit), mass (→ solar mass), and time (→ day), by means of → Kepler's third law. The dimensions of k^2 are
those of Newton's constant of gravitation: L ^3M ^-1T ^-2. Its value is: k = 0.01720209895.
→ Gaussian; → gravitational; → constant.
gerâneši (#)
Fr.: gravitationnel
Of or relating to or caused by → gravitation.
Adj. of → gravitation.
gravitational acceleration
شتاب ِگرانشی
šetâb-e gerâneši (#)
Fr.: accélération gravitationnelle
The acceleration caused by the force of gravity. At the Earth's surface it is determined by the distance of the object form the center of the Earth: g = GM/R^2, where G is the → gravitational
constant, and M and R are the Earth's mass and radius respectively. It is approximately equal to 9.8 m s^-2. The value varies slightly with latitude and elevation. Also known as the → acceleration of
→ gravitational; → acceleration.
gravitational attraction
درکشش ِگرانشی
darkešeš-e gerâneši
Fr.: attraction gravitationnelle
The force that pulls material bodies toward one another because of → gravitation.
→ gravitational; → attraction.
gravitational collapse
رمبش ِگرانشی
rombeš-e gerâneši (#)
Fr.: effondrement gravitationnel
Collapse of a mass of material as a result of the mutual → gravitational attraction of all its constituents.
→ gravitational; → collapse.
gravitational constant
پایای ِگرانشی
pâyâ-ye gerâneši (#)
Fr.: constante gravitationnelle
A fundamental constant that appears in → Newton's law of gravitation. It is the force of attraction between two bodies of unit mass separated by unit distance: G = 6.673 x 10^-8 dyn cm^2 g^-2 or
6.673 x 10^-8 cm^3s^-2g^-1, or 6.673 x 10^-11 N m^2 kg^-2 or 6.673 x 10^-11 m^3s^-2kg^-1. It was first measured in 1798 by Henry Cavendish (1731-1810), 71 years after Newton's death. Same as the →
Newtonian constant of gravitation.
→ gravitational; → constant.
gravitational contraction
ترنگش ِگرانشی
terengeš-e gerâneši
Fr.: contraction gravitationnelle
Decrease in the volume of an astronomical object under the action of a dominant, central gravitational force.
→ gravitational; → contraction.
gravitational coupling constant
پایای ِجفسری ِگرانشی
pâyâ-ye jafsari-ye gerâneši
Fr.: constante de couplage gravitationnel
The dimensionless gravitational constant defined as the gravitational attraction between pair of electrons and normally given by: α[G] = (Gm[e]^2) / (ħc) = (m[e] / m[P])^2 ~ 1.7518 × 10^-45, where ħ
is → Planck's reduced constant, c the → speed of light, m[e] is the → electron mass, and m[P] is the → Planck mass.
→ gravitational; → coupling; → constant.
gravitational encounter
رویارویی ِگرانشی
ruyâruyi-ye gerâneši
Fr.: rencontre gravitationnelle
An encounter in which two moving bodies alter each other's direction and velocity by mutual → gravitational attraction.
→ gravitational; → encounter.
gravitational energy
کاروژ ِگرانشی
kâruž-e gerâneši
Fr.: énergie gravitationnelle
Same as → gravitational potential energy.
→ gravitational; → energy.
gravitational equilibrium
ترازمندی ِگرانشی
tarâzmandi-ye gerâneši (#)
Fr.: équilibre gravitationnel
The condition in a celestial body when gravitational forces acting on each point are balanced by some outward pressure, such as radiation pressure or electron degeneracy pressure, so that no vertical
motion results.
→ gravitational; → equilibrium.
gravitational field
میدان ِگرانشی
meydân-e gerâneši (#)
Fr.: champ gravitationnel
The region of space in which → gravitational attraction exists.
→ gravitational; → field.
gravitational force
نیروی ِگرانشی
niru-ye gerâneši (#)
Fr.: force gravitationnelle
The weakest of the four fundamental forces of nature. Described by → Newton's law of gravitation and subsequently by Einstein's → general relativity.
→ gravitational; → force.
gravitational instability
ناپایداری ِگرانشی
nâpâydâri-ye gerâneši (#)
Fr.: instabilité gravitationnelle
The process by which fluctuations in an infinite medium of size greater than a certain length scale (the Jeans length) grow by self-gravitation.
→ gravitational; → instability.
gravitational interaction
اندرژیرش ِگرانشی
andaržireš-e gerâneši
Fr.: interaction gravitationnelle
Mutual attraction between any two bodies that have mass.
→ gravitational; → interaction.
gravitational lens
عدسی ِگرانشی
adasi-ye gerâneši (#)
Fr.: lentille gravitationnelle
A concentration of matter, such as a galaxy or a cluster of galaxies, that bends light rays from a background object, resulting in production of multiple images. If the two objects and the Earth are
perfectly aligned, the light from the distant object appears as a ring from Earth. This is called an Einstein Ring, since its existence was predicted by Einstein in his theory of general relativity.
→ gravitational; → lens.
gravitational lens equation
هموگش ِعدسی ِگرانشی
hamugeš-e adasi-ye gerâneši
Fr.: équation de lentille gravitationnelle
The main equation of gravitational lens theory that sets a relation between the angular position of the point source and the observable position of its image.
→ gravitational; → lens; → equation.
gravitational lensing
لنزش ِگرانشی
lenzeš-e gerâneši
Fr.: effet de lentille gravitationelle
The act of producing or the state of a → gravitational lens.
→ gravitational; → lensing. | {"url":"https://dictionary.obspm.fr/index.php?showAll=1&&search=&&formSearchTextfield=gravitational&&page=0","timestamp":"2024-11-08T01:49:08Z","content_type":"text/html","content_length":"30100","record_id":"<urn:uuid:05f36b29-b540-403f-9183-dd5845bcbbfb>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00649.warc.gz"} |
How do you teach angles in a fun way?
How do you teach angles in a fun way?
Fun ways to teach angles
1. Scratch animations. An angles unit is an ideal opportunity to introduce your students (and maybe yourself) to the basics of coding.
2. K’NEX models.
3. Estimating angles game.
4. Robotics.
5. Blindfold game.
6. Masking tape.
7. Explain Everything video explanations.
How will you explain to your students the concept of angles?
Angles are measured in degrees, which is a measure of circularity, or rotation. A full rotation, which would bring you back to face in the same direction, is 360°. A half-circle is therefore 180°,
and a quarter-circle, or right angle, is 90°. Two or more angles on a straight line add up to 180°.
What are the 5 types of angles?
Types of Angles – Acute, Right, Obtuse, Straight and Reflex Angles
• Acute angle.
• Right angle.
• Obtuse angle.
• Straight angle.
• Reflex angle.
What are 7 types of angles?
There are 7 types of angles. These are zero angles, acute angles, right angles, obtuse angles, straight angles, reflex angles, and complete angles.
How is the concept of angles applied in real life?
Angles are used in daily life. Engineers and architects use angles for designs, roads, buildings and sporting facilities. Carpenters use angles to make chairs, tables and sofas. Artists use their
knowledge of angles to sketch portraits and paintings.
What are the uses of angles?
Engineers use angle measurements to construct buildings, bridges, houses, monuments, etc. Carpenters use angle measuring devices such as protractors, to make furniture like chairs, tables, beds, etc.
The angle can be seen in the wall clocks of our homes, made by hands of clocks.
What are the 7 types of angles definition?
Acute Angle – An angle less than 90 degrees. Right Angle – An angle that is exactly 90 degrees. Obtuse Angle – An angle more than 90 degrees and less than 180 degrees. Straight Angle – An angle that
is exactly 180 degrees. Reflex Angle – An angle greater than 180 degrees and less than 360 degrees.
What is an angle for Grade 5?
In geometry, an angle can be defined as the figure formed by two rays meeting at a common end point. An angle is represented by the symbol ∠. Angles are measured in degrees, using a protractor.
What is an angle Grade 5?
How to teach students about the different types of angles?
Students will learn the characteristics of 4 different angles and use this information to identify and draw the angles. Tell students that today they will be learning about 4 different types of
angles. Begin your presentation with a right angle. Demonstrate how a right angle measures 90 degrees with a protractor.
How are the objectives of an angle lesson assessed?
The lesson objectives can be assessed by evaluating the Angle Worksheet (PDF) with the Angle Worksheet Key (PDF). Use the Assessment of Student Progress (PDF) to assess students’ overall abilities to
meet the lessons learning objectives which include identifying, drawing, and building various angles.
What’s the best way to draw angles and lines?
Give your students the Name That Angle worksheet. Instruct students to draw the following shapes: rectangle, square, triangle, trapezoid, kite, and rhombus. Have your students trace the parallel
lines on each shape in red and perpendicular lines on each shape in blue.
How to describe a right angle in geometry?
A right angle is an angle that measures 90 degrees. You should be able to describe six basic geometry terms after watching this video lesson: point, line, line segment, ray, angle and right angle. To
unlock this lesson you must be a Study.com Member. | {"url":"https://brainwritings.com/how-do-you-teach-angles-in-a-fun-way/","timestamp":"2024-11-02T20:46:12Z","content_type":"text/html","content_length":"43600","record_id":"<urn:uuid:45d7ef34-a95d-40f4-b231-44dbb475c0f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00525.warc.gz"} |
Centimetre–gram–second system of units
The centimetre–gram–second system of units (CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of
time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover electromagnetism.^[1]^[2]^[3]
The CGS system has been largely supplanted by the MKS system based on the metre, kilogram, and second, which was in turn extended and replaced by the International System of Units (SI). In many
fields of science and engineering, SI is the only system of units in use, but CGS is still prevalent in certain subfields.
In measurements of purely mechanical systems (involving units of length, mass, force, energy, pressure, and so on), the differences between CGS and SI are straightforward: the unit-conversion factors
are all powers of 10 as 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS unit of force is the dyne, which is defined as 1 g⋅cm/s^2, so the SI unit of force, the newton (1 kg⋅m/s^2), is equal to
100000 dynes.
On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields, voltage, and so on), converting between CGS and SI is less straightforward.
Formulas for physical laws of electromagnetism (such as Maxwell's equations) take a form that depends on which system of units is being used, because the electromagnetic quantities are defined
differently in SI and in CGS. Furthermore, within CGS, there are several plausible ways to define electromagnetic quantities, leading to different "sub-systems", including Gaussian units, "ESU",
"EMU", and Heaviside–Lorentz units. Among these choices, Gaussian units are the most common today, and "CGS units" is often intended to refer to CGS-Gaussian units.
The CGS system goes back to a proposal in 1832 by the German mathematician Carl Friedrich Gauss to base a system of absolute units on the three fundamental units of length, mass and time.^[4] Gauss
chose the units of millimetre, milligram and second.^[5] In 1873, a committee of the British Association for the Advancement of Science, including physicists James Clerk Maxwell and William Thomson
recommended the general adoption of centimetre, gram and second as fundamental units, and to express all derived electromagnetic units in these fundamental units, using the prefix "C.G.S. unit of
The sizes of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday objects are hundreds or thousands of centimetres long, such as humans, rooms and
buildings. Thus the CGS system never gained wide use outside the field of science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally
for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern SI standard.
Since the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide. CGS units have been deprecated in
favor of SI units by NIST,^[7] as well as organizations such as the American Physical Society^[8] and the International Astronomical Union.^[9] SI units are predominantly used in engineering
applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativistic electrodynamics, and astrophysics.^[10]^[11]
The units gram and centimetre remain useful as noncoherent units within the SI system, as with any other prefixed SI units.
Definition of CGS units in mechanics
In mechanics, the quantities in the CGS and SI systems are defined identically. The two systems differ only in the scale of the three base units (centimetre versus metre and gram versus kilogram,
respectively), with the third unit (second) being the same in both systems.
There is a direct correspondence between the base units of mechanics in CGS and SI. Since the formulae expressing the laws of mechanics are the same in both systems and since both systems are
coherent, the definitions of all coherent derived units in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units:
Thus, for example, the CGS unit of pressure, barye, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, pascal, is related to the SI base units of
length, mass, and time:
1 unit of pressure = 1 unit of force / (1 unit of length)^2 = 1 unit of mass / (1 unit of length × (1 unit of time)^2)
1 Ba = 1 g/(cm⋅s^2)
1 Pa = 1 kg/(m⋅s^2).
Expressing a CGS derived unit in terms of the SI base units, or vice versa, requires combining the scale factors that relate the two systems:
1 Ba = 1 g/(cm⋅s^2) = 10^−3 kg / (10^−2 m⋅s^2) = 10^−1 kg/(m⋅s^2) = 10^−1 Pa.
Definitions and conversion factors of CGS units in mechanics
Quantity Quantity symbol CGS unit name Unit symbol Unit definition In SI units
length, position L, x centimetre cm 1/100 of metre 10^−2 m
mass m gram g 1/1000 of kilogram 10^−3 kg
time t second s 1 second 1 s
velocity v centimetre per second cm/s cm/s 10^−2 m/s
acceleration a gal Gal cm/s^2 10^−2 m/s^2
force F dyne dyn g⋅cm/s^2 10^−5 N
energy E erg erg g⋅cm^2/s^2 10^−7 J
power P erg per second erg/s g⋅cm^2/s^3 10^−7 W
pressure p barye Ba g/(cm⋅s^2) 10^−1 Pa
dynamic viscosity μ poise P g/(cm⋅s) 10^−1 Pa⋅s
kinematic viscosity ν stokes St cm^2/s 10^−4 m^2/s
wavenumber k kayser cm^−1^[12] or K cm^−1 100 m^−1
Derivation of CGS units in electromagnetism
CGS approach to electromagnetic units
The conversion factors relating electromagnetic units in the CGS and SI systems are made more complex by the differences in the formulas expressing physical laws of electromagnetism as assumed by
each system of units, specifically in the nature of the constants that appear in these formulas. This illustrates the fundamental difference in the ways the two systems are built:
• In SI, the unit of electric current, the ampere (A), was historically defined such that the magnetic force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a
current of 1 ampere is exactly 2×10^−7 N/m. This definition results in all SI electromagnetic units being numerically consistent (subject to factors of some integer powers of 10) with those of
the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of
the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI
require an additional constant of proportionality (see Vacuum permeability) to relate electromagnetic units to kinematic units. (This constant of proportionality is derivable directly from the
above definition of the ampere.) All other electric and magnetic units are derived from these four base units using the most basic common definitions: for example, electric charge q is defined as
current I multiplied by time t, ${\displaystyle q=I\,t,}$ resulting in the unit of electric charge, the coulomb (C), being defined as 1 C = 1 A⋅s.
• The CGS system variant avoids introducing new base quantities and units, and instead defines all electromagnetic quantities by expressing the physical laws that relate electromagnetic phenomena
to mechanics with only dimensionless constants, and hence all units for these quantities are directly derived from the centimetre, gram, and second.
In each of these systems the quantities called "charge" etc. may be a different quantity; they are distinguished here by a superscript. The corresponding quantities of each system are related through
a proportionality constant.
Maxwell's equations can be written in each of these systems as:^[10]^[13]
System Gauss's law Ampère–Maxwell law Gauss's law for magnetism Faraday's law
${\displaystyle abla \cdot \mathbf {E} ${\displaystyle abla \times \mathbf {B} ^{\text{ESU}}-c^{-2}{\dot ${\displaystyle abla \cdot ${\displaystyle abla \times \mathbf {E} ^{\
CGS-ESU ^{\text{ESU}}=4\pi \rho ^{\text{ESU}}}$ {\mathbf {E} }}^{\text{ESU}}=4\pi c^{-2}\mathbf {J} ^{\text \mathbf {B} ^{\text{ESU}}= text{ESU}}+{\dot {\mathbf {B} }}^{\text
{ESU}}}$ 0}$ {ESU}}=0}$
${\displaystyle abla \cdot \mathbf {E} ${\displaystyle abla \times \mathbf {B} ^{\text{EMU}}-c^{-2}{\dot ${\displaystyle abla \cdot ${\displaystyle abla \times \mathbf {E} ^{\
CGS-EMU ^{\text{EMU}}=4\pi c^{2}\rho ^{\text {\mathbf {E} }}^{\text{EMU}}=4\pi \mathbf {J} ^{\text{EMU}}}$ \mathbf {B} ^{\text{EMU}}= text{EMU}}+{\dot {\mathbf {B} }}^{\text
{EMU}}}$ 0}$ {EMU}}=0}$
${\displaystyle abla \cdot \mathbf {E} ${\displaystyle abla \times \mathbf {B} ^{\text{G}}-c^{-1}{\dot ${\displaystyle abla \cdot ${\displaystyle abla \times \mathbf {E} ^{\
CGS-Gaussian ^{\text{G}}=4\pi \rho ^{\text{G}}}$ {\mathbf {E} }}^{\text{G}}=4\pi c^{-1}\mathbf {J} ^{\text{G}}}$ \mathbf {B} ^{\text{G}}=0} text{G}}+c^{-1}{\dot {\mathbf {B} }}^{\text
$ {G}}=0}$
${\displaystyle abla \cdot \mathbf {E} ${\displaystyle abla \times \mathbf {B} ^{\text{LH}}-c^{-1}{\dot ${\displaystyle abla \cdot ${\displaystyle abla \times \mathbf {E} ^{\
CGS-Heaviside–Lorentz ^{\text{LH}}=\rho ^{\text{LH}}}$ {\mathbf {E} }}^{\text{LH}}=c^{-1}\mathbf {J} ^{\text{LH}}}$ \mathbf {B} ^{\text{LH}}= text{LH}}+c^{-1}{\dot {\mathbf {B} }}^{\text
0}$ {LH}}=0}$
${\displaystyle abla \cdot \mathbf {E} ${\displaystyle abla \times \mathbf {B} ^{\text{SI}}-\mu _{0}\ ${\displaystyle abla \cdot ${\displaystyle abla \times \mathbf {E} ^{\
SI ^{\text{SI}}=\rho ^{\text{SI}}/\epsilon epsilon _{0}{\dot {\mathbf {E} }}^{\text{SI}}=\mu _{0}\mathbf {J} \mathbf {B} ^{\text{SI}}= text{SI}}+{\dot {\mathbf {B} }}^{\text{SI}}=
_{0}}$ ^{\text{SI}}}$ 0}$ 0}$
Electrostatic units (ESU)
In the electrostatic units variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form of Coulomb's law without a multiplying constant (and current is then defined as
charge per unit time):
${\displaystyle F={q_{1}^{\text{ESU}}q_{2}^{\text{ESU}} \over r^{2}}.}$
The ESU unit of charge, franklin (Fr), also known as statcoulomb or esu charge, is therefore defined as follows:^[14]
two equal point charges spaced 1 centimetre apart are said to be of 1 franklin each if the electrostatic force between them is 1 dyne.
Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne:
${\displaystyle \mathrm {1\,Fr=1\,statcoulomb=1\,esu\;charge=1\,dyne^{1/2}{\cdot }cm=1\,g^{1/2}{\cdot }cm^{3/2}{\cdot }s^{-1}} .}$
The unit of current is defined as:
${\displaystyle \mathrm {1\,Fr/s=1\,statampere=1\,esu\;current=1\,dyne^{1/2}{\cdot }cm{\cdot }s^{-1}=1\,g^{1/2}{\cdot }cm^{3/2}{\cdot }s^{-2}} .}$
In the CGS-ESU system, charge q is therefore has the dimension to M^1/2L^3/2T^−1.
Other units in the CGS-ESU system include the statampere (1 statC/s) and statvolt (1 erg/statC).
In CGS-ESU, all electric and magnetic quantities are dimensionally expressible in terms of length, mass, and time, and none has an independent dimension. Such a system of units of electromagnetism,
in which the dimensions of all electric and magnetic quantities are expressible in terms of the mechanical dimensions of mass, length, and time, is traditionally called an 'absolute system'.^[15]^:3
All electromagnetic units in the CGS-ESU system that have not been given names of their own are named as the corresponding SI name with an attached prefix "stat" or with a separate abbreviation
"esu", and similarly with the corresponding symbols.^[14]
Electromagnetic units (EMU)
In another variant of the CGS system, electromagnetic units (EMU), current is defined via the force existing between two thin, parallel, infinitely long wires carrying it, and charge is then defined
as current multiplied by time. (This approach was eventually used to define the SI unit of ampere as well).
The EMU unit of current, biot (Bi), also known as abampere or emu current, is therefore defined as follows:^[14]
The biot is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one centimetre apart in vacuum,
would produce between these conductors a force equal to two dynes per centimetre of length.
Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne:
${\displaystyle \mathrm {1\,Bi=1\,abampere=1\,emu\;current=1\,dyne^{1/2}=1\,g^{1/2}{\cdot }cm^{1/2}{\cdot }s^{-1}} .}$
The unit of charge in CGS EMU is:
${\displaystyle \mathrm {1\,Bi{\cdot }s=1\,abcoulomb=1\,emu\,charge=1\,dyne^{1/2}{\cdot }s=1\,g^{1/2}{\cdot }cm^{1/2}} .}$
Dimensionally in the CGS-EMU system, charge q is therefore equivalent to M^1/2L^1/2. Hence, neither charge nor current is an independent physical quantity in the CGS-EMU system.
All electromagnetic units in the CGS-EMU system that do not have proper names are denoted by a corresponding SI name with an attached prefix "ab" or with a separate abbreviation "emu".^[14]
Practical CGS units
The practical CGS system is a hybrid system that uses the volt and the ampere as the units of voltage and current respectively. Doing this avoids the inconveniently large and small electrical units
that arise in the esu and emu systems. This system was at one time widely used by electrical engineers because the volt and ampere had been adopted as international standard units by the
International Electrical Congress of 1881.^[16] As well as the volt and ampere, the farad (capacitance), ohm (resistance), coulomb (electric charge), and henry (inductance) are consequently also used
in the practical system and are the same as the SI units. The magnetic units are those of the emu system.^[17]
The electrical units, other than the volt and ampere, are determined by the requirement that any equation involving only electrical and kinematical quantities that is valid in SI should also be valid
in the system. For example, since electric field strength is voltage per unit length, its unit is the volt per centimetre, which is one hundred times the SI unit.
The system is electrically rationalized and magnetically unrationalized; i.e., 𝜆 = 1 and 𝜆′ = 4π, but the above formula for 𝜆 is invalid. A closely related system is the International System of
Electric and Magnetic Units,^[18] which has a different unit of mass so that the formula for 𝜆′ is invalid. The unit of mass was chosen to remove powers of ten from contexts in which they were
considered to be objectionable (e.g., P = VI and F = qE). Inevitably, the powers of ten reappeared in other contexts, but the effect was to make the familiar joule and watt the units of work and
power respectively.
The ampere-turn system is constructed in a similar way by considering magnetomotive force and magnetic field strength to be electrical quantities and rationalizing the system by dividing the units of
magnetic pole strength and magnetization by 4π. The units of the first two quantities are the ampere and the ampere per centimetre respectively. The unit of magnetic permeability is that of the emu
system, and the magnetic constitutive equations are B = (4π/10)μH and B = (4π/10)μ[0]H + μ[0]M. Magnetic reluctance is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits.
In all the practical systems ε[0] = 8.8542 × 10^−14 A⋅s/(V⋅cm), μ[0] = 1 V⋅s/(A⋅cm), and c^2 = 1/(4π × 10^−9 ε[0]μ[0]).
There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system.^[19] These include the Gaussian units and the Heaviside–Lorentz units.
Electromagnetic units in various CGS systems
In this table, c = 29979245800 is the numeric value of the speed of light in vacuum when expressed in units of centimetres per second. The symbol "≘" is used instead of "=" as a reminder that the
units are corresponding but not equal. For example, according to the capacitance row of the table, if a capacitor has a capacitance of 1 F in SI, then it has a capacitance of (10^−9 c^2) cm in ESU;
but it is incorrect to replace "1 F" with "(10^−9 c^2) cm" within an equation or formula. (This warning is a special aspect of electromagnetism units. By contrast it is always correct to replace,
e.g., "1 m" with "100 cm" within an equation or formula.)
Physical constants in CGS units
Advantages and disadvantages
Lack of unique unit names leads to potential confusion: "15 emu" may mean either 15 abvolts, or 15 emu units of electric dipole moment, or 15 emu units of magnetic susceptibility, sometimes (but not
always) per gram, or per mole. With its system of uniquely named units, the SI removes any confusion in usage: 1 ampere is a fixed value of a specified quantity, and so are 1 henry, 1 ohm, and 1
In the CGS-Gaussian system, electric and magnetic fields have the same units, 4π𝜖[0] is replaced by 1, and the only dimensional constant appearing in the Maxwell equations is c, the speed of light.
The Heaviside–Lorentz system has these properties as well (with ε[0] equaling 1).
In SI, and other rationalized systems (for example, Heaviside–Lorentz), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4π, those concerning
coils of current and straight wires contain 2π and those dealing with charged surfaces lack π entirely, which was the most convenient choice for applications in electrical engineering and relates
directly to the geometric symmetry of the system being described by the equation.
Specialized unit systems are used to simplify formulas further than either SI or CGS do, by eliminating constants through a convention of normalizing quantities with respect to some system of natural
units. For example, in particle physics a system is in use where every quantity is expressed by only one unit of energy, the electronvolt, with lengths, times, and so on all converted into units of
energy by inserting factors of speed of light c and the reduced Planck constant ħ. This unit system is convenient for calculations in particle physics, but is impractical in other contexts.
References and notes | {"url":"https://wikizero.com/en/Centimetre%E2%80%93gram%E2%80%93second_system_of_units","timestamp":"2024-11-04T23:47:42Z","content_type":"text/html","content_length":"219612","record_id":"<urn:uuid:389647c1-5e52-48f4-9f69-e4c66af0b666>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00233.warc.gz"} |
Differentiating 1D Integrals
Differentiating 1D Integrals #
by Shuang Zhao
In what follows, we discuss the differentiation of a simple Riemann integral $I(\theta)$ over some 1D interval $(a, b) \subseteq \real$:
$$\label{eqn:I} I(\theta) = \int_a^b f(x, \theta) \,\D x.$$
The Incomplete Solution #
The derivative of the integral in Eq. \eqref{eqn:I} with respect to $\theta$ can sometimes be obtained by exchanging the ordering of differentiation and integration:
$$\label{eqn:dI_0} \frac{\D}{\D\theta} I = \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right) \stackrel{\Large ?}{=} \int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x.$$
Precisely, the second equality in Eq. \eqref{eqn:dI_0} requires the integrand $f$ to be continuous^1 throughout the interval $(a, b)$.
Success Example #
We now provide a toy example where Eq. \eqref{eqn:dI_0} holds. Let $f(x, \theta) := x^2 \,\theta$. Consider the following integral:
$$ I = \int_0^1 (x^2 \,\theta) \,\D x. $$
Since $I = \left[ (x^3 \,\theta)/3 \right]_0^1 = \theta/3$, we know that
$$ \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{\theta}{3} \right) = {\color{blue}\frac{1}{3}}. $$
We now try calculating the same derivative $\D I/\D\theta$ using Eq. \eqref{eqn:dI_0}:
$$ \frac{\D I}{\D\theta} = \int_0^1 \frac{\D}{\D\theta} (x^2 \,\theta) \,\D x = \int_0^1 x^2 \,\D x = \left[ \frac{x^3}{3} \right]_0^1 = {\color{blue}\frac{1}{3}}, $$
which matches the manually calculated result above.
Failure Example #
We now show another toy example for which simply exchanging differentiation and integration outlined in Eq. \eqref{eqn:dI_0} fails. Let
$$\label{eqn:f_step} f(x, \theta) := \begin{cases} 1, & (x < \theta/2)\\ 1/2. & (x \geq \theta/2) \end{cases}$$
Then, for any $0 < \theta < 2$, it holds that
$$ \begin{split} I &= \int_0^1 f(x, \theta) \,\D x = \left( \int_0^{\theta/2} \D x \right) + \left( \int_{\theta/2}^1 \frac{1}{2} \,\D x \right)\\ &= \left[ x \right]_0^{\theta/2} + \left[ \frac{x}
{2} \right]_{\theta/2}^1 = \frac{\theta}{2} + \left( \frac{1}{2} - \frac{\theta}{4} \right) = \frac{1}{2} + \frac{\theta}{4}, \end{split} $$
$$\label{eqn:f_step_dI_manual} \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{1}{2} + \frac{\theta}{4} \right) = {\color{red}\frac{1}{4}}.$$
However, since the integrand $f$ is piecewise-constant in this example, we have $\D f/\D\theta \equiv 0$. Thus, Eq. \eqref{eqn:dI_0} in this example gives
$$ \int_0^1 \frac{\D}{\D\theta} f(x, \theta) \,\D x = \int_0^1 0 \,\D x = {\color{red}0}, $$
which does not match the manually calculated result in Eq. \eqref{eqn:f_step_dI_manual}.
The General Solution #
Examining The Previous Examples #
Before presenting the general expression of the derivative $\D I/\D\theta$, we first examine the examples shown above.
The Success Example #
We first examine the success example with the integrand $f(x, \theta) = x^2 \,\theta$. In the following, we show the graph of $f(x, \theta)$ for some fixed $\theta = \theta_0$ in the following: $I(\
theta_0) := \int_0^1 f(x, \theta) \,\D x$ equals the signed area (marked in light blue) of the region below the graph. Further, by adding some small $\Delta\theta > 0$ to $\theta_0$, we obtain the
graph of $f(x, \theta_0 + \Delta\theta)$ and the corresponding signed area $I(\theta_0 + \Delta\theta)$, both illustrated in red:
We recall that the derivative of $I$ with respect to $\theta$ is given by the rate at which $I$ changes with $\theta$. To calculate this rate, we examine the difference between $I(\theta_0 + \Delta\
theta)$ and $I(\theta_0)$:
$$\label{eqn:diffI0_0} I(\theta_0 + \Delta\theta) - I(\theta_0) = \int_0^1 \left(f(x, \theta_0 + \Delta\theta) - f(x, \theta_0)\right) \,\D x.$$
Geometrically, this difference equals the (signed) area of the orange region illustrated below:
At each fixed $0 < x < 1$, the integrand of Eq. \eqref{eqn:diffI0_0} satisfies that
$$ f(x, \theta_0 + \Delta\theta) - f(x, \theta_0) \approx \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta. $$
Base on this relation, we can rewrite the area difference \eqref{eqn:diffI0_0} as:
$$ I(\theta_0 + \Delta\theta) - I(\theta_0) \approx \int_0^1 \left( \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta \right) \D x = \Delta\theta \int_0^1 \left[ \frac
{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \D x. $$
In both equations above, the equalities become exact at the limit of $\Delta\theta \to 0$. By dividing both sides by $\Delta\theta$ and taking the limit of $\Delta\theta \to 0$, we have
$$ \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0} := \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta} = \int_0^1 \left[ \frac{\D}{\D\theta} f
(x, \theta) \right]_{\theta = \theta_0} \D x, $$
for any $0 < \theta_0 < 1$. This agrees with the incomplete solution expressed in Eq. \eqref{eqn:dI_0}.
The Failure Example #
So what has been the cause for the failure example? To be specific, what has been missing from the incomplete solution \eqref{eqn:dI_0}?
To understand what has been going on, we again examine the integrand $f(x, \theta)$ which, for this example, is the piecewise-constant function defined in Eq. \eqref{eqn:f_step}.
The following are the graphs of $f(x, \theta)$ for some fixed $\theta = \theta_0$ and $\theta = \theta_0 + \Delta\theta$ (for some small $\Delta\theta > 0$), respectively:
Further, the difference $I(\theta_0 + \Delta\theta) - I(\theta_0)$ between the signed areas below the two graphs is caused by the rectangle illustrated in orange:
Intuitively, in the success example, the change of signed area is caused by vertical shifts of the graph—which is captured by the incomplete solution \eqref{eqn:dI0}. On the other hand, in this
failure example, the change of signed area is caused by horizontal shifts of the graph _at jump discontinuities—which is missing from the incomplete solution!
We now calculate the signed area of the orange rectangle shown above. We first observe that the length of the rectangle’s vertical edge equals the difference $\Delta f \equiv 1 - 1/2 = 1/2$ of the
integrand $f(x, \theta)$ across the discontinuity point.
To calculate the length of the rectangle’s horizontal edge, we let $x(\theta) = \theta/2$ denote the jump discontinuity point of $f(x, \theta)$ defined in Eq. \eqref{eqn:f_step}. Then, the (signed)
length of the horizontal edge is simply $x(\theta_0 + \Delta\theta) - x(\theta_0)$.
Based on the observations above, we know that
$$ I(\theta_0 + \Delta\theta) - I(\theta_0) = \Delta f \,(x(\theta_0 + \Delta\theta) - x(\theta_0)). $$
Dividing both sides of this equation by $\Delta t$ and taking the limit $\Delta\theta \to 0$ produce:
$$ \begin{split} \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0} &= \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta}\\ &= \Delta f \,\lim_{\
Delta\theta \to 0}\frac{x(\theta_0 + \Delta\theta) - x(\theta_0)}{\Delta\theta} = \Delta f \left[ \frac{\D}{\D\theta} x(\theta) \right]_{\theta = \theta_0}. \end{split} $$
Therefore, we know that
$$ \frac{\D}{\D\theta} I(\theta) = \underbrace{\Delta f}_{=\, 1/2} \; \underbrace{\frac{\D}{\D\theta} x(\theta)}_{=\, 1/2} = {\color{red}\frac{1}{4}}, $$
matching the hand-derived result in Eq. \eqref{eqn:f_step_dI_manual}.
The Full Derivative #
Based on the observations above, we now present the general derivative of the 1D integral expressed in Eq. \eqref{eqn:I}:
$$\label{eqn:dI} \boxed{ \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right) = \underbrace{\int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x}_{\text{interior}} \,+\, \
underbrace{\sum_i \Delta f(x_i(\theta), \theta) \,\frac{\D}{\D\theta} x_i(\theta)}_{\text{boundary}}\,, }$$
which comprises:
• A interior component obtained by exchanging differentiation and integration operations—identical to Eq. \eqref{eqn:dI_0}.
• A boundary component involving a sum over all jump discontinuity points $\{ x_i(\theta) : i = 1, 2, \ldots \}$.
Remarks #
Precisely, $\Delta f(x, \theta)$ in the boundary component is defined as
$$ \Delta f(x, \theta) := \lim_{u \uparrow x} f(u, \theta) - \lim_{u \downarrow x} f(u, \theta), $$
where $\lim_{u \uparrow x}$ and $\lim_{u \downarrow x}$ denote one-sided limits with $u$ approaching $x$ from below (i.e., $u < x$) and above (i.e., $u > x$), respectively. For any fixed $\theta$, $\
Delta f(x, \theta)$ is nonzero (and well-defined) if and only if $x$ is a jump discontinuity point of $f(\cdot, \theta)$.
Lastly, when the endpoints $a$ and $b$ of the integral depend on $\theta$, they should be considered as jump discontinuities with $\Delta f(a, \theta) = -f(a, \theta)$ and $\Delta f(b, \theta) = f(b,
In the next section, we will present a generalization of Eq. \eqref{eqn:dI} that describes derivatives of Lebesgue integrals.
1. Unless otherwise stated, we use “continuous” to indicate the $C^0$ class. ↩︎ | {"url":"https://diff-render.org/docs/diff-render/basics/diff-int-1d/","timestamp":"2024-11-07T20:17:06Z","content_type":"text/html","content_length":"19699","record_id":"<urn:uuid:14deacd3-bd24-4414-a31d-da8d06a79387>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00638.warc.gz"} |
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Class 15: Public-Key Revolution (Diffie-Hellman Key Exchange)
Overview: In this part we introduce the idea of asymmetric encryption schemes in which the receiver has a public key and a private key. The public information allows anyone to encrypt a message that
the receiver can decrypt using their symmetric key. This is very different than using the same key to encrypt and decrypt.
So it's time to shift into a new mode of thinking. We'll end up talking about computational number theory and some beautiful mathematics. For now, let's jump in and start doing it.
First of all you'll notice that every one of our encryption schemes so far has had a shared secret. That is, some key which both encrypter and decrypter need for the scheme to succeed. How would that
secret key be shared with your partner? Briefcases and spies have done the job before, but we're in the internet age.
Today I want to look at a method for sharing a private key over a public channel (our class chat for example). I imagine yelling out this information to the whole room and letting our partners work
with us to get a secret shared.
A full conversation involves having four keys, a public and private pair for both Alice (sender) and Bob (recipient).
We won't yet understand every step of this process, but we'll have a good place to start our conversations.
The hard problem
Every public-key scheme is going to involve a "hard problem" that would crack the scheme. Ideally the designer of the scheme has set it up so that either your messages are secure OR the attackers
have made a practical solution to a very tough computer science / math problem.
The first hard problem we face is called the Discrete Log problem. Which is this: Given \(A = g^{\alpha} \pmod{p}\) and \(g\) and \(p\) find \(\alpha\).
Discrete Log Starter: Suppose \(22 = 3^{x} \pmod{31}\). Find \(x\).
Discrete Log Thinking: If you were to solve a larger version, can you tackle it any easier than trying every value of \(x\)?
Overview: This part shows the details of publicly exchanging a private key using the Diffie-Hellman Key Exchange. The world uses this idea to setup all of our HTTPS connections. Once that key is
exchanged we just switch to symmetric-key encryption with AES. So the public-key part is mostly just setting up the crypto we already know. Let's practice the transition.
This idea is important enough that we should try several approaches. Pay attention to which parts are secret and which parts are public.
1) Generate a public triplet
As the initiator (ALICE) we follow these steps to generate a discrete log problem and an answer (which we keep secret).
1. Generate a "Strong" prime (script below) \(p\)
2. Pick a "base" which can really just be the number \(g = 2\)
3. Generate a PRIVATE random number, \(a\), which shares no factors with \(p-1\) (recipe below)
4. Calculate the public exponent: \(A := g^a \pmod{p} \).
5. Publish your public key (triplet): \(p, g, A\) (DO NOT PUBLISH \(a\)!)
Generate a Public Key Triplet: also store the private key somewhere. Publish your triplet in the chat. You'll have to generate several private keys until the GCD is 1, make a while loop there.
The security of these scheme rests on an attacker's inability to figure out \(a\) given \(p, g, A\).
2) BOB's job: the recipient's role
Once Bob has your public-key triplet he can generate a new secret that can be publicly shared with ALICE.
Bob takes in \(p\), \(g\), and \(A\). Now Bob has to also generate a secret:
1. Generate a PRIVATE random number, \(b\), which shares no factors with \(p-1\).
2. Calculate the PUBLIC number \(B = g^b \pmod{p} \).
3. Calculate the PRIVATE shared secret \(K = A^b \pmod{p} \), note that this is really the number \(g^{ab} \pmod{p} \).
4. Publish to Alice: \(B\).
Be BOB: Take in the triplet \( (p, g, A) = (101, 2, 6) \) and generate a response \(B\) and a shared secret. (Little \(a\) was 70 if you want to confirm that you've got the mechanics down.)
3) Our prize: a shared secret
1. ALICE receives \(B\) (she doesn't know \(b\))
2. ALICE computes \(K = (B)^{a} \pmod{p} \) which BOB already knows.
3. ALICE and BOB rejoice in their sneaky cleverness to have shouted information and secretly communicated.
The secret sauce was that anyone can see \(A = g^a \pmod{p}\) and not know \(a\), anyone can see \(B = g^b \pmod{p}\) and now know \(b\). Now only ALICE can compute \(B^a \pmod{p}\) and only BOB can
compute \(A^b \pmod{p}\) since only ALICE knows \(a\) and only BOB knows \(b\). But \(B^a = (g^b)^a = g^{ab} \pmod{p} \) and \(A^b = (g^a)^b = g^{ab} \pmod{p} \).
Do a full exchange: Pretend to be Alice and Bob and generate a shared key. You could also do a swap in the chat channels.
CTF Problem for DHKE:
Overview: In this part we accomplish two things. Get the idea across that after a key exchange we can switch to AES. The other goal is to practice full strength Key swaps using real tools. Let's do
Preface: Use private key
When you've done this key exchange you end up with a shared secret which is likely to have more bits than the typical AES secret key (we need more bits for public-key security than private-key
So at this stage I suggest you switch to AES in an appropriate mode (like CTR). For converting your shared secret into the key for AES you can use a hashing algorithm on the resulting bytes.
Generating Safe Primes
OK, so we've done some amateur DHKE, some basic number theory, and we're finally ready to explore the way this is really done.
Analysis question: We know that the hard problem is reversing \(g^{x} \pmod{p}\). If you were choosing the \(g\) to use would you rather have \(|g| = p-1\) or \(|g| = 2\)?
Create a generator: If I gave you a prime \(p\) and asked you to give me a generator of the multiplicative group \(\mathbb{Z}_p^{*}\) what would you do?
In the last part we used a library to generate a safe prime. Here is a look under the hood.
Safe Prime Explore: the answer that is done most in practice is to generate a safe prime, that is, \(p = 2\cdot q + 1\) where \(q\) is also prime. How would you generate a safe prime?
So the big idea is that if we find a 'safe prime' then we can find a generator with great ease. This is because there are very few subgroups in \(\mathbb{Z}_p^{*}\) just a subgroup of order 2, order
\(q\), and order \(p-1\). That way if we just avoid the small subgroup we know we've got a brute-force space of at least size \(q\)!
Here is a safe-prime generating snippet:
SAGE Run it: That is some SAGE code, so run it at cloud.sagemath.com and see how long it takes when bits is small and now try it with bits at 1024 (the smallest "safe" prime size for DH). WARNING: be
prepared to stop the process!
Working with SSL parameters
Let's learn how to leave it to the professionals:
OpenSSL: in a cloud9 run the following to generate a DH strength prime and a generator of the group \(\mathbb{Z}_p^{*}\): openssl dhparam -out dh1024.pem 1024 (marvel at the speed).
Interpret it two ways: the first way to read it is openssl asn1parse -in dh1024.pem
By hand using Base64: now that prime is stored in .pem format we can get access to it by reading base64. import base64 and run base64.standard_b64decode on the parts that matter. This gives you raw
bytes. Your prime is at raw[6:6+129]. Get these bytes, convert to hex then an integer. The generator is probably the last byte (normally 2).
Confirm: use sage to confirm that \(p\) and \((p-1)/2\) are both prime.
Overview: This is a very useful trick for computational math, but it also unleashes a brutal attack on Diffie-Hellman and the general discrete log problem.
OK let's play a game. A random number is picked and you can only learn the remainder of that number modulo single digit moduli. Your goal is to find the number in the fewest number of guesses, here
Once we're done with that let's talk CRT, the Chinese Remainder Theorem.
There is a perfect mapping from \(\mathbb{Z}_N \leftrightarrow \mathbb{Z}_{q_1} \times \cdots \mathbb{Z}_{q_k}\) whenever the pairwise GCD of \(q_i\) is 1 and \(N = q_1 \cdots q_k\).
Here is an example:
Given that \(x \equiv 2 \mod{5}, x \equiv 1 \mod{3}\) then \(x\) must be equivalent to \(7 \mod{15}\).
SAGE Cloud: You could code your own in Python (like https://rosettacode.org/wiki/Chinese_remainder_theorem ) but as we move into Number Theory I want to show you Python with super-powered math. Head
to https://cloud.sagemath.com . Do the command CRT? once you've made a sage worksheet. Now compute the smallest positive number which is \(3 \mod{9}, 8\mod{13}, 6\mod{25}, 36\mod{121}\).
There are many great applications of the CRT, but in our case we're going to attack all of these schemes we've established when careful primes are not selected.
A Simple CRT Flag
Overview: This attack is important to understand because it doesn't show itself just by size of the key. You have to have more cleverness because of this attack so pay close attention.
I want to introduce a feasible attack on the discrete log problem. That is, given \(A := \alpha^x \mod{p}, p, \alpha\) find \(x\).
This attack will teach us about what makes a prime strong enough, cyclic groups, and the Chinese Remainder Theorem.
Real problem to solve: We are going to solve the following discrete log: \(p= 125301575591,\alpha = 115813337451, \alpha^x \mod{p} = 73973989900\). Compute \(x\). (Solve this after working the small
example below.)
Now here is what we want to do. The problem is, given \(p, g, h:= g^x \mod{p}\) find \(x\).
The big idea is this, the multiplicative subgroup of integers mod \(p\) has size \(p-1\). If we know the factors of \(p-1\) (and they are all small) then we can convert this into a smaller problem.
Imagine that \(q | p-1\) then \((g^{(p-1)/q})^x = h^{(p-1)/q}\) is another equation involving \(x\) but now the possible answers for \(x\) aren't mod \(p-1\) but are mod \(q\).
A small worked example
Given \(p = 31, g = 3, h = 26 = g^x\) find \(x\).
We could try every value of \(x\) from 1 to 30 until we got 26 mod 31. In this case that would be cheap and not a problem, BUT it won't scale to the larger problem.
So we start by factoring \(p-1 = 30 = 2 \cdot 3 \cdot 5\). We will convert the discrete log problem into a three smaller problems that we can Chinese Remainder to find the final solution.
Start with \(q = 2\) which divides \(p-1\). Since we are looking for \(x\) which satisfies the relationship that \(3^x \equiv 26 \pmod{31}\) then if we replace \(3\) and \(26\) by \(3^{15}\) and \(26
^{15}\) then we'll get another relationship \(3^{15x} \equiv 26^{15} \pmod{31}\).
If we look at this a little deeper we know that \(3^{30} \equiv 1 \pmod{31}\) based on what we know about cyclic groups. So this new relationship actually only gives us an answer mod 2. Here's what I
mean. Suppose \(x\) were odd, then \(3^{15x}\) is exactly equivalent to \(3^{15}\) and if \(x\) is even then \(3^{15x}\) is always equivalent to \(1\). So either \(3^{15x}\) is equivalent to \(3^{15}
\) or it is equivalent to \(1\). That means that we have learned a solution to the equation \(x \equiv r \pmod{2}\).
In this case we just check \(26^{15} \pmod{31}\) and we get \(30\) which matches \(3^{15} \pmod{31}\).
So we know that \(x \equiv 1 \pmod{2}\).
Now let's try \(q = 5\). We raise both \(3\) and \(26\) to the \((p-1)/5\)-th power. We get \(3^{6x} \equiv 26^{6} \equiv 1 \pmod{31}\). Now try every remainder of \(x \pmod{5}\) until we find the
right power.
\(3^{0} \equiv 1, 3^6 \equiv 16, 3^{12} \equiv 8, 3^{18} \equiv 4, 3^{24} \equiv 2 \pmod{31}\), so we now know that \(x \equiv 0 \pmod{5}\) and that \(x \equiv 1 \pmod{2}\) which tells us that \(x \
equiv 5 \pmod{10}\).
Why factors? Take a look at the results of the following quick loop. The number of 1s is 1 when we have a generator of \(Z_p^{*}\). What is the pattern?
for j in range(1, 31):
print j, [pow(3**j,i, 31) for i in range(30)].count(1)
Now you try: Using the last prime factor, 3, raise both \(3\) and \(26\) to the \( (p-1)/3 \) and deduce the remainder of \(x \pmod{3}\), and using all three clues deduce the value of \(x \pmod{30}
\). x is 2 mod 3, so we know that x is actually 5 to solve the problem mod 31.
Now write a program to solve the larger problem we opened with. SAGE (or even Wolfram Alpha) can factor \(p-1\) for you.
Overview: It's very important that you feel nervous when using public key crypto. The primes that you work with have to be picked carefully otherwise advanced attacks will undo you. So to deploy this
with confidence you need to know those best attacks and how to thwart them.
The follow up to Pohlig-Hellman is that even a large prime can fall if every factor of \(p-1\) is small.
The same is true when we get to the elliptic curve world.
So you must pick primes where \(p-1\) has at least one large prime factor.
We can use PyCrypto to generate strong primes, and when it comes to elliptic curves we can analyze the parameters of the chosen curves.
Almost Live CTF Problem:
This weekend the highest point crypto problem from one of them was the following pcap file: | {"url":"https://crypto.prof.ninja/class15/","timestamp":"2024-11-11T17:55:14Z","content_type":"text/html","content_length":"19418","record_id":"<urn:uuid:2fee588b-5749-48f0-8643-09d69f274085>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00804.warc.gz"} |
John Siegenthaler: Important differences between head and Delta-P
This month I want to deviate a bit from a purely renewable energy topic to one that’s important across the entire spectrum of hydronics technology. It’s a topic that likely gets exercised on a daily
basis in any engineering office where water-based HVAC systems are conceived.
Those who have worked with the technical side of hydronics have likely used terms such as: Pressure, differential pressure, head and head loss. These terms all have legitimate and specific meanings.
But when some of these words get scrambled into jargon, the result can be an undefined or meaningless term. One example of such a scramble is “head pressure.”
To see why the phrase “head pressure” is not a valid technical term, it’s important to understand both words.
Let’s start with head. Although I don’t know who coined this word for use in fluid mechanics, I do know where the concept that the word represents came from. It’s attributable to a man named Daniel
Bernoulli. In 1738, he published a book entitled “Hydrodynamica,” which presented a concept that is now embodied in what’s appropriately called Bernoulli’s equation. Anyone who has studied fluid
mechanics has surely come across this equation. It provides the basis for analyzing a wide range of situations, such as a pipeline carrying water from a reservoir to a city to the shape of airplane
wings to — you guessed it — fluid flow in hydronic systems.
Fundamentally, Bernoulli’s equation describes the mechanical energy present in a fluid and how that energy can be transformed as the fluid changes height, pressure and speed. The “head” of a fluid is
simply the total mechanical energy contained in that fluid.
In the case of a closed-loop hydronic system, head energy is added to the fluid by a circulator. Everything else the fluid flows through — piping, fittings, valves, heat emitters, etc. — removes
head energy from the fluid due to the friction present between moving fluid molecules, as well as between those molecules and the surfaces they come in contact with. This concept is illustrated in
Figure 1.
Invisible but real
We can’t “see” energy. We can’t see it with our bare eyes, or under a microscope. Think about it — have you ever seen a Btu, a Kilowatt•hour or a Joule of energy?
Neither have I.
Although we can’t see it directly, we can still detect when energy is added to or removed from a material. For example, consider water flowing into a boiler at 140° F, and leaving that boiler at 155°
F. If the piping into and out of the boiler were transparent, the water coming out would look identical to the water going in. Yet we know there’s more thermal energy in 155° water compared to 140°.
The indicator of that additional thermal energy is a temperature rise. When thermal energy is added to a material (and the material doesn’t change phase between being a solid, liquid or gas), the
temperature of that material increases. Conversely, when thermal energy is removed from a material, and the material remains in the same phase, its temperature decreases. Thus, a change in
temperature is the “evidence” that thermal energy, which we can’t directly see — has been added to or removed from the material.
When it comes to head energy and pressure, there’s an analogy to the relationship between thermal energy and temperature. A decrease in pressure along a horizontal piping path is the “evidence” that
head energy has been removed from a liquid. An increase in pressure is the “evidence” that head energy has been added to the liquid. We can’t see head energy, but we can detect it being added or
removed from a liquid by measuring changes in pressure. The constraint of a horizontal piping path in the above statement is to eliminate any pressure change in the fluid due to elevation change. The
drop in pressure due to head loss is still present in non-horizontal piping, but it would not reveal itself as the “sole” cause of the pressure difference between two points along a pipe.
Why feet?
In North America, the unit used to express head energy is “feet.” The word “feet” has undoubtedly caused a lot of confusion over the years. I know it confused me for a while. Why would energy be
expressed in units that are commonly used for distance? Here again, past practices have prevailed.
The unit of feet, abbreviated as ft. comes from a mathematical simplification of the units shown in Figure 2.
This arrangement of units would be properly stated as “foot-pounds per pound.”
The unit foot-pound, abbreviated as ft•lb is a valid unit of energy. As such, it can be converted into any other valid unit of energy. For example: 1 ft•lb = 0.0012850675 Btu.
Consider water flowing through an operating circulator. In this situation, the arrangement of units in Figure 2 can be interpreted as the number of ft•lb of mechanical energy added to each pound of
water passing through the circulator. Thus, a circulator that happens to be operating at say 10 feet of head is adding 10 ft•lb of mechanical energy to each pound of water passing through the
So why don’t we say it that way (e.g., the circulator is adding 10 ft•lb of mechanical energy to each pound of water passing through it?) It’s because mathematically, the units of lb in the top of
the fraction of Figure 2 cancels out with the unit of lb in the bottom of the fraction, and thus the only remaining unit is ft. It’s shorter to just state head in feet rather than fl•lb/lb.
If I had a seat at the table when this simplification became the “standard” in the industry, probably back sometime in the 1800s, my vote would have been to keep it ft•lb/lb. It’s longer, but it
better represents the concept of energy per unit weight of liquid.
The relationship
So how does one determine the amount of head energy added or removed from a fluid based on observed change in pressure? Answer: Use Formula 1:
Formula 1:
H = head (added or removed) in units of (feet);
∆P = change in pressure in units of (psi);
D = density of the fluid in units of (lb/ft3); and
144 = a number needed for the units to work correctly.
Formula 1 can be used to calculate the head energy added to the fluid when a pressure increase occurs — such as a pressure increase measured across an operating circulator. The formula can also be
used to calculate the head energy removed from the fluid when a pressure decrease occurs, such as across any component or group of components connected together in the circuit.
Water at 60° F has a density of about 62.4 lb/ft3. This makes the fraction of (144/62.4) equal to approximately 2.31. However, the density of water changes significantly with temperature. The density
of other liquids, such as solutions of glycol-based antifreeze, is also different from that of water, and also dependent on fluid temperature. Thus, stating that the head energy exchanged is 2.31
times the pressure change is only an approximation. It gets you in the ballpark, but the best accuracy is still attained when you use Formula 1 along with the density of the fluid. In a hydronic
circuit, you can determine the density based on the average temperature of the liquid flowing through that circuit.
Slinging slang
So back to the jargon of head pressure. Based on what we just discussed, these two words, paired together, are analogous to heat temperature, a term that has no meaning in our industry, or any other
Our industry uses plenty of jargon. For example, we might say that the output of a boiler is 80,000 Btu, when what we mean is 80,000 Btu/h. We might state that the electrical energy used by a small
circulator is 50 watts, when what we really mean is that the power demand of the circulator is 50 watts. We may describe a circulator operating with a head pressure of 10 feet when what we mean is a
head of 10 feet.
Jargon is usually acceptable, and perhaps even a bit “admirable” when describing hardware. For example, how many mere mortals know what is meant by a blind flange, street ell or bullhead tee?
However, when learning to manage the physics that determines how the system operates, jargon often clouds understanding. That leads to uncertainty, lowered confidence and even finger crossing when
making design decisions. We’ve all been there at times, and it makes us (or should make us) uncomfortable.
After reading this, some may be thinking that I’m “nitpicking” about words that most of us already sort of understand. Why not be specific and take away the words “sort of” in the previous sentence.
It’s precise and professional. | {"url":"https://www.pmmag.com/articles/106090-john-siegenthaler-important-differences-between-head-and-delta-p","timestamp":"2024-11-05T23:35:42Z","content_type":"text/html","content_length":"101630","record_id":"<urn:uuid:c64321bb-554a-41ad-83f1-851d5ee503d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00389.warc.gz"} |
Total Insert
[Create Worksheet]
What is the sum of these integers?
1. Read the given question.
2. Solve the problem mentally or by using a scratch paper.
3. Enter the answer you got.
EZSchool^ ® is federally registered and protected trademark. Copyright @ 1998-2024 Asha Dinesh | {"url":"https://www.ezschool.com/Games/Grades7-12/Math/Integers/AddIntegers/TotalInsert/","timestamp":"2024-11-07T20:38:05Z","content_type":"application/xhtml+xml","content_length":"15250","record_id":"<urn:uuid:17b62d29-9a99-4b1a-9009-578cc23aa3b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00308.warc.gz"} |
Reference Database
Search Site
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Prime Curios!
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Submit primes This is the Prime Pages' interface to our BibTeX database. Rather than being an exhaustive database, it just lists the references we cite on these pages. Please let me know of any
errors you notice.
References: [ Home | Author index | Key index | Search ]
All items with author Hardy (sorted by date) | {"url":"https://t5k.org/references/refs.cgi?author=Hardy","timestamp":"2024-11-07T02:58:54Z","content_type":"text/html","content_length":"5646","record_id":"<urn:uuid:3638a6f5-e112-4288-b353-3550ab2a880c>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00572.warc.gz"} |
Algebra 2 with TI-nspire: Semester 1
by Brendan Kelly (Author), Teresa Kelly (Editor), Taisa Dorney (Illustrator), Michelle Junkin (Illustrator), Chris Francis (Illustrator)
Perfect Paperback: 224 pages
Publisher: Brendan Kelly Publishing Inc.; First edition (February 22, 2010)
Language: English
ISBN-10: 1895997321
ISBN-13: 978-1895997323
This book develops the first half of a full course in Algebra 2 using the TI-nspire technology (version 2). The topics are introduced in a highly motivational way, using cartoons, human interest
items, and real-world applications that connect the mathematics to the student's interests. Each lesson begins with a motivational introduction, followed by worked examples that showcase the
solutions with and without the TI-nspire technology. Keying sequences are provided for both TI-nspire and TI-nspire CAS. Each unit begins with a display of the TI-nspire menus that will be used
throughout the unit. Only the appropriate menus and commands are employed so that the mathematics is not subordinated to the technological procedures. There are complete solutions to all the
The table of contents is as follows:
Unit 1: Numbers & Number Systems Unit Preview: The Home Menu The Fundamental Theorem of Arithmetic Greatest Common Divisor & Least Common Multiple From Integers to Rational Numbers The Discovery of
Irrational Numbers The Golden Ratio & Rational Approximations Complex Numbers: Rectangular Form Complex Numbers: Operations
Unit 2: Sequences, Series & Functions Unit Preview: Defining & Graphing Sequences Using Formulas to Define Sequences Using Recursion to Define Sequences The Sum of an Arithmetic Series The Sum of a
Geometric Series Sums of Infinite Series Applications of Sequences: Future Value Applications of Sequences: Present Value
Unit 3: Matrices Unit Preview: Matrices & Matrix Transformations Matrices & Matrix Addition Products of Matrices Matrix Transformations Successive Transformations The Determinant of a Matrix The
Inverse of a Matrix
Unit 4: Linear Systems Unit Preview: Six Ways to Solve Linear Systems Solving a Linear System Using a Table or Graph Solving a Linear System Using Algebra Analyzing 2 x 2 Linear Systems Solving
Systems of Linear Inequalities Linear Programming Solving 3 x 3 Linear Systems
Unit 5: Quadratic Functions & Equations Unit Preview: Five Ways to Solve a Quadratic Equation Quadratic Growth: From Tables to Graphs Analyzing Quadratic Functions in Vertex Form Analyzing Quadratic
Functions in Standard Form The Roots of a Quadratic Equation Quadratic Inequalities Using Quadratic Functions to Model Data
Unit 6: Polynomials & Polynomial Equations Unit Preview: The Max, Min & Solve Commands From Monomials to Polynomials Products & Powers of Polynomials Factoring Polynomials The Remainder Theorem & The
Factor Theorem The Fundamental Theorem of Algebra Transforming Polynomial Functions Modeling Data with Polynomial Functions Answers to the Exercises Learning Outcomes for Algebra 2 with TI-nspire:
Semester 1 Learning Outcomes for Algebra 2 with TI-nspire: Semester 2 TI-nspire Functions & Programs
By Barbara Henley
Rating: ★★★★★
I have several of Brendan Kelly's books and am familiar with other works of his in the educational arena. He is by far one of the best educators I have seen, and I come from a huge family of
educators. His books would be well worth the price-even at double what he charges. The books are written plainly with step by step directions, and is written in a humorous, fun, and an easy to
understand way. I highly recommend any of his books.
By Margaret Thornton
Rating: ★★★★★
Brendan Kelly has written some of the finest math books ever seen. Whether you are a student, teacher, or a casual reader you need these books. There are new interesting things on every page. They
would be worth reading even without the aid of a TI Nspire CAS. With it they are dynamite.
Related Products
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Increase productivity with TI-Nspire CX graphing calculator that provides algebraic capability to symbolically solve equations, factor & expand variable expressions, complete the square, find
antiderivatives & exact solutions in irrational forms. | {"url":"https://www.calculatordeal.com/algebra-2-with-ti-nspire-semester-1/","timestamp":"2024-11-08T20:30:46Z","content_type":"text/html","content_length":"34207","record_id":"<urn:uuid:26ee2a55-e61e-46d5-876b-5fdbe5d67bcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00523.warc.gz"} |
Proof of Step 1
A given point a[0] on the unit sphere E uniquely picks out a unit vector from the origin to a[0] which in turn uniquely picks out a ray in R3 through the origin and a[0]. We here work with unit
vectors, since this involves no loss of generality. We write a[0], a[1], … for points and u(a[0]), u(a[1]), … for the corresponding unit vectors. We call a KS diagram realizable on E, if there is a
1:1 mapping of points of E, and thus of vectors in R3, to vertices of the diagram such that the orthogonality relations in the diagram — namely, vertices joined by a straight line represent mutually
orthogonal points — are satisfied by the corresponding vectors.
We now show (see Kochen and Specker 1967: , Redhead 1987: 126):
If vectors u(a[0]) and u(a[9]), corresponding to points a[0] and a[9] of the following ten-point KS graph Γ[1] are separated by an angle θ with 0 ≤ θ ≤ sin^−1(1/3), then Γ[1] is realizable.
Figure 4: Ten-point KS graph Γ[1]
Suppose that θ, the angle between u(a[0]) and u(a[9]), is any acute angle. Since u(a[8]) is orthogonal to u(a[0]) and u(a[9]), and u(a[7]) also is orthogonal to u(a[9]), u(a[7]) must lie in the plane
defined by u(a[0]) and u(a[9]). Moreover, the direction of u(a[7]) can be chosen such that, if φ is the angle between u(a[0]) and u(a[7]), then φ = π/2 − θ.
Now, let u(a[5]) = i and u(a[6]) = k and choose a third vector j such that i, j, k form a complete set of orthonormal vectors. Then u(a[1]), being orthogonal to i, may be written as:
u(a[1]) = (j + xk) (1 + x^2)^−½
for a suitable real number x, and similarly u(a[2]), being orthogonal to k, may be written as:
u(a[2]) = (i + yj) (1 + y^2)^−½
for a suitable real number y. But now the orthogonality relations in the diagram yield:
u(a[3]) = u(a[5]) × u(a[1]) = (-xj + k) (1 + x^2)^−½
u(a[4]) = u(a[2]) × u(a[6]) = (yi − j) (1 + y^2)^−½
Now, u(a[0]) is orthogonal to u(a[1]) and u(a[2]), so:
u(a[0]) = u(a[1]) × u(a[2]) / ( | u(a[1]) × u(a[2]) | ) = (-xyi + xj − k) (1 + x^2 + x^2 y^2)^−½
Similarly, u(a[7]) is orthogonal to u(a[3]) and u(a[4]), so:
u(a[7]) = u(a[4]) × u(a[3]) / ( | u(a[4]) × u(a[3]) | ) = (-i − yj − xyk) (1 + y^2 + x^2 y^2)^−½
Recalling now that the inner product of two unit vectors just equals the cosine of the angle between them, we get:
u(a[0]) u(a[7]) = cos φ = xy[(1 + x^2 + x^2 y^2) (1 + y^2 + x^2 y^2)]^−½
sin θ = xy[(1 + x^2 + x^2 y^2) (1 + y^2 + x^2 y^2)]^−½
This expression achieves a maxium value of 1/3 for x = y = ±1. Hence, the diagram is realizable, if 0 ≤ θ ≤ sin^−1(1/3), or, equivalently if 0 ≤ sin θ ≤ 1/3. | {"url":"https://seop.illc.uva.nl/entries/kochen-specker/step1.html","timestamp":"2024-11-11T23:14:50Z","content_type":"text/html","content_length":"17290","record_id":"<urn:uuid:a697d083-4277-4fe1-bd09-dc71d5390314>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00168.warc.gz"} |
Ternary Aspects
Additional aspects used in astrology, although less common, include the 7th-Harmonic, 9th-Harmonic, 10th-Harmonic, 11th-Harmonic, 14th-Harmonic, 16th-Harmonic, and 24-Harmonic aspects. These aspects
are derived from dividing the circle of the Zodiac into segments of seven, nine, ten, eleven, fourteen, sixteen, and twenty-four, respectively.
The septile (abbreviated as Sep) represents an angle of approximately 51.428571°, which is one-seventh of the circle of the zodiac. When aspected, it is believed to signify irrational relations
between its constituent components but also reveal the hidden underlying nature and deeper destiny of those components.
The triseptile (abbreviated as TSp) corresponds to an angle of approximately 154.285714°, which is roughly 3/7 of the circle of the zodiac.
The novile (abbreviated as Nov), also referred to as a nonagon, corresponds to an angle of 40°, which is one-ninth of the circle of the zodiac. It is believed that the novile represents a
constriction between aspects that can be unlocked and utilized as a catalyst for self-enhancement.
The binovile (abbreviated as BNv) corresponds to an angle of 80°.
The quadnovile (abbreviated as QNv), also called a quadrinovile, represents an angle of 160°.
The decile, also referred to as a semi-quintile, corresponds to an angle of 36°, which is one-tenth of the circle of the zodiac. This aspect is believed to bestow the ability to assist others.
The tridecile, alternatively called the sesquiquintile, represents an angle of 108°. This aspect is believed to imbue individuals with social creativity or evoke a sense of withdrawal and
introspection necessary for external originality. It is also referred to as the quintile-and-a-half.
The vigintile, also known as the semi-decile, corresponds to an angle of 18°, which is one-twentieth of the circle of the zodiac, or half of a decile.
The undecile (abbreviated as Und), also referred to as the undecim, represents one-eleventh of the zodiac circle or an angle of approximately 32.727272° (32°43’38”). Additionally, there are the
biundecile (also known as biundecim) at 65.454545° (65°27’16”), the triundecile (also known as triundecim) at 98.181816° (98°10’55”), the quadriundecile (also known as quadriundecima) at 130.90909°
(130°54’33”), and the quinqueundecile (also known as quinqueundecim) at 163.636363° (163°38’11”). These represent, respectively, two-elevenths, three-elevenths, four-elevenths, and five-elevenths of
the zodiac circle. The undecile is associated with social consciousness and the ability to seek help beyond oneself.
The semiseptile, also known as quattuordecimal (derived from the Latin word for fourteen, quattuordecim), corresponds to an angle of approximately 25.714286° (or 25°42’51” in degree-minute-second
format). It represents one-fourteenth of the circle of the zodiac, or half of a septile. This aspect is believed to involve relinquishing what has been completed in order to transition to the next
cycle of activity.
The tresemiseptile, also referred to as the sesquiseptile, corresponds to an angle of approximately 77.142858° (or 77°08’34”). This angle represents three-fourteenths of the circle of the zodiac.
The quinsemiseptile represents an angle of approximately 128.57143° (or 128°34’17”), which corresponds to five-fourteenths of the circle of the zodiac.
The semioctile represents an angle of 22.5° (or 22°30′), which is one-sixteenth of the circle of the zodiac.
The sesquioctile corresponds to an angle of 67.5° (or 67°30′), which is equivalent to three-sixteenths of the zodiac circle. It can also be described as a semisquare (octile) and a half.
The quattuorvigintile represents an angle of 15°, which is one-twenty-fourth of the zodiac circle.
The squile corresponds to an angle of 75°, which is equivalent to 5/24 of the zodiac circle. It is considered a hybrid between a square and a sextile.
The squine represents an angle of 105°, which is equivalent to 7/24 of the zodiac circle. It is considered a hybrid between a square and a trine.
The quindecile, also known as the johndro or contraquindecile, corresponds to an angle of 165°. It is believed to be associated with an unrelenting, headstrong determination, often accompanied by
disruption and upheaval. | {"url":"https://synnisigns.com/aspects/ternary-aspects/","timestamp":"2024-11-13T08:32:03Z","content_type":"text/html","content_length":"312312","record_id":"<urn:uuid:18e52a20-5a49-4b51-817d-17ed51506498>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00347.warc.gz"} |
Polar Coordinates 🌀
Cartesian Coordinate System
A coordinate system allows us to use numbers to determine the position of points in a 2D or 3D space. The most popular coordinate system is probably the Cartesian coordinate system. It allows you to
locate each point by a pair of numerical coordinates (x, y). There is a very good chance that you have used this system. It is everywhere — SVG, Canvas, WebGL and even Sketch & Illustrator.
Figure 1: Cartesian coordinate system
Polar Coordinate System
Polar coordinate system is a 2D coordinate system in which each point is determined by r & θ. Where r is the distance from the origin and θ is the angle from the x-axis.
Figure 2: Polar coordinate system
Converting Between Polar and Cartesian Coordinates
Trigonometry! Love it or hate, it is everywhere. You can convert a point from polar coordinates (r, θ) to Cartesian coordinates (x, y) using the equations below. These come in really handy since most
tools only accept x & y as point locations.
const x = r * Math.cos(theta);
const y = r * Math.sin(theta);
Below I have a few animations by Dave Whyte aka bees & bombs. They all have one thing in common, they use polar coordinates to generate a pattern.
Figure 3: Patterns generated using polar coordinates
Cartesian coordinates are a great choice for placing things evenly on a rectangular grid. However, if you want to distribute things evenly around a circle then polar coordinates are generally the
better option. Keeping the radius constant we compute the angle for each point as angle = 360° * index / number of sides.
Figure 4: Placing items evenly around a circle
Below is the JavaScript implementation of this idea. The points function also accepts an optional offset argument to shift the points along the perimeter of the circle. For example, the outermost
circle of points in Figure 4 can be generated using points(12, 200). The next one inwards using points(12, 175, 15). The one after that using points(12, 150, 30) and so on. My splash CodePen is was
built using this.
function points(count, radius, offset = 0) {
const angle = 360 / count;
const vertexIndices = range(count);
return vertexIndices.map(index => {
return {
theta: offset + degreesToRadians(offset + angle * index),
r: radius,
// number => [0, 1, 2, ... number]
function range(count) {
return Array.from(Array(count).keys());
function degreesToRadians(angleInDegrees) {
return (Math.PI * angleInDegrees) / 180;
Polygon Generator
A regular polygon is a polygon that is equiangular i.e., all its angles are equal and all its sides have the same length. This means that all the vertices of a regular polygon are points evenly
spaced on a circle. And isn’t it handy that we just created a function that generates exactly this!
Figure 5: Drawing a polygon by connecting polar coordinates
To generate an SVG polygon we will generate the list of of points using the points function and then simply connect the dots. With SVG we have two options for this: <polygon> or <path>. The example
below generates the points attribute for a <polygon> element.
* Usage with Vanilla JS/DOM:
* polygonEl.setAttribute('points', polygon((5, 64, 18)));
* Usage with React:
* <polygon points={polygon((5, 64, 18)} />
* Usage with View:
* <polygon :points="polygon((5, 64, 18)" />
function polygon(noOfSides, circumradius, rotation) {
return points(noOfSides, circumradius, rotation)
.join(' ');
function toCartesian({ r, theta }) {
return [r * Math.cos(theta), r * Math.sin(theta)];
Seems simple but, you can do a lot of interesting things with it. Here are a couple of examples:
Gems generated using polar coordinates
Relative Polar Coordinates
When you define a point as (r, θ), by default, this is relative to the origin (0, 0). We can define points relative to other points by shifting the origin. This is often used for defining the
position of curve handles relative to a vertex.
const x = cx + r * Math.cos(theta);
const y = cy + r * Math.sin(theta);
Figure 6: Relative polar coordinates
We can modify the polygon generator to allow us to draw a polygon centred at any location in the SVG canvas.
function polygon(noOfSides, circumradius, rotation, [cx = 0, cy = 0]) {
return points(noOfSides, circumradius, rotation)
.map(pt => toCartesian(pt, [cx, cy]))
.join(' ');
function toCartesian({ r, theta }, [cx, cy]) {
return [cx + r * Math.cos(theta), cy + r * Math.sin(theta)];
Another common application of polar coordinates is to rotate things around a point. Here (cx, cy) is the point about which you want to rotate.
x = cx + r * Math.cos(theta);
y = cy + r * Math.sin(theta);
// somewhere in an animation loop
window.setInterval(() => {
}, 1000 / 60);
Polar curves
We started by looking at individual points. Then we grouped a few points into a set to define shapes. For this we used the polygon generator function to compute the location of each vertex of the
shape. We can write similar functions using other mathematical equations. Allowing us to generate more complex shapes and curves.
Two dimensional curves are described by equations of the type y = f(x). For example the equation of a circle is x^2 + y^2 = r^2. We can generate the set of points, called locus, by iterating x and
computing the corresponding y value or vice-versa. Therefore, each point will be of the form (x, f(x)) or (g(y), y).
With polar coordinates we can similarly draw polar curves. For example, the polar equation of a circle is r = 2 * cos(0). The points on a polar curve have the form (r(0), 0).
// examples of fn:
// circle : 2 * Math.cos(theta)
// blob thing : a * (1 - Math.cos(theta) * Math.sin(3 * theta))
const r = fn(theta);
const x = cx + r * Math.cos(theta);
const y = cy + r * Math.sin(theta);
All the diagrams in this post were created using a language called eukleides. It is a fantastic tool for making geometric drawings. Just look at this declarative API 😍
c = circle(point(3, 0), 3)
P = point(c, 170°)
M = point(0, 0)
N = point(6, 0)
draw (M.N.P)
label M, P, N right, 0.6 | {"url":"https://varun.ca/polar-coords/?ref=prototypr","timestamp":"2024-11-05T09:24:00Z","content_type":"text/html","content_length":"106911","record_id":"<urn:uuid:971ba285-66ab-4ae7-a3fa-fa6fb3d1aefd>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00121.warc.gz"} |
Euclid's Proof of the Pythagorean Theorem
The following window shows a geometrical proof of Pythagoras' Theorem. The three buttons, NEXT, BACK, RESTART, allow you to go through the steps of the proof. As well, if you would like to repeat the
action of the diagram simply click on the image. (The text can be retyped by clicking on the text box).
This Java applet was written by Jim Morey at the Universtiy of British Columbia. His home page. Take a look at the poorly documented program (Morey's description) and its helpers Banner and
fillTriangle. It was hacked from the hotjava people. | {"url":"https://math.hawaii.edu/~ralph/Pythagoras/index.html","timestamp":"2024-11-11T09:38:35Z","content_type":"text/html","content_length":"1325","record_id":"<urn:uuid:abcc84cc-93fe-4217-aa6d-95b17b32511a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00852.warc.gz"} |
AVL Tree in Data Structures with Examples
06 Jun 2024
24.2K Views
43 min read
AVL Tree in Data Structures: An Overview
We know that the
Binary Search Tree
cannot guarantee logarithmic complexity. If the elements are inserted in a sorted increasing order, the tree becomes skewed, so the worst-case
time complexity
for insert/search operations becomes
. To overcome this limitation of binary search trees, AVL Trees came into existence. In this
DSA tutorial
, we will discuss AVL trees, balance factors, rotations of AVL trees, operations, etc.
To further enhance your understanding and application of AVL Trees, consider enrolling in the Best Data Structures And Algorithms Course to gain comprehensive insights into effective data structure
utilization for improved problem-solving and time management.
What is an AVL Tree in Data Structures?
AVL tree in data structures is a popular self-balancing binary search tree where the difference between the heights of left and right subtrees for any node does not exceed unity. It was introduced by
Georgy Adelson-Velsky and Evgenii Landis in 1962, hence the name AVL. It automatically adjusts its structure to maintain the minimum possible height after any operation with the help of a balance
factor for each node.
Balance Factor in AVL Tree in Data Structures
The balance factor of a node in an AVL Tree is a numerical value that represents the difference in height between the left and right subtrees of that node. It is the extra information used to
determine the tree's balance. The balance factor is calculated as follows:
Balance Factor = height(left subtree) - height(right subtree)
bf = hl − hr,
where bf = balance factor of a given node, hl = height of the left subtree, and hr = height of the right subtree.
The value of the balance factor always lies between -1 and 1.
∣bf∣ = ∣hl − hr∣ ≤ 1
1. If the balance factor of any node = 1, the left sub-tree is one level higher than the right sub-tree i.e. the given node is left-heavy.
2. If the balance factor of any node = 0, the left sub-tree and right sub-tree are of equal height.
For leaf nodes, the balance factor is 0 as they do not contain any subtrees.
3. If the balance factor of any node = -1, the left sub-tree is one level lower than the right sub-tree i.e. the given node is right-heavy.
Hence, in this way, the self-balancing property of an AVL tree is maintained by the balance factor. Thus, we can find an unbalanced node in the tree and locate where the height-affecting operation
was performed that caused the imbalance of the tree.
AVL Tree Rotations
AVL tree rotation is a fundamental operation used in self-balancing binary search trees, specifically in AVL trees. Due to any operations like insertion or deletion, if any node of an AVL tree
becomes unbalanced, specific tree rotations are performed to restore the balance.
The tree rotations involve rearranging the tree structure without changing the order of elements. The positions of the nodes of a subtree are interchanged. There are four types of AVL rotations:
1. Left Rotation (LL Rotation)
In a left rotation, a node's right child becomes the new root, while the original node becomes the left child of the new root. The new root's left child becomes the original node's right child.
2. Right Rotation (RR Rotation)
In a right rotation, a node's left child becomes the new root, while the original node becomes the right child of the new root. The new root's right child becomes the original node's left child.
3. Left-Right Rotation (LR Rotation)
An LR rotation is a combination of a left rotation followed by a right rotation. It is performed when the left subtree of a node is unbalanced to the right, and the right subtree of the left child of
that node is unbalanced to the left.
4. Right-Left Rotation (RL Rotation)
An RL rotation is a combination of a right rotation followed by a left rotation. It is performed when the right subtree of a node is unbalanced to the left, and the left subtree of the right child of
that node is unbalanced to the right.
Read More - Data Structure Interview Questions for Experienced
Standard Operations on AVL Trees in Data Structures
1. Insertion: A newNode is always inserted as a leaf node with a balance factor equal to 0. After each insertion, the ancestors of the newly inserted node are examined because the insertion only
affects their heights, potentially inducing an imbalance. This process of traversing the ancestors to find the unbalanced node is called retracing.
Algorithm for Insertion in an AVL Tree
Step 1: START
Step 2: Insert the node using BST insertion logic.
Step 3: Calculate and check the balance factor of each node.
Step 4: If the balance factor follows the AVL criterion, go to step 6.
Step 5: Else, perform tree rotations according to the insertion done. Once the tree is balanced go to step 6.
Step 6: END
Let's understand with an example
1. Let the initial tree be:Let the node to be inserted be:
2. Go to the appropriate leaf node to insert a newNode using the following recursive steps. Compare newKey with rootKey of the current tree.
1. If newKey < rootKey, call the insertion algorithm on the left subtree of the current node until the leaf node is reached.
2. Else if newKey > rootKey, call the insertion algorithm on the right subtree of the current node until the leaf node is reached.
3. Else, return leafNode.
3. Compare leafKey obtained from the above steps with newKey:
1. If newKey < leafKey, make newNode as the leftChild of leafNode.
2. Else, make newNode as rightChild of leafNode.
4. Update balanceFactor of the nodes.
5. If the nodes are unbalanced, then rebalance the node.
1. If balanceFactor > 1, which means the height of the left subtree is greater than that of the right subtree. So, do a right rotation or left-right rotation
1. If newNodeKey < leftChildKey do the right rotation.
2. Else, do left-right rotation.
2. If balanceFactor < -1, it means the height of the right subtree is greater than that of the left subtree. So, do a right rotation or right-left rotation
1. If newNodeKey > rightChildKey do a left rotation.
2. Else, do right-left rotation
6. The final tree is:
2. Deletion:A node is always deleted as a leaf node. After deleting a node, the balance factors of the nodes get changed. To rebalance the balance factor, suitable rotations are performed.
Algorithm for Deletion in an AVL Tree
Step 1: START
Step 2: Find the node in the tree. If the element is not found, go to step 7.
Step 3: Delete the node using BST deletion logic.
Step 4: Calculate and check the balance factor of each node.
Step 5: If the balance factor follows the AVL criterion, go to step 7.
Step 6: Else, perform tree rotations to balance the unbalanced nodes. Once the tree is balanced go to step 7.
Step 7: END
Let's understand with an example
1. Locate nodeToBeDeleted (recursion is used to find nodeToBeDeleted in the code used below).
2. There are three cases for deleting a node:
1. If nodeToBeDeleted is the leaf node (ie. does not have any child), then remove nodeToBeDeleted.
2. If nodeToBeDeleted has one child, then substitute the contents of nodeToBeDeleted with that of the child. Remove the child.
3. If nodeToBeDeleted have two children, find the in-order successor w of nodeToBeDeleted (ie. node with a minimum value of key in the right subtree).
☆ Substitute the contents of nodeToBeDeleted with that of w.
☆ Remove the leaf node w.
3. Update balanceFactor of the nodes.
4. Rebalance the tree if the balance factor of any of the nodes is not equal to -1, 0, or 1.
1. If balanceFactor of currentNode > 1,
1. If balanceFactor of leftChild >= 0, do right rotation.
2. Else do left-right rotation.
2. If balanceFactor of currentNode < -1,
1. If balanceFactor of rightChild <= 0, do left rotation.
2. Else do right-left rotation.
5. The final tree is:
Implementation of AVL Tree in Different Programming Languages
import sys
# Create a tree node
class TreeNode(object):
def __init__(self, key):
self.key = key
self.left = None
self.right = None
self.height = 1
class AVLTree(object):
# Function to insert a node
def insert_node(self, root, key):
# Find the correct location and insert the node
if not root:
return TreeNode(key)
elif key < root.key:
root.left = self.insert_node(root.left, key)
root.right = self.insert_node(root.right, key)
root.height = 1 + max(self.getHeight(root.left),
# Update the balance factor and balance the tree
balanceFactor = self.getBalance(root)
if balanceFactor > 1:
if key < root.left.key:
return self.rightRotate(root)
root.left = self.leftRotate(root.left)
return self.rightRotate(root)
if balanceFactor < -1:
if key > root.right.key:
return self.leftRotate(root)
root.right = self.rightRotate(root.right)
return self.leftRotate(root)
return root
# Function to delete a node
def delete_node(self, root, key):
# Find the node to be deleted and remove it
if not root:
return root
elif key < root.key:
root.left = self.delete_node(root.left, key)
elif key > root.key:
root.right = self.delete_node(root.right, key)
if root.left is None:
temp = root.right
root = None
return temp
elif root.right is None:
temp = root.left
root = None
return temp
temp = self.getMinValueNode(root.right)
root.key = temp.key
root.right = self.delete_node(root.right,
if root is None:
return root
# Update the balance factor of nodes
root.height = 1 + max(self.getHeight(root.left),
balanceFactor = self.getBalance(root)
# Balance the tree
if balanceFactor > 1:
if self.getBalance(root.left) >= 0:
return self.rightRotate(root)
root.left = self.leftRotate(root.left)
return self.rightRotate(root)
if balanceFactor < -1:
if self.getBalance(root.right) <= 0:
return self.leftRotate(root)
root.right = self.rightRotate(root.right)
return self.leftRotate(root)
return root
# Function to perform left rotation
def leftRotate(self, z):
y = z.right
T2 = y.left
y.left = z
z.right = T2
z.height = 1 + max(self.getHeight(z.left),
y.height = 1 + max(self.getHeight(y.left),
return y
# Function to perform right rotation
def rightRotate(self, z):
y = z.left
T3 = y.right
y.right = z
z.left = T3
z.height = 1 + max(self.getHeight(z.left),
y.height = 1 + max(self.getHeight(y.left),
return y
# Get the height of the node
def getHeight(self, root):
if not root:
return 0
return root.height
# Get balance factore of the node
def getBalance(self, root):
if not root:
return 0
return self.getHeight(root.left) - self.getHeight(root.right)
def getMinValueNode(self, root):
if root is None or root.left is None:
return root
return self.getMinValueNode(root.left)
def preOrder(self, root):
if not root:
print("{0} ".format(root.key), end="")
# Print the tree
def printHelper(self, currPtr, indent, last):
if currPtr != None:
if last:
indent += " "
indent += "| "
self.printHelper(currPtr.left, indent, False)
self.printHelper(currPtr.right, indent, True)
myTree = AVLTree()
root = None
nums = [22, 14, 72, 44, 25, 63, 98]
for num in nums:
root = myTree.insert_node(root, num)
myTree.printHelper(root, "", True)
key = 25
root = myTree.delete_node(root, key)
print("After Deletion: ")
myTree.printHelper(root, "", True)
// Create node
class Node {
int item, height;
Node left, right;
Node(int d) {
item = d;
height = 1;
// Tree class
class AVLTree {
Node root;
int height(Node N) {
if (N == null)
return 0;
return N.height;
int max(int a, int b) {
return (a > b) ? a : b;
Node rightRotate(Node y) {
Node x = y.left;
Node T2 = x.right;
x.right = y;
y.left = T2;
y.height = max(height(y.left), height(y.right)) + 1;
x.height = max(height(x.left), height(x.right)) + 1;
return x;
Node leftRotate(Node x) {
Node y = x.right;
Node T2 = y.left;
y.left = x;
x.right = T2;
x.height = max(height(x.left), height(x.right)) + 1;
y.height = max(height(y.left), height(y.right)) + 1;
return y;
// Get balance factor of a node
int getBalanceFactor(Node N) {
if (N == null)
return 0;
return height(N.left) - height(N.right);
// Insert a node
Node insertNode(Node node, int item) {
// Find the position and insert the node
if (node == null)
return (new Node(item));
if (item < node.item)
node.left = insertNode(node.left, item);
else if (item > node.item)
node.right = insertNode(node.right, item);
return node;
// Update the balance factor of each node
// And, balance the tree
node.height = 1 + max(height(node.left), height(node.right));
int balanceFactor = getBalanceFactor(node);
if (balanceFactor > 1) {
if (item < node.left.item) {
return rightRotate(node);
} else if (item > node.left.item) {
node.left = leftRotate(node.left);
return rightRotate(node);
if (balanceFactor < -1) {
if (item > node.right.item) {
return leftRotate(node);
} else if (item < node.right.item) {
node.right = rightRotate(node.right);
return leftRotate(node);
return node;
Node nodeWithMinimumValue(Node node) {
Node current = node;
while (current.left != null)
current = current.left;
return current;
// Delete a node
Node deleteNode(Node root, int item) {
// Find the node to be deleted and remove it
if (root == null)
return root;
if (item < root.item)
root.left = deleteNode(root.left, item);
else if (item > root.item)
root.right = deleteNode(root.right, item);
else {
if ((root.left == null) || (root.right == null)) {
Node temp = null;
if (temp == root.left)
temp = root.right;
temp = root.left;
if (temp == null) {
temp = root;
root = null;
} else
root = temp;
} else {
Node temp = nodeWithMinimumValue(root.right);
root.item = temp.item;
root.right = deleteNode(root.right, temp.item);
if (root == null)
return root;
// Update the balance factor of each node and balance the tree
root.height = max(height(root.left), height(root.right)) + 1;
int balanceFactor = getBalanceFactor(root);
if (balanceFactor > 1) {
if (getBalanceFactor(root.left) >= 0) {
return rightRotate(root);
} else {
root.left = leftRotate(root.left);
return rightRotate(root);
if (balanceFactor < -1) {
if (getBalanceFactor(root.right) <= 0) {
return leftRotate(root);
} else {
root.right = rightRotate(root.right);
return leftRotate(root);
return root;
void preOrder(Node node) {
if (node != null) {
System.out.print(node.item + " ");
// Print the tree
private void printTree(Node currPtr, String indent, boolean last) {
if (currPtr != null) {
if (last) {
indent += " ";
} else {
indent += "| ";
printTree(currPtr.left, indent, false);
printTree(currPtr.right, indent, true);
public static void main(String[] args) {
AVLTree tree = new AVLTree();
tree.root = tree.insertNode(tree.root, 22);
tree.root = tree.insertNode(tree.root, 14);
tree.root = tree.insertNode(tree.root, 72);
tree.root = tree.insertNode(tree.root, 44);
tree.root = tree.insertNode(tree.root, 25);
tree.root = tree.insertNode(tree.root, 63);
tree.root = tree.insertNode(tree.root, 98);
System.out.println("Before Deletion: ");
tree.printTree(tree.root, "", true);
tree.root = tree.deleteNode(tree.root, 25);
System.out.println("After Deletion: ");
tree.printTree(tree.root, "", true);
#include <iostream>
using namespace std;
class Node {
int key;
Node *left;
Node *right;
int height;
int max(int a, int b);
// Calculate height
int height(Node *N) {
if (N == NULL)
return 0;
return N->height;
int max(int a, int b) {
return (a > b) ? a : b;
// New node creation
Node *newNode(int key) {
Node *node = new Node();
node->key = key;
node->left = NULL;
node->right = NULL;
node->height = 1;
return (node);
// Rotate right
Node *rightRotate(Node *y) {
Node *x = y->left;
Node *T2 = x->right;
x->right = y;
y->left = T2;
y->height = max(height(y->left),
height(y->right)) +
x->height = max(height(x->left),
height(x->right)) +
return x;
// Rotate left
Node *leftRotate(Node *x) {
Node *y = x->right;
Node *T2 = y->left;
y->left = x;
x->right = T2;
x->height = max(height(x->left),
height(x->right)) +
y->height = max(height(y->left),
height(y->right)) +
return y;
// Get the balance factor of each node
int getBalanceFactor(Node *N) {
if (N == NULL)
return 0;
return height(N->left) -
// Insert a node
Node *insertNode(Node *node, int key) {
// Find the correct postion and insert the node
if (node == NULL)
return (newNode(key));
if (key < node->key)
node->left = insertNode(node->left, key);
else if (key > node->key)
node->right = insertNode(node->right, key);
return node;
// Update the balance factor of each node and
// balance the tree
node->height = 1 + max(height(node->left),
int balanceFactor = getBalanceFactor(node);
if (balanceFactor > 1) {
if (key < node->left->key) {
return rightRotate(node);
} else if (key > node->left->key) {
node->left = leftRotate(node->left);
return rightRotate(node);
if (balanceFactor < -1) {
if (key > node->right->key) {
return leftRotate(node);
} else if (key < node->right->key) {
node->right = rightRotate(node->right);
return leftRotate(node);
return node;
// Node with minimum value
Node *nodeWithMimumValue(Node *node) {
Node *current = node;
while (current->left != NULL)
current = current->left;
return current;
// Delete a node
Node *deleteNode(Node *root, int key) {
// Find the node and delete it
if (root == NULL)
return root;
if (key < root->key)
root->left = deleteNode(root->left, key);
else if (key > root->key)
root->right = deleteNode(root->right, key);
else {
if ((root->left == NULL) ||
(root->right == NULL)) {
Node *temp = root->left ? root->left : root->right;
if (temp == NULL) {
temp = root;
root = NULL;
} else
*root = *temp;
} else {
Node *temp = nodeWithMimumValue(root->right);
root->key = temp->key;
root->right = deleteNode(root->right,
if (root == NULL)
return root;
// Update the balance factor of each node and
// balance the tree
root->height = 1 + max(height(root->left),
int balanceFactor = getBalanceFactor(root);
if (balanceFactor > 1) {
if (getBalanceFactor(root->left) >= 0) {
return rightRotate(root);
} else {
root->left = leftRotate(root->left);
return rightRotate(root);
if (balanceFactor < -1) {
if (getBalanceFactor(root->right) <= 0) {
return leftRotate(root);
} else {
root->right = rightRotate(root->right);
return leftRotate(root);
return root;
// Print the tree
void printTree(Node *root, string indent, bool last) {
if (root != nullptr) {
cout << indent;
if (last) {
cout << "R----";
indent += " ";
} else {
cout << "L----";
indent += "| ";
cout << root->key << endl;
printTree(root->left, indent, false);
printTree(root->right, indent, true);
int main() {
Node *root = NULL;
root = insertNode(root, 22);
root = insertNode(root, 14);
root = insertNode(root, 72);
root = insertNode(root, 44);
root = insertNode(root, 25);
root = insertNode(root, 63);
root = insertNode(root, 98);
printTree(root, "", true);
root = deleteNode(root, 25);
cout << "After deleting " << endl;
printTree(root, "", true);
| L----14
| R----25
After deleting
| L----14
Complexity Analysis of AVL Tree Operations
1. Time Complexity:
Operations Best Case Average Case Worst Case
Insertion O (log n) O (log n) O (log n)
Deletion O (log n) O (log n) O (log n)
Traversal O (n) O (n) O (n)
Search O (log n) O (log n) O (log n)
2. Space Complexity: Space complexity is the same as that of Binary Search Trees i.e., O(n) as AVL Trees don't modify the data itself.
Applications of AVL Trees
1. It is used to index huge records in a database and also search in that efficiently.
2. For all types of in-memory collections, including sets and dictionaries, AVL Trees are used.
3. Database applications, where insertions and deletions are less common but frequent data lookups are necessary.
4. Software that needs optimized search.
5. It is applied in corporate areas and storyline games.
Advantages of AVL Trees
1. AVL trees are capable of self-balancing.
2. It cannot be skewed.
3. Compared to Red-Black Trees, it offers faster lookups.
4. Superior searching efficiency compared to other trees, such as the binary tree.
5. Height is limited to log(n), where n is the number of nodes in the tree as a whole.
Disadvantages of AVL Trees
1. Implementing it is challenging.
2. Certain procedures have high constant factors.
3. Compared to Red-Black trees, AVL trees are less common.
4. As additional rotations are conducted, AVL trees offer complex insertion and removal procedures because of their rather rigid balance.
5. Requires to process more for balancing.
AVL Trees are a great tool to help structure and organize data. By keeping track of the difference in heights between two nodes, AVL trees can save you from calculating these values manually.
Furthermore, as it is self-balancing, you need not worry about the tree becoming unbalanced as it grows. While traversing may take longer due to all the re-balancing, this method gives you greater
control over your data sets than many regular BSTs. To implement your theoretical knowledge about AVL Trees, consider our
Data Structures Certification
The balance factor of a node in an AVL Tree is a numerical value that represents the difference in height between the left and right subtrees of that node.
There are four types of AVL tree rotations.
Space complexity
is the same as that of Binary Search Trees i.e., O(n) as AVL Trees don't modify the data itself. | {"url":"https://www.scholarhat.com/tutorial/datastructures/avl-tree-in-data-structures","timestamp":"2024-11-05T22:08:46Z","content_type":"text/html","content_length":"183005","record_id":"<urn:uuid:88e74210-282a-456d-890f-66877e89d191>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00392.warc.gz"} |
How Do We Solve Equations With Exponential Expressions Brainly - Tessshebaylo
How Do We Solve Equations With Exponential Expressions Brainly
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Linear combination of atomic orbitals molecular orbital method
Linear Combination of Atomic Orbitals Molecular Orbital Method
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What is an Inductor – Definition, Symbol, Types, and Applications
In this article, we will learn about the definition, symbol, types, and applications of the inductor.
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In electrical and electronics, an inductor is a passive circuit component that is designed to store electrical energy in the form of a magnetic field.
In other words, a circuit element whose terminal voltage is directly proportional to the derivative of current with respect to time is called an inductor.
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`\"Inductance",L=(μN^2 a)/l" "…(1)`
Where, N is the number of turns in the inductor coil, l is the mean length of the inductor, a is the area of cross-section, and µ is the permeability of the core.
The commercially available inductors come in different values and types. Typical practical inductors have inductance values ranging from a few micro-henry (µH) to tens of henrys (H). Inductors are
almost used in every electrical and electronic circuit. Inductors may be classified as fixed inductors or variable inductors. The circuit symbols of the fixed and variable inductors are shown in
An inductor whose inductance is independent of current is known as a linear inductor, and an inductor whose inductance varies with current is called a non-linear inductor. The voltage-current
relationship of a linear inductor is a straight line as shown in figure-3. The slope of the curve line gives the inductance (L) of the linear inductor.
Factors Affecting Inductance
From equation (1), we can see that the inductance of an inductor depends upon the following four factors:
• The inductance of an inductor can be increased by increasing the number of turns (N) in the coil.
• Inductance can be increased by using a core of high permeability (µ).
• Inductance can be increased by increasing the cross-sectional area (a) of the wire of the coil.
• Inductance can be increased by reducing the length (l) of the inductor coil.
Current-Voltage Relationship of Inductor
If an electric current is allowed to pass through an inductor, then it is found that the voltage across the inductor is directly proportional to the time rate of change of the current, i.e.
`\⟹v(t)=L (di(t))/dt" "…(2)`
Where, L is the inductance of the inductor, measured in Henry (H).
`\di(t)=1/L v(t) dt`
Integrating on both sides gives,
`\i(t)=1/L ∫_(-∞)^tv(τ)dτ" "…(3)`
`\i(t)=1/L ∫_(t_0)^tv(τ)dτ+i(t_0 )" "…(4)`
Where, i(t[0]) is the initial current, i.e. current for -∞ < t < t[0], and i(-∞) = 0, because there must be a time in the past when there was no current in the inductor.
Energy Stored in Inductor
The inductor is mainly designed to store electrical energy in the form of a magnetic field. The expression of energy stored in the inductor can be derived as follows-
The power supplied to the inductor is given by,
Thus, the energy store is given by,
`\w(t)=∫_(-∞)^t p(τ) dτ`
`\⟹w(t)=∫_(-∞)^t v(τ)i(τ)dτ`
`\⟹w(t)=L∫_(-∞)^t (di(τ))/(dτ) i(τ)dτ`
`\⟹w(t)=L∫_(-∞)^t i(τ) di(τ)`
`\⟹w(t)=L[(i^2 (t))/2]_(-∞)^t`
`\⟹w(t)=1/2 Li^2 (t)-1/2 Li^2 (-∞)`
`\∴w(t)=1/2 Li^2 (t)" "…(5)`
From equation (5), we can see that the total energy stored in the inductor is always positive (or zero). Therefore, an inductor is a passive circuit component.
Characteristics of an Ideal Inductor
The following are the important characteristics of an ideal inductor:
• There is no voltage across an inductor if the current flowing through it remains constant with respect to time. Therefore, a pure inductor acts as short-circuit on the application of DC.
• An inductor can store a finite amount of energy even if the voltage across the inductor is zero.
• The current through an inductor can never be changed abruptly, i.e. in zero time.
• An ideal inductor never dissipates energy, but only stores it in the form of a magnetic field.
Circuit Model for a Practical Inductor
An ideal inductor does not have any resistance, but a practical inductor has a significant resistive component. It is because the inductor is made of a conductive material like copper which has some
resistance. This resistance of the inductor is called the winding resistance (R[w]), and it appears in series with the inductor coil. Due to the winding resistance, a practical inductor store as well
dissipate the energy. Also, a practical inductor has a winding capacitance (C[w]) which appears in parallel with the series combination of the coil and winding resistance. Although, the winding
resistance and capacitance are very small, thus they can be neglected in most practical applications.
Applications of Inductors
The inductor is one of the most extensively used circuit components in electrical and electronic circuits. The following are some important applications of the inductor-
• Inductors are used in tuning circuits to select the desired frequency.
• Inductors are used in different contactless sensors like proximity sensors.
• Inductors are also used to store electrical energy in the form of a magnetic field.
• Inductors are used to form windings of electrical machines like motors, generators, transformers, measuring instruments, etc.
• Inductors are used in electronic filter circuits.
• Inductors are also used in chokes for blocking the alternating current flow and passing the direct current.
• Inductors are also used in ferrite beads to reduce radio frequency interference. The ferrite beads are used in computer cables and mobile charging cables.
• Inductors are also used in electromagnetic relays.
Now, we can conclude this article with the following points-
• An inductor is a passive circuit component.
• The inductor is a two-terminal circuit element that opposes any change in the magnitude of current through it.
• The property of an inductor by which it opposes any change in the current is called its inductance.
• An inductor can either be fixed or variable.
• A pure inductor stores electrical energy in the form of a magnetic field, but does not dissipate.
• The current through an inductor cannot change instantaneously. Thus, an inductor opposes an abrupt change in the current through it.
• A practical or non-ideal inductor shows very small resistive and capacitive effects as well.
• In the steady state condition, the inductor acts like a short circuit.
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Algorithm Theory
Graduate Course - Winter Term 2022/23
Fabian Kuhn
Course Description
The design and analysis of algorithms is fundamental to computer science. In this course, we will study efficient algorithms for a variety of basic problems and, more generally, investigate advanced
design and analysis techniques. Central topics are algorithms and data structures that go beyond what has been considered in the undergraduate course Algorithms and Datastructures. Basic algorithms
and data structures knowledge, comparable to what is done in Algorithms and Datastructures is therefore assumed. The topics of the course include (but are not limited to):
• Divide and conquer: geometrical divide and conquer, fast fourier transformation
• Randomization: median, randomized quicksort, probabilistic primality testing
• Amortized analysis: Fibonacci heaps, union-find data structures
• Greedy algorithms: minimum spanning trees, scheduling, matroids
• Dynamic programming: matrix chain product problem, edit distance, longest common subsequence problem
• Graph algorithms: network flows, combinatorial optimization problems on graphs
Exam Information
• Review: 04.10.23, 3:00 pm in room 00-015, building 106.
• Repeat-Exam: 18.08.2023, 2pm, Building 101 Room 00-026.
• Review: 09.05.22, 4:00 pm in room 00-007, building 106.
• Date and time. 22.03.2023, 9:00 - 11:00 am (2 hours). Please arrive by 8:45 am.
• Mode. The exam will be written.
• Location. Building 101 Room 00-026 & HS 00-036. Enter only after the student ID has been checked at the entrance.
• Allowed material. 6 A4 pages (corresponds to 3 double-sided A4 sheets) of handwritten notes.
• Preparation. Old exams are available further below. We may also offer an exam Q&A session to discuss previous exam tasks. We will update you on this soon. On Zulip you can make suggestions which
exam tasks we should cover.
Lectures: Tuesdays, 16:15 - 18:00, 101-00-026
Besides the lecture on Tuesday, we provide the lecture recordings from winter term 2021/2022.
Important note: The recordings can only be accessed from within the university network. This can be done by establishing a VPN tunnel to the university network.
Exercise Tutorials: Thursdays, 14:15 - 16:00, 101-00-026
Exercises: There will be an exercise sheet every week which you should work on at home. We accept and also recommend to submit the exercise sheets in groups of at most four people. The exercises will
be published every Tuesday. They are due one week later (on Wednesday 11:59 pm).
Instant Messenger: We will offer an instant messaging platform (Zulip) for all students to discuss all topics related to this lecture, where you are free to discuss among yourself or pose questions
to us. Please consider the section below on technical information.
Format: All submissions must be in, or converted to pdf format and be written in English. We strongly recommend to prepare your solutions with Latex for best readability (being able to work with
latex is a good skill to have anyway). Solutions prepared with Word or similar text editors are ok. Scans or photos of handwritten solutions in pdf format are ok as well, but must be well readable!
Submission Guidelines: The exercises will be conducted online with the course management system Daphne. The solution of each exercise must be uploaded to your SVN repository each one in a separate
folder named exercise-XY, where XY is the exercise number (with a leading 0 if that number is smaller than 10). More on the submission via SVN in the technical information section.
Team Submission: Teams will be allowed. We recommend to build teams of three members. Teams may consist of at most four members. In case you submit your solution as part of a team, each team member
must still submit a copy of the solution pdf to their respective SVN-repository (c.f. technical information). The members of the teams must be clearly marked on the top of the solution pdf.
Exercise sheet Assigned Due Solution
Exercise 01 (O-Notation, Divide & Conquer) 18.10. 26.10. Solution 01
Exercise 02 (FFT) 26.10. 09.11. Solution 02
Exercise 03 (Greedy Algorithms) 09.11. 16.11. Solution 03
Exercise 04 (Dynamic Programming) 16.11. 23.11. Solution 04
Exercise 05 (Amortized Analysis, Union Find) 15.11. 23.11. Solution 05
Exercise 06 (Fibonacci Heaps) 30.11. 07.12.
Exercise 07 (Maximum Flow) 07.12. 14.12. Solution 07
Exercise 08 (Maximum Flow 2) 14.12. 21.12. Solution 08
Exercise 09 (updated) (Matchings and Flows) 21.12. 11.01. Solution 09
Exercise 10 (Matching and Randomization) 10.01. 18.01. Solution 10
Exercise 11 (Randomization II) 18.01. 25.01. Solution 11
Exercise 12 (RandomizationIII) 25.01. 01.02. Solution 12
Exercise 13 (Approximation Algorithms) 01.02. 08.02. Solution 13 (updated)
Exercise 14 (Online Algorithms) 08.02. 15.02. Solution 14
Technical Information
Zulip: The Zulip data is available here.
Important note: This website can only be accessed from within the university network. This can be done by establishing a VPN tunnel to the university network.
An introduction to Zulip is given here: Getting Started (in particular, note that Zulip supports Latex)
Daphne: The exercises are submitted via Daphne. Please use this link to register for the Algorithm Theory course with your rz-account. On Daphne, an overview of the points you achieved so far is
given. Exercises are also published there.
Subversion (SVN): After registration, a SVN repository will be created for you with an URL of the following form:
You should do a "checkout" of your SVN-repository (using the URL described above) to get a local copy on your pc. With the command "update" you synchronize your local copy with the current version on
the server. Through the command "commit" you upload your local files to your repository on the server. A more detailed overview can be found here. There are different subversion clients you can use.
For Windows Tortoise-SVN is recommended.
Note that every solution of an exercise sheet must be uploaded ("committet") to a separate folder named exercise-XY, where XY is the exercise number (with a leading 0 if smaller than 10). This folder
will be automatically locked (that is, no commits are possible anymore) after the submission deadline on the following Tuesday at 4 pm sharp!
Past Exams
• Jon Kleinberg and Éva Tardos: Algorithm Design, Addison Wesley
• Thomas H. Cormen, Charles E. Leiserson, Robert L. Rivest, and Cliford Stein: Introduction to Algorithms, MIT Press
• Thomas Ottmann and Peter Widmayer: Algorithmen und Datenstrukturen, Spektrum Akademischer Verlag | {"url":"https://ac.informatik.uni-freiburg.de/teaching/ws22_23/algo_theo.php","timestamp":"2024-11-08T17:49:48Z","content_type":"application/xhtml+xml","content_length":"19199","record_id":"<urn:uuid:0ed16025-21e3-4bd3-a98c-13646a4fb46b>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00696.warc.gz"} |
Price per Count Calculator
The Price per Count Calculator can calculate the price for each count based on the total price of all the counts.
To calculate the price per count, we divide the total price of all the counts by the number of counts.
Please enter the total price of all the counts and the number of counts so we can calculate the price per count:
Price per Diaper Calculator
Here is a similar calculator you may find interesting. | {"url":"https://pricecalculator.org/per/price-per-count-calculator.html","timestamp":"2024-11-12T05:29:34Z","content_type":"text/html","content_length":"6471","record_id":"<urn:uuid:352e8898-caf9-40ab-b0f0-2fb2b1b15b06>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00569.warc.gz"} |
Average Order Value in context of sales revenue
13 Oct 2024
Title: The Impact of Average Order Value (AOV) on Sales Revenue: A Theoretical Analysis
Abstract: This article explores the concept of Average Order Value (AOV) and its significance in the context of sales revenue. We examine the formula for calculating AOV, its relationship with sales
revenue, and the implications for businesses seeking to optimize their pricing strategies.
Introduction: In today’s competitive business landscape, companies are constantly seeking ways to maximize their sales revenue while minimizing costs. One key metric that can help achieve this goal
is Average Order Value (AOV). AOV represents the average amount spent by a customer in a single transaction, and it plays a crucial role in determining a company’s overall sales revenue.
Formula for Calculating AOV: The formula for calculating AOV is as follows:
AOV = Total Revenue / Number of Orders
AOV = (Total Revenue) / (Number of Orders)
• AOV represents the Average Order Value
• Total Revenue refers to the total sales revenue generated by a company over a given period
• Number of Orders represents the number of transactions made by customers during the same period
Relationship between AOV and Sales Revenue: As evident from the formula, AOV is directly proportional to sales revenue. When AOV increases, it implies that customers are spending more on average in
each transaction, leading to higher total revenue for the company.
Implications for Businesses: Understanding the relationship between AOV and sales revenue can have significant implications for businesses seeking to optimize their pricing strategies. By focusing on
increasing AOV through targeted marketing campaigns, product bundling, or upselling/cross-selling initiatives, companies can potentially boost their sales revenue without necessarily increasing the
number of orders.
Conclusion: In conclusion, Average Order Value (AOV) is a critical metric that businesses should consider when evaluating their pricing strategies. By calculating and analyzing AOV, companies can
gain valuable insights into their customers’ spending habits and make informed decisions to optimize their sales revenue.
Related articles for ‘sales revenue’ :
• Reading: Average Order Value in context of sales revenue
Calculators for ‘sales revenue’ | {"url":"https://blog.truegeometry.com/tutorials/education/00164617175da023930885280e542be7/JSON_TO_ARTCL_Average_Order_Value_in_context_of_sales_revenue.html","timestamp":"2024-11-03T00:22:44Z","content_type":"text/html","content_length":"15914","record_id":"<urn:uuid:7cd9e0c6-4768-4fd7-bf19-ffccfa8d9183>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00404.warc.gz"} |
Finding a Possible Universal Set given Two of Its Subsets
Question Video: Finding a Possible Universal Set given Two of Its Subsets
Given that π = {7, 11, 22}, π = {1, 11, 21, 22}, and π and π are subsets of some universal set π , which of the following could be true? [A] π = {22} [B] π = {1, 11, 21} [C] π
= {1, 7, 11, 21} [D] π = {1, 7, 11, 21, 22}
Video Transcript
Given that π is a set containing the elements seven, 11, and 22, π is a set containing the elements one, 11, 21, and 22, and π and π are subsets of some universal set π , which of the
following could be true? π is a set containing the element 22, π is a set containing the elements one, 11, and 21, π is a set containing the elements one, seven, 11, and 21, or π is a set
containing the elements one, seven, 11, 21, and 22.
This problem tests whether weβ re able to say what a possible universal set might contain if we know some of its subsets. The question tells us that the universal set that weβ re talking about is π
, and π and π are the subsets. Letβ s represent this as a Venn diagram. Letβ s use this green square to represent the universal set π . This orange circle can represent set π . Notice
that weβ ve drawn the circle completely inside the square. This is because π is a subset of the universal set π . Everything in π is inside π . And the numbers in π are seven, 11, and
And we can also draw a pink circle to represent set π , which is also a subset of the universal set. Now, when we look at the elements of set π , we can see that two of the numbers are also in
set π . So, when we draw our pink circle, weβ re going to have to make sure it overlapped with the orange one. So, the two numbers that both sets share are 11 and 22. And then, as weβ ve said
already, set π contains the number seven too, and set π contains the numbers one and 21. So, this Venn diagram represents the question.
Now, weβ re given four statements about the universal set, π . And weβ re asked which of the following could be true. Each statement gives the letter π and then tells us what the elements of
that universal set are. So, the correct answer is the one that contains all the elements. When we look inside our green square, we can see the numbers one, seven, 11, 21, and 22. This means that the
correct answer is the one that says π equals a set containing the elements one, seven, 11, 21, and 22, the five numbers that we can see inside the set.
Notice the word could in the last question. Which of the following could be true? The reason why it could be true, and is not definitely true, is that the question doesnβ t tell us that π and π
are the only things within the universal set π . If they were the only sets within π , then this answer is correct. But of course, if there was another set there, then we would need to include
the numbers in that too. Universal sets contain everything within them.
So, given that π is a set containing the elements seven, 11, and 22, and π is a set containing the elements one, 11, 21, and 22. And given that both π and π are subsets of a universal
set, which weβ ll call π . The statement that could be true is that π is a set containing the elements one, seven, 11, 21, and 22. | {"url":"https://www.nagwa.com/en/videos/723187483540/","timestamp":"2024-11-03T03:41:51Z","content_type":"text/html","content_length":"246120","record_id":"<urn:uuid:8a479723-39ea-49ca-a29b-ecd0f0766dfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00447.warc.gz"} |
Ch-1 Real Numbers - Edu Spot- NCERT Solution, CBSE Course, Practice Test
Ch-1 Real Numbers
Euclid’s Division Lemma
It is basically the restatement of the usual division system. The formal statement for this is-
For each pair of given positive integers a and b, there exist unique whole numbers q and r which satisfies the relation
a = bq + r, 0 ≤ r < b, where q and r can also be Zero.
where ‘a’ is a dividend, ‘b' is divisor, ‘q’ is quotient and ‘r’ is remainder.
∴ Dividend = (Divisor x Quotient) + Remainder
Natural Numbers
Non-negative counting numbers excluding zero are known as natural numbers.
i.e. 5, 6, 7, 8, ……….
Whole numbers
All non-negative counting numbers including zero are known as whole numbers.
i.e. 0, 1, 2, 3, 4, 5, …………….
All negative and non-negative numbers including zero altogether known as integers.
i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
An algorithm gives us some definite steps to solve a particular type of problem in a well-defined manner.
A lemma is a statement which is already proved and is used for proving other statements.
Euclid’s Division Algorithm
This concept is based on Euclid’s division lemma. This is the technique to calculate the HCF (Highest common factor) of given two positive integers m and n,
To calculate the HCF of two positive integers’ m and n with m > n, the following steps are followed:
Step 1: Apply Euclid’s division lemma to find q and r where m = nq + r, 0 ≤ r < n.
Step 2: If the remainder i.e. r = 0, then the HCF will be ‘n’ but if r ≠ 0 then we have to apply Euclid’s division lemma to n and r.
Step 3: Continue with this process until we get the remainder as zero. Now the divisor at this stage will be HCF(m, n). Also, HCF (m, n) = HCF (n, r), where HCF (m, n) means HCF of m and n.
The Fundamental Theorem of Arithmetic
We can factorize each composite number as a product of some prime numbers and of course, this prime factorization of a natural number is unique as the order of the prime factors doesn’t matter.
• HCF of given numbers is the highest common factor among all which is also known as GCD i.e. greatest common divisor.
• LCM of given numbers is their least common multiple.
• If we have two positive integers ‘m’ and ‘n’ then the property of their HCF and LCM will be:
HCF (m, n) × LCM (m, n) = m × n.
Rational Numbers
The number ‘s’ is known as a rational number if we can write it in the form of m/n where ‘m' and ‘n’ are integers and n ≠ 0, 2/3, 3/5 etc.
Rational numbers can be written in decimal form also which could be either terminating or non-terminating. E.g. 5/2 = 2.5 (terminating) and
Irrational Numbers
The number ‘s’ is called irrational if it cannot be written in the form of m/n, where m and n are integers and n≠0 or in the simplest form, the numbers which are not rational are called irrational
numbers. Example - √2, √3 etc.
• If p is a prime number and p divides a^2 , then p is one of the prime factors of a^2 which divides a, where a is a positive integer.
• If p is a positive number and not a perfect square, then √n is definitely an irrational number.
• If p is a prime number, then √p is also an irrational number.
Rational Number and their Decimal Expansions
• Let y be a real number whose decimal expansion terminates into a rational number which we can express in the form of a/b, where a and b are coprime, and the prime factorization of the denominator
b has the powers of 2 or 5 or both like 2^n5^m, where n, m are non-negative integers.
• Let y be a rational number in the form of y = a/b, so that the prime factorization of the denominator b is of the form 2^n5^m, where n, m are non-negative integers then y has a terminating
decimal expansion.
• Let y = a/b be a rational number, if the prime factorization of the denominator b is not in the form of 2^n2^m, where n, m are non-negative integers then y has a non-terminating repeating decimal
□ The decimal expansion of every rational number is either terminating or a non-terminating repeating.
□ The decimal form of irrational numbers is non-terminating and non-repeating.
Some Important Quetions for class 10 Maths
Question 3.
Prove that the product of two consecutive positive integers is divisible by 2.
Let n and n + 1 are two consecutive positive integer
We know that n is of the form n = 2q and n + 1 = 2q + 1
n (n + 1) = 2q (2q + 1) = 2 (2q^2 + q)
Which is divisible by 2
If n = 2q + 1, then
n (n + 1) = (2q + 1) (2q + 2)
= (2q + 1) x 2(q + 1)
= 2(2q + 1)(q + 1)
Which is also divisible by 2
Hence the product of two consecutive positive integers is divisible by 2
Question 4.
Prove that the square of any positive, integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.
Let a be any positive integer
Let it be in the form of 3m or 3m + 1
Let a = 3q, then
Question 5.
Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers ?
We know that a composite number is that number which can be factorize. It has more factors other than itself and one
Now, 7 x 11 x 13 + 13 = 13 (7 x 11 + 1) = 13 x 78
Which is composite number
7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 = 5(7 x 6 x 4 x 3 x 2 x 1 + 1)
= 5 x 1009
Which is a composite number
NCERT SOLUTIONS CH - 1 MCQ SOLUTIONS CH - 1
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Welcome to JuliaAtoms!
The JuliaAtoms GitHub organisation collects a few Julia packages that are useful for calculations within Atomic Physics.
At the moment, the following packages are available (all of them under development, i.e. no stability promised yet):
• A library used to define electronic configurations constructed from electronic orbitals in spherical symmetry.
• Data structures for representing atoms in a product space of orbitals and a radial grid. The radial grid can be any implementation of the ContinuumArrays.jl interface, but AtomicStructure.jl has
only been tested with CompactBases.jl so far. It also contains a submodule for the solution of integro-differential eigenproblems, in a self-consistent manner, as well as using manifold
optimization routines from Optim.jl.
• A library for setting up the energy expression of a system built up from a set of configurations. At the moment, the implementation is geared towards atomic systems (in that it uses data
structures from AtomicLevels.jl), but it is applicable to other systems as well, such as molecules.
• A library for the special case of energy expressions in spherical symmetry, but more importantly, also the computation of tensor matrix elements between spin-orbitals.
• Contains some of the analytically known results for atomic hydrogen, or more generally, a one-electron system in spherical symmetry.
• A library that implements the calculation of the Coulombic repulsion between pairs of electrons, also known as the Slater integrals.
As mentioned above, the radial problem is implemented using
• This library implements various basis sets of compact support, such as finite-difference, finite-element discrete-variable representation, and B-splines, all with their respective benefits and
drawbacks for discretization of partial differential/integro-differential equations. | {"url":"http://juliaatoms.org/dev/index.html","timestamp":"2024-11-09T17:22:30Z","content_type":"text/html","content_length":"13958","record_id":"<urn:uuid:3fa600ad-f815-41f6-8ca2-80d2a4deab42>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00500.warc.gz"} |
Which gas do plants give off?
Which gas do plants give off?
Plants use photosynthesis to capture carbon dioxide and then release half of it into the atmosphere through respiration. Plants also release oxygen into the atmosphere through photosynthesis.
Is photosynthesis important to humans?
Photosynthesis And Cellular Respiration It also produces oxygen which is the gas we need to live. Photosynthesis creates clean air for humans to breathe. It also allows plants to grow, which feeds
How do humans use photosynthesis?
Humans have to grow, hunt, and gather food, but many living things aren’t so constrained. Plants, algae and many species of bacteria can make their own sustenance through the process of
photosynthesis. They harness sunlight to drive the chemical reactions in their bodies that produce sugars.
How energy from the sun is passed to you?
humans get energy indirectly from the sun, but directly through plants. plants get energy from the sun through photosynthesis, and we eat the plants.
How much oxygen do plants release?
Humans consume around 50 liters of oxygen per hour and each plant leaf gives off about five milliliters of oxygen per hour.
Do plants give off energy?
Plants use photosynthesis to convert light energy to chemical energy, which is stored in the bonds of sugars they use for food. The only byproducts of photosynthesis are protons and oxygen. “This is
potentially one of the cleanest energy sources for energy generation,” Ryu said.
How do plants get their energy?
Plants convert energy from sunlight into sugar in a process called photosynthesis. Photosynthesis uses energy from light to convert water and carbon dioxide molecules into glucose (sugar molecule)
and oxygen (Figure 2).
What is the importance of photosynthesis in the ecosystem?
It provides energy for nearly all ecosystems. By transforming light energy into chemical energy, photosynthesis provides the energy used by organisms, whether those organisms are plants,
grasshoppers, wolves, or fungi.
Why do plants make me happy?
Plants bring feelings of vitality and improve the state of mind. The subliminal effect of plants has an effect that lifts the spirit and brings happiness. An environment that includes natural
elements and plants brings a positive outlook on life and boosts people into feeling more alive and active.
What do plants give us?
Plants provide us with food, fiber, shelter, medicine, and fuel. The basic food for all organisms is produced by green plants. In the process of food production, oxygen is released. This oxygen,
which we obtain from the air we breathe, is essential to life.
Why do plants give off oxygen?
Plants produce oxygen as a waste product of making sugar using sunlight, carbon dioxide, and water. If a plant needs energy, but doesn’t have sunlight, then it can burn the sugar that it made back
when it had sunlight, and doing so requires oxygen.
Can money plant be kept in bedroom?
Placing a money plant in the bedroom helps avoid arguments and cure sleeping disorders. The very important benefit of money plant is that it attracts wealth, hence the name money plant. Vastu experts
suggest keeping a money plant in the house helps remove financial obstacles and brings prosperity & good luck.
What animal uses photosynthesis?
The sea slugs live in salt marshes in New England and Canada. In addition to burglarizing the genes needed to make the green pigment chlorophyll, the slugs also steal tiny cell parts called
chloroplasts, which they use to conduct photosynthesis.
Why do plants need energy?
To do that, plants need water and sun to make their own food energy called: (Photosynthesis). They need energy because they are living things. Living things require energy to carry out a number of
processes, including growth and reproduction. For this reason, energy is very important.
Do we eat sunlight?
Yes, definitely. Almost all living things on earth are fueled by the sun either directly or indirectly. (The exception is organisms that live near volcanic vents deep in the ocean.) Energy from the
sun is taken in by plants and eaten by animals.
Do plants give off oxygen?
Breathing Easier During photosynthesis, plants absorb carbon dioxide and release oxygen.
What does photosynthesis produce?
During the process of photosynthesis, cells use carbon dioxide and energy from the Sun to make sugar molecules and oxygen. These sugar molecules are the basis for more complex molecules made by the
photosynthetic cell, such as glucose.
How does energy from the sun become food?
In this case plants convert light energy (1) into chemical energy, (in molecular bonds), through a process known as photosynthesis. Most of this energy is stored in compounds called carbohydrates.
The plants convert a tiny amount of the light they receive into food energy.
Which plants are good for home?
10 Best Vastu Plants for Home
1. Money Plant. Money plant is considered one of the best fortune bringing plants to be placed at home.
2. Tulsi. Tulsi is believed to be a goddess itself and is considered the queen of the herbs.
3. Neem Tree.
4. Lucky Bamboo Plant.
5. Citrus Plant.
6. Aloe Vera.
7. Banana Tree.
8. Lily Plant.
What is the importance of photosynthesis Class 7?
Photosynthesis helps to maintain a balance between oxygen and carbon dioxide in the atmosphere as it absorbs carbon dioxide and release oxygen. Sunlight is necessary for photosynthesis. Thus sun is
the ultimate source of energy for all living organism. Our earth is the unique planet, where photosynthesis takes place. | {"url":"https://www.presenternet.com/which-gas-do-plants-give-off/","timestamp":"2024-11-06T04:16:03Z","content_type":"text/html","content_length":"45130","record_id":"<urn:uuid:33edf95a-3f6d-4975-8ede-272886f0483a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00725.warc.gz"} |
RRB JE 1st June 2019 Shift-3
Compare static friction and sliding friction.
Graafian follicles are characteristically found in the -
Two squares differ in areas by 32 cm². If the difference in their sides is 4 cm, what are the sides of the two squares?
Who was invited by Lord Wavell to form the interim Government in India in 1946?
In this question, two statements are given followed by two conclusions. Choose the conclusion(s) which best fit(s) logically.
1) Some towns are cities.
2) All cities are homes.
I. Some towns are homes.
II. Some homes are cities.
Five years ago, the average age of a couple was 24. At present, the average of the couple and a child is 20. What is the child's age?
Find the missing group of alphabets in the following series. ABC, EFG, IJK, (…), UVW
Two boys A and B start from the opposite direction of 14 km apart. A is facing east and B isfacing west. A reaches the distance of 5 km towards east and B reaches a distance of 2 kmtowards west. What
is the distance between the two boys?
The ratio of salaries of P and Q last year is 4 : 5. The ratio of last year salary and the presentsalary of P is 3: 5 and for Q this ratio is 2 : 3. If their total salary at present is Rs. 6800, what
is salary of Q ?
If $$x^{4}+\frac{1}{x^{4}}$$=47 ,then find the value of x+$$\frac{1}{x}$$ | {"url":"https://cracku.in/rrb-je-1st-june-2019-shift-3-question-paper-solved?page=4","timestamp":"2024-11-11T23:42:50Z","content_type":"text/html","content_length":"157502","record_id":"<urn:uuid:65332b3a-3206-4f83-b4bc-18d01e9ce656>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00496.warc.gz"} |
Irrational number
Short description: Number that is not a ratio of integers
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers
cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that
they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples
of itself.
Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two.^[1] In fact, all square roots of natural
numbers, other than of perfect squares, are irrational.^[2]
Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end
with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal
expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics.
Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.
As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.^[3]
Ancient Greece
The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),^[4] who probably discovered them while identifying sides of the
pentagram.^[5] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the
other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by
demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is
impossible. His reasoning is as follows:
• Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by c:b.
• Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).
• By the Pythagorean theorem: c^2 = a^2+b^2 = b^2+b^2 = 2b^2. (Since the triangle is isosceles, a = b).
• Since c^2 = 2b^2, c^2 is divisible by 2, and therefore even.
• Since c^2 is even, c must be even.
• Since c is even, dividing c by 2 yields an integer. Let y be this integer (c = 2y).
• Squaring both sides of c = 2y yields c^2 = (2y)^2, or c^2 = 4y^2.
• Substituting 4y^2 for c^2 in the first equation (c^2 = 2b^2) gives us 4y^2= 2b^2.
• Dividing by 2 yields 2y^2 = b^2.
• Since y is an integer, and 2y^2 = b^2, b^2 is divisible by 2, and therefore even.
• Since b^2 is even, b must be even.
• We have just shown that both b and c must be even. Hence they have a common factor of 2. However, this contradicts the assumption that they have no common factors. This contradiction proves that
c and b cannot both be integers and thus the existence of a number that cannot be expressed as a ratio of two integers.^[6]
Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while
out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be
reduced to whole numbers and their ratios.'^[7] Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very
serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory.
The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This was brought to light by Zeno of Elea, who questioned
the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for "whole numbers represent
discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects",^[8] but Zeno found that in fact "[quantities] in general are not discrete collections
of units; this is why ratios of incommensurable [quantities] appear... .[Q]uantities are, in other words, continuous".^[8] What this means is that contrary to the popular conception of the time,
there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be
split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is
halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes, which
demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not
regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to
The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the
distinction between magnitude and number. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say,
continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5".^[9] Numbers are composed of some smallest, indivisible unit, whereas magnitudes are
infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of
its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a
number. "Eudoxus' theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios".^[10] This
incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios
that a strong mathematical foundation of irrational numbers was created.^[11]
As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still
incompatible with the concept of incommensurability, Greek focus shifted away from numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic
conceptions were reformulated into geometric terms. This may account for why we still conceive of x^2 and x^3 as x squared and x cubed instead of x to the second power and x to the third power. Also
crucial to Zeno's work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization
that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this
necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the
earlier decision to rely on deductive reasoning for proof".^[12] This method of exhaustion is the first step in the creation of calculus.
Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17.^[13]
Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the
Samhitas, Brahmanas, and the Shulba Sutras (800 BC or earlier). (See Bag, Indian Journal of History of Science, 25(1-4), 1990).
It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers
such as 2 and 61 could not be exactly determined.^[14] Historian Carl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true".^[15]
Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square
Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century Bhāskara II evaluated some of
these formulas and critiqued them, identifying their limitations.
During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain
irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the Yuktibhāṣā.^[17]
Middle Ages
In the Middle ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects.^[18] Middle Eastern mathematicians also merged the concepts of "
number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of
continuous magnitude.^[19] In his commentary on Book 10 of the Elements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He
provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:^[19]
"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is
impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."
In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also
introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:^[19]
"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form
of square roots, cube roots and fourth roots.^[20] In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he
considered multiplication, division, and other arithmetical functions.^[19] Iranian mathematician, Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating
that if a definite quantity is:^[19]
"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time this (latter) magnitude comprises a half, a third, or a
quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three-fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to
this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (1/n), or parts (m/n) of a given magnitude, it is
irrational, i.e. it cannot be expressed other than by means of roots."
Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. Al-Hassār, a Moroccan mathematician from Fez specializing in
Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he
writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, [math]\displaystyle{ \frac{3 \quad 1}{5 \quad 3} }[/math]."^[21] This same fractional notation
appears soon after in the work of Leonardo Fibonacci in the 13th century.^[22]
Modern period
The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. The completion of the theory of complex numbers in the 19th century
entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory
of irrationals, largely ignored since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor
(Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set
forth by Salvatore Pincherle in 1880,^[23] and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor,
and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain
characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray.
Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence
through the writings of Joseph-Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e^n is irrational if n is rational (unless n = 0).^[24] While Lambert's proof is often called incomplete, modern assessments
support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the Bessel–Clifford function, provided a proof to show that π^2 is
irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their
existence by a different method, which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann
(1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary
by Adolf Hurwitz and Paul Gordan.^[25]
Square roots
The square root of 2 was likely the first number proved irrational.^[26] The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers that are not perfect
squares are irrational and a proof may be found in quadratic irrationals.
General roots
The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization into primes. Using it we can show
that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator that does not divide into the numerator
whatever power each is raised to. Therefore, if an integer is not an exact kth power of another integer, then that first integer's kth root is irrational.
Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction that log[2] 3 is irrational (log[2] 3 ≈ 1.58 > 0).
Assume log[2] 3 is rational. For some positive integers m and n, we have
[math]\displaystyle{ \log_2 3 = \frac{m}{n}. }[/math]
It follows that
[math]\displaystyle{ 2^{m/n}=3 }[/math]
[math]\displaystyle{ (2^{m/n})^n = 3^n }[/math]
[math]\displaystyle{ 2^m=3^n. }[/math]
The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will
be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log[2] 3 is rational (and so expressible as a quotient of
integers m/n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log[2] 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.
Cases such as log[10] 2 can be treated similarly.
• number theoretic distinction: transcendental/algebraic
Almost all irrational numbers are transcendental and all real transcendental numbers are irrational (there are also complex transcendental numbers): the article on transcendental numbers lists
several examples. So e^ r and π^ r are irrational for all nonzero rational r, and, e.g., e^π is irrational, too.
Irrational numbers can also be found within the countable set of real algebraic numbers (essentially defined as the real roots of polynomials with integer coefficients), i.e., as real solutions of
polynomial equations
[math]\displaystyle{ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\;, }[/math]
where the coefficients [math]\displaystyle{ a_i }[/math] are integers and [math]\displaystyle{ a_n \ne 0 }[/math]. Any rational root of this polynomial equation must be of the form r /s, where r is a
divisor of a[0] and s is a divisor of a[n]. If a real root [math]\displaystyle{ x_0 }[/math] of a polynomial [math]\displaystyle{ p }[/math] is not among these finitely many possibilities, it must be
an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that x[0] = (2^1/2 + 1)^1/3 is an irrational root of a polynomial with integer
coefficients: it satisfies (x^3 − 1)^2 = 2 and hence x^6 − 2x^3 − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and x[0], being greater than 1, is
neither of these), so x[0] is an irrational algebraic number.
Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3π + 2, π + √2 and e
√3 are irrational (and even transcendental).
Decimal expansions
The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), unlike any rational number. The same is true for binary, octal or
hexadecimal expansions, and in general for expansions in every positional notation with natural bases.
To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, there can never be a remainder greater than or equal to m. If 0 appears
as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and
then the decimal expansion repeats.
Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:
[math]\displaystyle{ A=0.7\,162\,162\,162\,\ldots }[/math]
Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In
this example we would multiply by 10 to obtain:
[math]\displaystyle{ 10A = 7.162\,162\,162\,\ldots }[/math]
Now we multiply this equation by 10^r where r is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10^3:
[math]\displaystyle{ 10,000A=7\,162.162\,162\,\ldots }[/math]
The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A
and 10A have .162162162... after the decimal point.
Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:
[math]\displaystyle{ 9990A=7155. }[/math]
[math]\displaystyle{ A= \frac{7155}{9990} = \frac{53}{74} }[/math]
is a ratio of integers and therefore a rational number.
Irrational powers
Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a^b is rational:^[27]
Consider √2^√2; if this is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2^√2 and b = √2. Then a^b = (√2^√2)^√2 = √2^√2·√2 = √2^2 = 2, which is rational.
Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √2^√2 is transcendental, hence irrational. This theorem states that if a and b are both
algebraic numbers, and a is not equal to 0 or 1, and b is not a rational number, then any value of a^b is a transcendental number (there can be more than one value if complex number exponentiation is
An example that provides a simple constructive proof is^[28]
[math]\displaystyle{ \left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3. }[/math]
The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, [math]\displaystyle{ \log_{\sqrt{2}}3 }[/math], is irrational. This is so
because, by the formula relating logarithms with different bases,
[math]\displaystyle{ \log_{\sqrt{2}}3=\frac{\log_2 3}{\log_2 \sqrt{2}}=\frac{\log_2 3}{1/2} = 2\log_2 3 }[/math]
which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then [math]\displaystyle{ \log_2 3 = m/2n }[/math] hence [math]\displaystyle{ 2^{\log_2 3}=
2^{m/2n} }[/math] hence [math]\displaystyle{ 3=2^{m/2n} }[/math] hence [math]\displaystyle{ 3^{2n}=2^m }[/math], which is a contradictory pair of prime factorizations and hence violates the
fundamental theorem of arithmetic (unique prime factorization).
A stronger result is the following:^[29] Every rational number in the interval [math]\displaystyle{ ((1/e)^{1/e}, \infty) }[/math] can be written either as a^a for some irrational number a or as n^n
for some natural number n. Similarly,^[29] every positive rational number can be written either as [math]\displaystyle{ a^{a^a} }[/math] for some irrational number a or as [math]\displaystyle{ n^{n^
n} }[/math] for some natural number n.
Open questions
It is not known if [math]\displaystyle{ \pi+e }[/math] (or [math]\displaystyle{ \pi-e }[/math]) is irrational. In fact, there is no pair of non-zero integers [math]\displaystyle{ m, n }[/math] for
which it is known whether [math]\displaystyle{ m\pi+ n e }[/math] is irrational. Moreover, it is not known if the set [math]\displaystyle{ \{\pi, e\} }[/math] is algebraically independent over [math]
\displaystyle{ \Q }[/math].
It is not known if [math]\displaystyle{ \pi e,\ \pi/e,\ \pi^e,\ \pi^\sqrt{2},\ \ln\pi, }[/math] Catalan's constant, or the Euler–Mascheroni constant [math]\displaystyle{ \gamma }[/math] are
irrational.^[30] It is not known if either of the tetrations [math]\displaystyle{ ^n\pi }[/math] or [math]\displaystyle{ ^n e }[/math] is rational for some integer [math]\displaystyle{ n \gt 1. }[/
In constructive mathematics
In constructive mathematics, excluded middle is not valid, so it is not true that every real number is rational or irrational. Thus, the notion of an irrational number bifurcates into multiple
distinct notions. One could take the traditional definition of an irrational number as a real number that is not rational.^[31] However, there is a second definition of an irrational number used in
constructive mathematics, that a real number [math]\displaystyle{ r }[/math] is an irrational number if it is apart from every rational number, or equivalently, if the distance [math]\displaystyle{ \
vert r - q \vert }[/math] between [math]\displaystyle{ r }[/math] and every rational number [math]\displaystyle{ q }[/math] is positive. This definition is stronger than the traditional definition of
an irrational number. This second definition is used in Errett Bishop's proof that the square root of 2 is irrational.^[32]
Set of all irrationals
Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.
Under the usual (Euclidean) distance function [math]\displaystyle{ d(x, y) = \vert x - y \vert }[/math], the real numbers are a metric space and hence also a topological space. Restricting the
Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed, the induced metric is not complete. Being a G-delta set—i.e., a
countable intersection of open subsets—in a complete metric space, the space of irrationals is completely metrizable: that is, there is a metric on the irrationals inducing the same topology as the
restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction
expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.
Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen groups so the space is zero-dimensional
See also
• Proof that π is irrational
1. ↑ The 15 Most Famous Transcendental Numbers. by Clifford A. Pickover. URL retrieved 24 October 2007.
2. ↑ Jackson, Terence (2011-07-01). "95.42 Irrational square roots of natural numbers — a geometrical approach" (in en). The Mathematical Gazette 95 (533): 327–330. doi:10.1017/S0025557200003193.
ISSN 0025-5572. https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/9542-irrational-square-roots-of-natural-numbers-a-geometrical-approach/6B9D8EBFDCC016013D303AA78973429F.
3. ↑ Cantor, Georg (1955). Philip Jourdain. ed. Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0-486-60045-1. https://archive.org/details/
4. ↑ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics 46 (2): 242–264. doi:10.2307/1969021.
5. ↑ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal 11 (5): 312–316. doi:10.2307/3026893.
6. ↑ Kline, M. (1990). Mathematical Thought from Ancient to Modern Times, Vol. 1. New York: Oxford University Press (original work published 1972), p. 33.
7. ↑ Kline 1990, p. 32.
8. ↑ ^8.0 ^8.1 Kline 1990, p. 34.
9. ↑ Kline 1990, p. 48.
10. ↑ Kline 1990, p. 49.
11. ↑ Charles H. Edwards (1982). The historical development of the calculus. Springer.
12. ↑ Kline 1990, p. 50.
13. ↑ Robert L. McCabe (1976). "Theodorus' Irrationality Proofs". Mathematics Magazine 49 (4): 201–203. doi:10.1080/0025570X.1976.11976579. .
14. ↑ T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, in Selin, Helaine; D'Ambrosio, Ubiratan, eds (2000). Mathematics Across Cultures: The History of Non-western
Mathematics. Springer. ISBN 1-4020-0260-2. .
15. ↑ Boyer (1991). "China and India". A History of Mathematics (2nd ed.). p. 208. ISBN 0471093742. OCLC 414892. "It has been claimed also that the first recognition of incommensurables appears in
India during the Sulbasutra period, but such claims are not well substantiated. The case for early Hindu awareness of incommensurable magnitudes is rendered most unlikely by the lack of evidence
that Indian mathematicians of that period had come to grips with fundamental concepts."
16. ↑ Datta, Bibhutibhusan; Singh, Awadhesh Narayan (1993). "Surds in Hindu mathematics". Indian Journal of History of Science 28 (3): 253–264. https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/
Vol28_3_2_BDatta.pdf. Retrieved 18 September 2018.
17. ↑ Katz, V. J. (1995). "Ideas of Calculus in Islam and India". Mathematics Magazine 63 (3): 163–174. doi:10.2307/2691411.
18. ↑ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/
HistTopics/Arabic_mathematics.html ..
19. ↑ ^19.0 ^19.1 ^19.2 ^19.3 ^19.4 Matvievskaya, Galina (1987). "The theory of quadratic irrationals in medieval Oriental mathematics". Annals of the New York Academy of Sciences 500 (1): 253–277.
doi:10.1111/j.1749-6632.1987.tb37206.x. Bibcode: 1987NYASA.500..253M. See in particular pp. 254 & 259–260.
20. ↑ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer. ISBN 1-4020-0260-2
. .
21. ↑ Cajori, Florian (1928). A History of Mathematical Notations (Vol.1). La Salle, Illinois: The Open Court Publishing Company. pg. 269.
22. ↑ (Cajori 1928)
23. ↑ Salvatore Pincherle (1880). "Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principii del prof. C. Weierstrass". Giornale di Matematiche: 178–254, 317–320. https://
24. ↑ Lambert, J. H. (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes, circulaires et logarithmiques" (in fr). Mémoires de l'Académie royale des sciences de Berlin:
265–322. http://www.kuttaka.org/~JHL/L1768b.pdf.
25. ↑ Gordan, Paul (1893). "Transcendenz von e und π". Mathematische Annalen (Teubner) 43 (2–3): 222–224. doi:10.1007/bf01443647. https://zenodo.org/record/1428218.
26. ↑ Fowler, David H. (2001), "The story of the discovery of incommensurability, revisited", Neusis (10): 45–61
27. ↑ George, Alexander; Velleman, Daniel J. (2002). Philosophies of mathematics. Blackwell. pp. 3–4. ISBN 0-631-19544-0. http://condor.depaul.edu/mash/atotheamg.pdf.
28. ↑ Lord, Nick, "Maths bite: irrational powers of irrational numbers can be rational", Mathematical Gazette 92, November 2008, p. 534.
29. ↑ ^29.0 ^29.1 Marshall, Ash J., and Tan, Yiren, "A rational number of the form a^a with a irrational", Mathematical Gazette 96, March 2012, pp. 106-109.
30. ↑ Albert, John. "Some unsolved problems in number theory". Department of Mathematics, University of Oklahoma. http://www.math.ou.edu/~jalbert/courses/openprob2.pdf. (Senior Mathematics Seminar,
Spring 2008 course)
31. ↑ Mark Bridger (2007). Real Analysis: A Constructive Approach through Interval Arithmetic. John Wiley & Sons. ISBN 978-1-470-45144-8.
32. ↑ Errett Bishop; Douglas Bridges (1985). Constructive Analysis. Springer. ISBN 0-387-15066-8.
Further reading
• Adrien-Marie Legendre, Éléments de Géometrie, Note IV, (1802), Paris
• Rolf Wallisser, "On Lambert's proof of the irrationality of π", in Algebraic Number Theory and Diophantine Analysis, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyter
External links
Original source: https://en.wikipedia.org/wiki/Irrational number. Read more | {"url":"https://handwiki.org/wiki/Irrational_number","timestamp":"2024-11-12T03:45:56Z","content_type":"text/html","content_length":"128937","record_id":"<urn:uuid:78d3d954-656b-4ca9-b690-96c08fc3f312>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00256.warc.gz"} |
Free Operational Amplifiers Books Download | Ebooks Online
Operational Amplifiers Books
There are many downloadable free Operational Amplifiers books, available in our collection of books. Which are available in the form of PDF, Online Textbooks, eBooks and lecture notes. These books
cover basics, beginner, and advanced concepts and also those who looking for introduction to the same. | {"url":"https://www.freebookcentre.net/Electronics/Operational-Amplifiers-Books.html","timestamp":"2024-11-04T01:37:19Z","content_type":"text/html","content_length":"32899","record_id":"<urn:uuid:aadf229e-8594-42ae-a81f-b4dd9fcc5272>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00746.warc.gz"} |
A metal container on the form of a cylinder surmounted by a hem-Turito
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A metal container on the form of a cylinder surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7 m and the internal radius is 3.5 m, Calculate the internal volume
of the container in cubic meter ?
Volume of cylinder
The correct answer is: 359.18
• We have given a cylinder is surmounted by a hemisphere with same radius. Height of the cylinder is 7mnd internal radius is 3.5m
• We have to find the internal volume of the container.
Step 1 of 1:
The radius of the cylinder is 3.5m and the height is 7m
So, The volume of cylinder will be
= 269.39
And the volume of the hemisphere will be
So, Total volume of the solid will be
= 359.18m^3
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Trading Stocks Generate Its Own Problems – Part II
Jan. 16, 2019
In Part I, it was shown that a trading strategy could be expressed as a simple equation. The outcome of that equation gave the sum of all winning and losing trades. You would have a number of losing
trades, and the rest would have an average profit, making the strategy worthwhile or not.
The equation was: Ʃ x[i] = (n - λ)x[+] + λx[-] where λ was the number of losing trades, x[+] the average profit per winning trade, while x[-] was the average loss per losing trade. The equation says
that of all the trades taken; there will be λ losing trades with an average loss per trade of x[-]. The remaining trades will be at an average profit.
Each time a simulation is performed, those numbers become available for analysis and can help to statistically describe what the trading strategy has done over the period where it was applied. To
know the average profit or loss per trade is simple: you take the sum of all profits and losses Ʃ x[i] and divide it by n, the total number of trades. You know the average profit or loss will tend to
some constant the more the number n of trades is large. A direct consequence of the Law of large numbers. You also know you will not win all the trades all the time, hence λ.
Put the equation in Excel; it is not that complicated. Take some of your own simulation results. What it will show are some of the limits your trading strategy might have. Without knowing these
limits, what kind of expectation can you extract from your strategy?
Following is a snapshot that could get you started. I used Leo C's tearsheet (Link no longer available due to Quantopian shutdown) as an example since it had "round_trips=True" in its backtest
analysis. Thanks, Leo.
Strategy Equation
(click to enlarge)
It is presented with a general section and an example from the cited Quantopian tearsheet. The first column increases the number of trades by a factor of 10. The win and loss rates determine the
percentage of trades that finished with a profit or loss. The average win and average loss columns give the average amount won or lost per trade. The rest of the columns take the total profit Ʃ x[i]
to give the CAGR corresponding to the number of years it could have taken to get there. The yellow cells are used to build scenarios based on the equation at the top of the chart; the rest are
There Is Math To This Game
Whether we like it or not, there is math to this game.
The equation above prevails no matter what is the composition of your trading strategy. The above table says a lot. The more trades are performed (while keeping the same edge), and the shorter the
time interval used to do the job, the better the CAGR.
Thereby saying, the number n of executed trades matters. And how long it takes to execute these trades matters too. It is as if you were in this race to terminate your n trades as fast as you
possibly can.
Performance degrades very fast should you take longer to execute the same outcome, as illustrated in the following chart, which uses the “n = 10,000” line. It does say that the longer you take to
execute all those trades, the lower the CAGR will be. And it degrades fast. Note that using another line will only change the scale, not the shape of the curve.
Strategy Equation CAGR
(click to enlarge)
What should we take out of this? We should look at the numbers and figure out how we could improve on them no matter what our trading strategy may be. With all other things being equal, we could look
at the problem from the point of view of one variable at a time. We increase the number of trades over the same time period, and it improves the picture. We reduce λ the number of losing trades, and
we improve the end results. We compact those trades into shorter time intervals, which will also improve performance. We positively increase the spread (the strategy's edge) between our average win
and average loss, and it will also improve overall results.
These measures did not deal with the nature of the trading strategy, only its math and how it will end. The what we do to accomplish this task could be anything that shows its mark in a backtest,
even if it is not related to the way we usually operate, as long as there is some logical reason for it to do what it does.
It might not even matter if you are operating off the fumes of white noise to get your performance as long as you get it, and there is some rationale that can justify your method of play. If you want
to game the game, go ahead, but know why you are doing it while keeping a long-term vision of the goals you want to reach. Gambling your way out based on your know-how is also admissible. Your
trading account will not be able to tell the difference either way or which trading methods you used. In fact, it will not even care what you used to make the account grow. It will only tally the
results one trade at a time.
If you modify your code and the number of trades is somehow reduced, the other variables in the equation will have to work harder to compensate. And if they do not, the strategy will degrade even
faster. You are not in a search for some equilibrium, the real task is to maximize the outcome Ʃ x[i] using whatever you have available and do the job as fast as you possibly can.
A simple question like: how do I increase the number of trades becomes important? You know it will have an impact if you do so, then it becomes your task to make it happen within the limitations of
your trading account. A simple solution would be to do more of whatever your strategy is already doing which, on average, provided you with your positive edge.
For instance, under the same trading conditions as Leo's tearsheet, one could find ways to increase the number of trades per year. From the tearsheet charts, trading volume, exposure, and number of
trades are relatively constant. Therefore, and due to the size of the sample, we could use those numbers as averages to make projections on an annual basis.
Doing More
A few lines were added to the first chart, as shown below:
Strategy Equation Enhanced
(click to enlarge)
It starts by converting to a per-year basis (see "IF per year" line), where the number of trades becomes the average per year. Each year, the average outcome is added to its performance. We can still
see the CAGR degrading over the years. This view is more realistic than the previous one since they do match what is coming out of the tearsheet. Therefore, the strategy, with no fault of its own,
will see its CAGR degrade with time as if the strategy were breaking down when all that is needed would be to compensate for the deterioration.
To compensate for the CAGR degradation, it appears sufficient, in this case, to increase the number of trades by 2.3% per year. This is not a major move. It is adding over its first year 261 similar
trades to its 11,353 (about 1 trade per day).
If the 2.3% increase was sufficient to maintain the long-term CAGR, going to 5.0% is enough to see the CAGR rise over the years. This is an expanding CAGR. The more you trade, the more you get while
doing the same things as before.
It is easy to compensate for long-term CAGR degradation. It can be done on any trading strategy. And if you wanted more, increase the number of trades even more as illustrated in the 10.0% line.
It is all one's choice. You code your trading strategy to do what you want it to do while faced with uncertainty. Except that, you should want to organize everything in such a way that, in the end,
you will win no matter what.
This is far-reaching. For all those that think that all trading strategies fail, well, think again, it is not necessarily true, except if you want it to. But, let it be said that it is your choice to
compensate for return degradation or not. If you do not intend to do it, then be ready to accept the consequences.
Not only can you compensate for the inherent CAGR decay seen in most trading strategies, but you can also reach the expanding CAGR level by finding more ways to increase Ʃ x[i] and thereby compensate
even more. This is covered in more detail in my book Building Your Stock Portfolio.
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How much is a yard of gravel - Civil Sir
How much is a yard of gravel
How much is a yard of gravel | how much is a yard of gravel weigh | how much is a yard of gravel cover | how much is a yard of gravel cost | how much is 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 yards of gravel
weigh, cover and cost.
Gravel is one of the most important building material collected from river basin, mountains, rocks, small rocks, pebbles, loose and dry sand, aggregate and pea gravel. Pea gravel is chosen best for
walkway because it is round in shape of small size which make it most comfortable to work on.
How much area does a yard of gravel cover? It will depends on the size of the stone, the amount of dust content and the thickness of the layer and the how level the surface is to be covered.
Gravel is made of rock is to build roads, path, driveway, Patio, pedestrian, pavement, pathways, roadways, landscaping, and etc.
Gravel is categorised according to their size, size more than 5mm put in the category of gravel, it is formed of igneous rocks. It is categorised as fine gravel (4 – 8 mm), medium gravel (8 – 16 mm),
coarse gravel (16 – 32 mm), pebbles (32 – 64 mm), cobbles (64 – 256 mm) and Boulder more than 256mm.
How much is a yard of gravel
You are looking to buying gravel and crushed stone for your construction work, if you want to apply them in normal depth 50 mm for your driveway and 35 mm depth normally in pedestrian pathway, you
need to know how much area will cover by 1 ton of gravel and you should purchase and you can load in your vehicle.
Most of gravel supplier which are available nearly to you, will give you option to deliver gravel and crushed stone at your homes, for this they should cost some money for transportation. If you have
a truck or vehicle that you can use to bring gravel to your destination or construction site, then it is cheaper and faster option for you.
In this regarding, how much is a yard of gravel, how much is a yard of gravel weigh, how much is a yard of gravel cover, how much is a yard of gravel cost, Knowing about full detail analysis, then
you should keep reading.
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How much is a yard of gravel
Weight of gravel depends on the rocks type, loose and dense condition, compact, moisture content, dry and wet condition, others inorganic mixed in the gravel. For estimating purpose, contractor’s and
Builder’s would take weight of gravel as 3000lb (pounds) per yard or 1.5 short tons per yard, and which is equivalent to 110 lb per cubic feet.
1 yard of gravel look likes by 3 feet long by 3 feet wide and by 3 feet height, which is equal as 27 cubic feet as (length× width× height= 27).
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically weighs about 3,000lb (pounds) or 1.5 tons, and will cover approximately 162 square feet at 2 inches
thick, and will cost between $15 to $75 per yard, with an average cost of $40 per yard. If you buy gravel bag, 54 bags of 50 lb gravel to cover the same area as a cubic yard.
How much is a yard of gravel weigh
A cubic yard of gravel, which visually is 3 foot long by 3 foot wide by 3 foot tall, is typically weighs about 3,000lb (pounds) or 1.5 tons, which is approximately equal as 110 lbs per cubic foot. A
cubic yard of dry gravel is typically weighs about 2970 pounds or 1.5 tons. A cubic yard of wet gravel is typically weighs about 3375 pounds or 1.7 tons. Moisture is prime factors that determining
the weight of gravel.
How much is a yard of gravel cover
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically covers 162 square feet (18 square yards or 15 square meters) at recommended depth of 2 inches thick,
324 square feet at 1 inch thick, 108 square feet (12 square yards or 10 square meters) at 3 inches thick, or 81 square feet (9 square yards or 7.5 square meters) at 4 inches thick.
How much is a yard of gravel cost
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically costs ranges from $15 to $75, with an average of $40 per yard. The costs of gravel ranges from $10 to
$50 per ton, or $15 to $75 per cubic yard, or $1 to $3 per square foot, or $1350 per truckload depending on the rock types, volume, and travel distance. Gravel spreading costs $12 per yard or $46 per
How much is 2 yards of gravel
2 cubic yards of gravel is typically weighs around 6,000 pound (3 tons) and which will cover 200 square feet area at 3 inches deep, and will cost an average of $80 for 2 cubic yards of gravel
(national an average cost will be $40 per yard).
How much is 3 yards of gravel
3 cubic yards of gravel is typically weighs around 9,000 pound (4.5 tons) and which will cover 300 square feet area at 3 inches deep, and will cost an average of $120 for 3 cubic yards of gravel
(national an average cost will be $40 per yard).
How much is 4 yards of gravel
4 cubic yards of gravel is typically weighs around 12,000 pound (6 tons) and which will cover 400 square feet area with a standard depth of 3 inches thick, and will cost an average of $160 for 4
cubic yards of gravel (national an average cost will be $40 per yard).
How much is 5 yards of gravel
5 cubic yards of gravel is typically weighs around 15,000 pounds or 7.5 tons, and which will cover 500 square feet area with a standard depth of 3 inches thick, and will cost an average of $200 for 5
cubic yards of gravel (national an average cost will be $40 per yard).
How much is 6 yards of gravel
6 cubic yards of gravel, is typically weighs around 18,000lb (pounds) or 9 tons, and which will cover 972 square feet area with a standard depth of 2 inches thick, and will cost an average of $240
for 6 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 7 yards of gravel
7 cubic yards of gravel, is typically weighs around 21,000lb (pounds) or 10.5 tons, and which will cover 1,134 square feet (126 square yards, or 105 m2) area with a standard depth of 2 inches thick,
and will cost an average of $280 for 7 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 8 yards of gravel
8 cubic yards of gravel, is typically weighs around 24,000lb (pounds) or 12 tons, and which will cover 1,296 square feet area (144 square yards, or 120 m2) with a standard depth of 2 inches thick,
and will cost an average of $320 for 8 cubic yards of gravel (national an average cost will be $40 per yard).
“How many bags of concrete is in a yard
“How to figure yards, cubic feet or bag of dirt
“How to figure yards of gravel
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“How much does a yard of mulch cover
How much is 9 yards of gravel
9 cubic yards of gravel, is typically weighs around 27,000lb (pounds) or 13.5 tons, and which will cover 1,458 square feet area (162 square yards, or 135 m2) with a standard depth of 2 inches thick,
and will cost an average of $3600 for 9 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 10 yards of gravel
10 cubic yards of gravel, is typically weighs around 30,000lb (pounds) or 15 tons, and which will cover 1,620 square feet area (180 square yards, or 150 m2) with a standard depth of 2 inches thick,
and will cost an average of $400 for 10 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 12 yards of gravel
12 cubic yards of gravel, is typically weighs around 36,000lb (pounds) or 18 tons, and which will cover 1,944 square feet area (216 square yards, or 180 m2) with a standard depth of 2 inches thick,
and will cost an average of $480 for 12 cubic yards of gravel (national an average cost will be $40 per yard).
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically weighs about 3000lb (pounds) or 1.5 tons, and will cover approximately 162 square feet at 2 inches
thick, and will cost between $15 to $75 per yard, with an average cost of $40 per yard. If you buy gravel bag, 54 bags of 50 lb gravel to cover the same area as a cubic yard. | {"url":"https://civilsir.com/how-much-is-a-yard-of-gravel/","timestamp":"2024-11-08T13:59:05Z","content_type":"text/html","content_length":"97172","record_id":"<urn:uuid:ff6eefbf-0027-4a55-b681-8a65c264d09e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00862.warc.gz"} |
This AdaptableUnaryFunction computes the square-free part of a polynomial of type PolynomialTraits_d::Polynomial_d up to a constant factor.
A polynomial \( p\) can be factored into square-free and pairwise coprime non-constant factors \( g_i\) with multiplicities \( m_i\) and a constant factor \( a\), such that \( p = a \cdot g_1^{m_1} \
cdot ... \cdot g_n^{m_n}\), where all \( g_i\) are canonicalized.
Given this decomposition, the square free part is defined as the product \( g_1 \cdot ... \cdot g_n\), which is computed by this functor.
See also | {"url":"https://doc.cgal.org/5.5.2/Polynomial/classPolynomialTraits__d_1_1MakeSquareFree.html","timestamp":"2024-11-06T13:49:37Z","content_type":"application/xhtml+xml","content_length":"13549","record_id":"<urn:uuid:543d16e6-fc0c-47f5-ae4b-77ca0aa1e7d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00447.warc.gz"} |
Games Built the Computer: Babbage, Lovelace and the Dawn of the Ludic Age
Samuel Pizelo
This article argues that games were used as modeling technologies for the earliest symbolic computational device, the Analytical Engine of Charles Babbage and Ada Lovelace. Consequently, it is argued
that the history of computing technology is one part of a longer history of games as modeling technologies. Before Babbage first wrote on the theory of computation in the 1820s, he had spent nearly a
decade developing a “geometry of situation” through the study of games of skill, inspired by the work of German polymath Gottfried Wilhelm Leibniz. Babbage employed this new geometry to describe the
operations of mechanical computers in space and over time in symbolic language. I argue that Babbage’s earlier study of games provided crucial tools and concepts for his later project of making a
symbolic computational device. I also examine the discussion of games in Babbage and Lovelace’s earliest correspondence to argue that games continued to model crucial innovations in their design of
the Analytical Engine. Examples of this include the use of punch card programs and the development of an “anticipating carry” to speed up computation. I demonstrate how Babbage and Lovelace relied on
historically specific forms of games such as chess, solitaire and tic-tac-toe to develop a symbolic language describing the relationship between space, time and mechanism. This elemental
correspondence between game form and computational architecture can provide computer game scholars with new ways of describing the relationship between computers and games. Recognizing the historical
role of games as models foregrounds their ongoing epistemological influence.
Keywords: game history, models, algorithms, history of computing, Ludic Age, temporality, spatiality, game theory, mathematics, Charles Babbage, Ada Lovelace, Gottfried Wilhelm Leibniz
Ada Lovelace prophesized the dawn of the information age in 1843, with her published translation of the “Sketch of the Analytical Engine invented by Charles Babbage.” Lovelace’s lengthy notes on the
translation displayed an intimate familiarity with the proposed Engine, including a lucid speculation on the capacities of symbolic computation in the arts and sciences and a demonstration of a
written program for computing the Bernoulli Numbers -- earning her the moniker of the “first programmer” (Toole, 1998). The Analytical Engine was qualitatively different from any prior computational
device, such as Babbage’s 1822 Difference Engine, because it did not merely execute a particular form of linear operation -- the calculation of the differences between sequences of numbers -- but was
“an embodying of the science of operations” itself through its ability to interpret symbolic operations (Lovelace, 1843, p. 161, emphasis in original). Electronic computers characterized the
so-called technosciences by the end of the Second World War, but Babbage’s and Lovelace’s vision for symbolic computation would not be actualized until the late 1940s -- more than a century after
Lovelace’s notes. Almost two centuries later, we are living the reality of this early vision.
What made the symbolic revolution possible? Babbage’s earlier research provides clues: in the winter of 1822, mere months after completing work on his first Difference Engine, he penned two texts
expressing the desire for a more expansive device that was able to follow multiple sequential instructions. What usually escapes mention in histories of computation is that, in both texts, Babbage
explicitly credits his realization of a need for this general calculating engine to his earlier mathematical analysis of games. In fact, a survey of Babbage’s early work on games provides ample
evidence that the “science of operations” described by Lovelace was developed through an intimate study of the algorithmic logics of games. The symbolic revolution was first a ludic revolution.
In what follows, I will look to Babbage’s early publications, manuscripts, and notes as well as his earliest correspondence with Lovelace to trace a history of symbolic computation that began as the
study of games. I will argue that Babbage and Lovelace used games as models of algorithmic operations, spatialized computation and predictive reasoning. These three aspects of games later became the
three greatest innovations of the Analytical Engine: the science of operations, the spatial mapping of computation through punched card programming and the predictive mechanism of the “anticipating
carry.” I follow a growing cadre of scholars who attend to the use of games as models and the epistemological consequences of these ludic practices [1]. By revisiting this early moment in the history
of computation, I hope to provide insight into the historical consequences of game modeling as a means of describing the relationship between space, time and action. As scholars like McKenzie Wark,
Alexander Galloway and Eric Zimmerman argue, game logics structure every aspect of life under late capitalism (Wark, 2007; Galloway, 2006; Zimmerman, 2015). Thus, a history of the diffusion of game
logics is also a history of the present.
A New Science of Algorithms
Babbage first recognized the need for a new engine when he confronted a problem that his first engine could not solve. Completed in June of 1822, Babbage’s Difference Engine was designed to calculate
tables of logarithms for navigation and astronomy -- a tedious and expensive task that previously required roomfuls of trained workers. Later that year, Babbage reflected on his achievement, noting
that he had stumbled upon an entirely new “species” of mathematical problem that could only be solved algorithmically because its “analytical laws are unknown” (Babbage, 2010, p. 216). It was not the
Difference Engine itself that inspired this new species of mathematics, however, but his study of the game of chess. He makes this connection explicit:
That the mere consideration of a mechanical engine should have suggested these inquiries, is of itself sufficiently remarkable; but it is still more singular, that amongst researches of so very
abstract a nature, […] I should have met with and overcome a difficulty […] in attempting the solution of a problem connected with the game of chess. (Babbage, 2010, p. 222)
Babbage constructed several examples of this new species of problem in order to illustrate a single principle: most empirical sequences of numbers do not change uniformly and predictably. Instead,
these sequences folded back on themselves and exhibited conditional logics -- for instance, calculate the exponents of 3 but increase the number by ten every time there is a “2” in the tens place [2]
. This essentially algorithmic approach to mathematics was wholly foreign to the European tradition, which privileged deductive analysis and proofs (Grattan-Guinness, 1992). Babbage realized that
solving these algorithmic problems would necessitate a new kind of mechanical computer -- one that could be given a series of instructions without a determinate solution and iterate through the
sequence until it computed the desired entry. Babbage’s mathematical study of chess inspired an entirely new algorithmic approach to European mathematics and laid the foundation for a theory of
computation that would occupy the remainder of his career.
What might seem at first like a chance discovery was in reality a systematic research program spanning the first decade of Babbage’s mathematical career. Inspired by the work of German polymath
Gottfried Wilhelm Leibniz, Babbage developed a “geometry of situation” that would allow for the mathematical study of the relationships between objects in space and through time, without the need to
locate objects by their Cartesian coordinates. Leibniz’s “situations” were fully relativistic configurations of objects as a network of relations -- and indeed, his situational geometry would
influence the development of graph theory, combinatorics, General Relativity, and topology (De Risi, 2007; De Risi, 2018). Babbage first mentions the geometry of situation in a preface to an 1813
text by the Analytical Society, his mathematical reformist club at Cambridge. In this preface, Babbage describes the geometry of situation as a theory at the boundaries of mathematics that is still
in its infancy due to “the great difficulty of reducing its conditions into symbolic language” (Babbage and Herschel, 2013, p. xx). He notes only three applications of this theory -- Leibniz’s use of
it to describe the board game solitaire, and Leonhard Euler’s and Alexandre-Theophile Vandermonde’s later elaboration of it through the study of the “Knight’s Tour” problem in the game of chess. By
1817, Babbage would join these venerable mathematicians in the study of this new geometry through games.
Figure 1. A representation of the Knight’s Tour around every square of the chessboard (Jaenisch 1862, Plate XVIII). Click image to enlarge.
Babbage’s short article on the Knight’s Tour problem laid the foundations for a unique “algorithmic” approach to mathematics that would characterize the rest of his career (Grattan-Guinness, 1992).
Babbage was clear about the enormous implications of this work, noting that a solution to this puzzle would allow for “a general process applicable to [an entire] class of equations” which “open a
wide field of analytical inquiry,” and present the possibility of novel approaches using mechanical calculation (Babbage, 2010, p. 218). This problem -- which dates to at least the ninth century in
India -- was first mathematically formalized by Euler in 1766 and was further generalized by Vandermonde shortly afterwards [3]. Euler asked: Can a proof be constructed describing a route for a chess
knight to land on each square of the board exactly once before returning to its starting square (Euler, 1766, p. 310)? Babbage’s approach was to construct an entirely new method of symbolic notation
describing the set of knight’s moves as an ordinal sequence and appending any other move possibility to the actual move made. For instance, move 11 “communicates with” move 32 because the knight
could have gone to either of these two squares interchangeably and still completed its tour (Babbage 1817, p.73). This symbolic notation allowed Babbage to establish a relationship between the spaces
to which the knight jumped and the time sequence in which it made those jumps. It was a new language of space, time and action.
As can be seen in the quote above, Babbage openly credited his interest in this new “species” of problem to the study of chess. But even this sweeping proclamation does not fully capture the extent
of his interest in games. In 1820, he published an essay concerning games of chance in which he notes that the series of problems he is considering are “not themselves dependent on chance” (Dubbey,
2004, p. 153). Instead, they are further elaborations of the geometry of situation. In the essay, he describes certain situations in games of chance where probability theory fails because it does not
account for the “order of succession” in which events happen -- from repeated coin tosses to the selection of colored lottery balls from an urn (Babbage 1820, p. 154). To solve these problems, he
again uses a symbolic description of event sequences that accounts for multiple possible “moves” (heads or tails for a coin or the different colored lottery balls). In the conclusion, Babbage notes
that these problems “afford an instance of the immediate application of some very abstract propositions of analysis to a subject of constant occurrence, which […] I have preferred treating in
particular instances, instead of investigating in its most general form” (Ibid., p. 177). His work on chess was merely one instance of a larger research program to construct a new geometry through
the study of games.
While Babbage made good on his promise not to publish a general study of the geometry of situation, this was not due to any resolve on his part, but to historical exigency. He produced a manuscript
in 1821-22 dedicated to the study of mathematical analysis (now housed in the British Library) that concludes with an essay dedicated to games; this essay, Babbage notes, “consists of a variety of
problems requiring the invention of new modes of analysis” (Babbage, 1997). The manuscript utterly baffled his friend and editor of the Edinburgh Journal of Science, David Brewster, who told Babbage
that he would not find a single reader who could understand it (Dubbey, 2004) However, the essays in this manuscript were widely known and quite influential among fellow English mathematical
reformers at the time (Lambert, 2021). The analysis contained therein was of sufficient quality that historian J.M. Dubbey would conclude: “If he had developed the very fruitful ideas contained in
the book […] it might well have been that mathematical philosophy, modern algebra, the theory of games and stochastic mathematics would have developed many decades before they actually did” (Dubbey,
2004, pp. 129-30). Babbage himself makes clear, however, that these ideas were further developed under the guise of a general theory of symbolic computation.
Babbage’s unpublished essay on games was an ambitious attempt to develop a complete geometry of situation through the analysis of games of all kinds. He begins by describing a class of questions
entirely different from other modes of mathematical analysis, for which all current methods are insufficient: “The class of questions to which I allude chiefly comprise such as are referable to the
Geometry of Situation and have very frequently arisen from games of skill” (Ibid., p. 126). Babbage’s stated objective in this essay is to develop a symbolic language for representing sequences and
relative positions in games, and to outline the various difficulties and possibilities present in the various types of game. He alludes to Leibniz by suggesting that games share a privileged
relationship with human ingenuity, and thus merit the organized pursuit of mathematicians [4]. He then divides games into categories, each with their own mathematical approach: games of pure chance
are the province of the mathematics of probability; individual games like solitaire can be solved by reversing the series of moves taken (as Leibniz had previously described); games of mixed strategy
and games of pure strategy presented a unique problem of describing symbolically the position of each move in a series relative to an opponent’s moves.
Figure 2. A later sketch by Babbage of his tic-tac-toe program [5]. Click image to enlarge.
Anticipating his design for a game-playing automaton later in life, Babbage pursues a solution to strategy games by presenting an equation of how to solve the game of tic-tac-toe. He finds a novel
use for some of his earlier symbolic algebra by describing the full sequence of nine moves in a game of tic-tac-toe, where each move is described with respect to every other possible move. In other
words, Babbage independently developed a search tree (see Figure 2) [6]. While the notation is unwieldy and the actual solution of an optimal strategy for tic-tac-toe was not attempted, the
revolutionary nature of this project should not be understated. Dubbey describes it as “the first recorded stochastic process in the history of mathematics” (Dubbey, 2004, p. 129). Babbage is
similarly confident in the efficacy of this new notational system. When he returns to the topic later in life, in his Passages from the Life of a Philosopher, he presents this system as a proof that
an optimal strategy for all games of skill can be determined through computation (Babbage, 1864, pp. 465-471). In fact, this stochastic representation of tic-tac-toe makes it clear that Babbage had
used games to devise general algorithms for mechanical computations before he first presented work on this new approach to computation in 1822. It was a program for the first ever computer game
(Monnens, 2013).
Babbage’s subsequent reference to this project suggests that he had already conceived of a mechanical application to the geometry of situation in games. He analyzes games once again in 1821 while
presenting his paper, “On the Influence of Signs in Mathematical Reasoning.” Games of chance and games of skill figure prominently in this essay, which argues for the efficacy of symbolic analysis in
mathematics (Babbage, 1826b). He begins his discussion of games of skill by noting their importance for determining solutions to problems “where two parties successively make choice either of things
or of situations” (Ibid., p. 19). Babbage argues that the only possible solutions for this class of problems result from an approach like his function mentioned above, which “enables us to delay the
decision of the individuals actually selected until the conclusion; and thus by other means, to satisfy the other conditions of the problem” (Ibid., p. 20). In other words, he turns an undefined
problem of assigning value to a sequence of choices into a finite calculation problem of determining how to arrive at a winning position in a game of skill.
It seems clear from Babbage’s subsequent discussion that his approach to this problem was intended to promote the unification of games and computation. He tips his hand when noting that he doubted
this method could actually be successful, “until some more condensed method of indicating and to a certain extent also of executing such operations, shall have been contrived” (Ibid., p. 20). At the
time he first presented this work, Babbage was actively devising a new method of mechanical notation (see discussion below) and a new means of calculating problems through mechanical computation.
Thus, this statement is less wistful speculation and more a reference to his ongoing research program. The connection between games and mechanical calculation is reinforced in his conclusion: “The
cause, on which its successful application depends, seems to be the power which it gives of uniting together a number of cases totally distinct, and of expressing them all by the same formula”
(Ibid., p. 21). While Babbage restricts his early research of the geometry of situation to the analysis of abstract “situations” using symbolic mathematics, he would next develop a notation system
for the function of mechanisms that borrowed from this earlier abstract geometry. As is clear from the statement above, he recognizes that this ludic geometry could be extended into a description of
the relationships between any phenomena. The space, time and action sequences in game situations could function as a model of all situations.
The Perfection of Mechanical Reason
Babbage’s early efforts to develop a symbolic language for algorithmic sequences would later be instantiated into the design of the Analytical Engine. Influenced by the punch card programming of the
Jacquard loom, he developed his own card programming system that would engage or disengage the various gears of the Engine depending on the patterns of holes in the card. Lovelace would later
identify this punch card system as the chief innovation of the Engine due precisely to its symbolic representation of arithmetical relationships:
The bounds of arithmetic were however outstepped the moment the idea of applying the cards had occurred; […] In enabling mechanism to combine together general symbols in successions of unlimited
variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science. A new, a vast, and a
powerful language is developed […] for the purposes of mankind than the means hitherto in our possession have rendered possible. (Lovelace, 1843, p. 163)
Lovelace’s speculation on the power of symbolic computation was eerily prescient of its actual contributions to the sciences. As she describes, the innovative use of punch cards to represent numbers
as variables (see Figure 3) unlocked an entirely new approach to mathematics. Operation cards could be inserted into the “store” of the Engine governed the different operations to be carried out
(addition, multiplication, squaring) and variable cards specifying the operand numbers were then placed in the “mill.” Each card was stamped with a grid of holes that addressed the various gears and
cogs of the machine -- in other words, the cards were abstract representations of the spatial and temporal relationships between the many components of the Engine.
Figure 3. Variable cards (left) and operation cards (right) for the Analytical Engine [7]. Click image to enlarge.
Babbage’s use of punch cards depended conceptually upon his prior analysis of situations in games. But he also made an intermediate innovation that allowed him to symbolically address the
interlocking parts of a machine in the first place. He called it the “Mechanical Notation” -- a symbolic system of notation that describes the relative position of each piece of a machine at a given
moment in time. This notation, which he first introduced in 1826, was intended to resolve the “difficulty of retaining in the mind all the contemporaneous and successive movements of a complicated
machine, and the still greater difficulty of properly timing movements which had already been provided for” (Babbage, 1826a, p. 250). To illustrate the efficacy of this notation system, Babbage uses
it to denote the movements of the components of a clock as they interact through time. Hinting at the greater importance of this work, Babbage concludes by noting: “The signs, if they have been
properly chosen, and if they should be generally adopted, will form as it were an universal language […] and I have myself experienced the advantages of its application to my own calculating engine,
when all other methods appeared nearly hopeless” (Ibid., p. 261). In early examples of this notation, the relative position of gears over time is depicted on the squares of a grid (see Figure 5).
Figure 4. Symbols substituted for positions in Babbage’s Knight’s Tour article, (Babbage, 1817). Click image to enlarge.
Figure 5. Spatial orientation of gears at different positions in time -- diagram of arithmetical operations in Babbage’s Difference Engine 1, 1834 (Babbage, 1989:2, p. 147). Click image to enlarge.
This suggestion of a universal symbolic notation again alludes to the work of Leibniz, which proved a persistent influence on Babbage’s work. A universal symbolic language was a centerpiece of
Leibniz’s life’s work -- what Leibniz called a “universal characteristic&rdquo -- and found its way into nearly all his other endeavors, from formal logic and philosophy to linguistics and political
science (Leibniz, 1989, pp. 221-228.). Babbage’s invocation of Leibniz’s universal language in the context of mechanics is not an idle one: Leibniz developed the geometry of situation with this very
usage in mind. In his earliest collected reference to the geometry of situation -- a letter to mathematician Christiaan Huygens in 1679 -- he describes its potential for perfecting the science of
I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometrical or linear and which will express situation [situs] directly […] And I believe that
I have found the way and that we can represent figures and even machines and movements by characters, as algebra represents numbers or magnitudes. […] If it were completed in the way in which I
think of it, one could carry out the description of a machine, no matter how complicated, in characters which would be merely the letters of the alphabet, and so provide the mind with a method of
knowing the machine and all its parts, their motion and use, distinctly and easily without the use of any figures or models and without the need of imagination. (Leibniz, 1989, p. 250)
The uncanny similarity between Babbage’s rationale for his notation and Leibniz’s own reasoning becomes even more apparent when each discuss the possibilities of this new geometrical symbolism. Both
of them see the abstract representation of mechanism as a way of optimizing the performance of mechanical calculations through mathematical analysis (Ibid., pp. 250-51). In other words, the geometry
of situation represents a general theory of computation that allows one to solve problems algorithmically through the symbolic representation of mechanism.
The evidence for the connection between the geometry of situation and mechanical notation is in an essay by Vandermonde to which Babbage continually referred across his work. In “Remarques sur les
Problèmes de Situation” (“Remarks on the Problems of Situation”), Vandermonde places Euler’s solution to the Knight’s Tour in the context of Leibniz’s broader study of situation in games. The
notation described by Vandermonde, “made up of numbers which do not represent quantities, but ranks in space,” is logically consistent with Babbage’s more elaborate formula describing a solution to
tic-tac-toe (Translation mine. Vandermonde, 1771, p. 566). It also closely resembles the complete version of Babbage’s Mechanical Notation described in 1851, where each piece (P) or arm (A) are
assigned a number (e.g., ^3P), described as a set of component “parts” (e.g., ^3P[1], ^3P[2], ^3P[3]) which are themselves represented according to their location at each period in time (e.g., ^3P^1,
^3P^2, ^3P^3) (Babbage, 1989:3, p. 224). This precise symbolic description of interlocking mechanisms situated in time and space represents a development of the earlier geometry of situation pursued
by Leibniz, Euler and Vandermonde. For Babbage, as for these other three, games served as models of spatiotemoral situations found in every other aspect of life.
In Babbage’s first description of the Analytical Engine in 1837, he is explicit about the stakes of this project, framing it as “the complete control which mechanism now gives us over number”
(Babbage, 2013 [1837], p. 54). This control was achieved (in theory) by a meticulous project of making each physical component addressable by other components at any point in time (Dhaliwal, 2022b;
Bromley, 1998). Babbage repeatedly emphasizes the importance of relying on “the perfect security and certainty” of the machine’s actions (Babbage, 2013 [1837], pp. 36-7; 48-9). It should be noted
that these philosophical ambitions continually ran aground when Babbage attempted to actually instantiate his theory into working mechanical devices (Lindgren, 1990; Jones, 2016). But it is important
to recognize that this vision of the mechanical perfection of reason, which occupied the better part of Babbage’s life, was the theoretical outcome of a study of games inspired by the writings of
When Leibniz formed the Berlin Academy of Sciences in 1710, the organization produced a journal of scientific writings, entitled the Miscellanea Berolinensia Ad Incrementum Scientiarum (The Berlin
Miscellany for the Furtherance of the Sciences), to mark its founding. Two of Leibniz’s essays bookend the diverse topics discussed in this text. Placed prominently in the first section is his essay
advocating for the formation of a general science of games, and the volume ends with his description of his “stepped reckoner” -- one of the earliest mechanical computers. The tissue connecting these
concerns is the study of “games of situation” -- Leibniz’s term for strategy games [8]. Charles Babbage was the conscious successor to this tradition. His contributions to the history of computation
were undergirded by a new sort of mathematics premised on the rules and regularities found in games.
Figure 6. Mathematical treatment of solitaire in Leibniz’s notebooks [9]. Click image to enlarge.
Leibniz not only constructed his geometry of situation around the game of solitaire, but also invented his own game called “reverse solitaire” to model his theory of the creation of the universe
(Leroux, 2015). In this reversal of the traditional rules, the board begins with a single marble at the center, and each leap adds rather than removes a marble from the board. He speculated that the
algorithmic steps involved in producing geometrical figures could constitute its own form of geometry (Leibniz, 1710). Coincidentally, the game of solitaire also inspired Lovelace’s first experiments
with algorithms three years before her first computer program. On February 16, 1840, Lovelace wrote a letter to Babbage alluding to a prior conversation with him concerning his study of chess and
suggesting that the game of solitaire could also be a topic of mathematical analysis -- unaware that Leibniz’s study of solitaire prompted Babbage’s entire work on games. She explains:
My Dear Mr Babbage. Have you ever seen a game, or rather puzzle, called Solitaire? […] I want to know if the problem admits of being put into a mathematical Formula, & solved in this manner. […]
There must be a definite principle, a compound I imagine of numerical & geometrical properties, on which the solution depends, & which can be put into symbolic language. […] I have numbered the
holes in my drawing for the sake of convenience of reference. (Lovelace, 1840, pp. 82-83)
Lovelace’s description of an algorithm for solving solitaire evinces the fact that, as with Babbage, she first modeled programs as abstract sequences of moves on a game board before penning
algorithms for the Analytical Engine. Further archival research has identified pen sketches by Babbage of several of Leonhard Euler’s situational puzzles, with attempted solutions traced in pencil by
Lovelace (Hollings, Martin & Rice, 2018, pp. 89-96). These findings provide us a rare view of the two inventors communicating through play. Lovelace hints at this earlier connection between games and
computers in her 1842 notes when she compares the store of the Engine to “a pile of rather large draughtsmen heaped perpendicularly one above another” (Lovelace, 1843, p. 165). Her subsequent
education in mathematics afforded her a symbolic language for formalizing the first computer algorithms, but only after she had already explored these spatiotemporal relationships in ludic practice.
Leibniz developed his geometry of situation in the wake of European probability theory and the ideology of certainty and reason that followed in its wake. Probability theory itself was modeled upon
games of chance: mathematics prodigy Blaise Pascal corresponded with fellow mathematician Pierre Fermat on the equitable outcome of an interrupted dice game and through so doing imagined the universe
itself as a fair game (Campe 2013). As Pascal later summarized, “the uncertainty of fortune is so restrained by the equity of reason, that each of two players can always be assigned exactly what is
rightly due. […] the doubtful outcomes of the lot […] has not been able to escape the dominion of reason” (Quoted in: Franklin, 2001, p. 312). This intellectual development -- what historian of
science Ian Hacking has called the “taming of chance” -- had a seismic impact on the knowledges and societies of early modern Western Europe [10]. In his exacting specifications for a computational
machine that eliminated any uncertainty of outcome, Babbage saw himself as an evangelist of this dominion of reason.
While Pascal described reason’s conquest of chance, Babbage saw in computation the potential to transmute space into time and back: “[I]t appears that the whole of the conditions which enable a
finite machine to make calculations of unlimited extent are fulfilled in the Analytical Engine […] I have converted the infinity of space, which was required by the conditions of the problem, into
the infinity of time” (Babbage, 2013 [1837], p. 13). This pronouncement posits the geometry of situation’s logical conclusion, which forged a relationship between space and time through the function
of mechanism. Merely three years after her first scribblings on gameboards and puzzles, Lovelace fully understood the implications of this new science: it could describe “those unceasing changes of
mutual relationship which, visibly or invisibly, […] are interminably going on in the agencies of the creation we live amidst” (Lovelace, 1843, p. 163). With this she declared the fulfillment of
Leibniz’s dream.
Anticipation and Learning
To fully appreciate the constitutive role that games played in Babbage’s and Lovelace’s computational thinking, it is important to distinguish which formal aspects of games led to which developments.
As Babbage clarified in his brief taxonomy of games quoted above: games of chance help to express probabilities, individual (puzzle) games like solitaire provide an opportunity for descriptions of
sequence, but only games of skill like chess or tic-tac-toe model conditional if-then logics and branching complexity. Perhaps for this reason, Babbage maintained an interest in games of skill until
the end of his life. One of his last attempts to secure funding for the assembly of the Analytical Engine was a planned construction of a game-learning automaton -- intended initially to play chess,
but later conceived as a tic-tac-toe automaton due to the complexity of the former. In his memoir, Passages from the Life of a Philosopher, he describes a basic conditional algorithm for a machine
that can account for the gameplay moves of an opponent and look ahead two or three moves to calculate the outcome of these moves (Babbage, 1864, p. 465). Summarizing the central principles of the
automaton, he notes: “Now I have already stated that in the Analytical Engine I had devised mechanical means equivalent to memory, also that I had provided other means equivalent to foresight, and
that the Engine itself could act on this foresight” (Ibid., p. 467). What characterized the automaton’s function were the two traditionally human capacities of memory and foresight, which are
necessitated by the conditional logics and branching complexity of games of skill.
The other means of foresight to which Babbage refers is the mechanism of the “anticipating carry” (Ibid., p. 63). While Lovelace declared the punch cards to be the central invention of the Analytical
Engine, Babbage himself was proudest of the anticipating carry. He realized when measuring the calculation time of the Engine that by far the least efficient process was the sequential transfer of
digit “carries” from gear to gear -- for example, when 9,999 adds up to 10,000, the final unit would have to register across five different gears. His mechanical solution was a wire connecting all
the gears that slotted into place when a given gear shifted to 9. When any of these 9s increased further, this wire would simultaneously propagate the carry across the entire sequence of 9s. The
efficiency gain of the anticipating carry was so great that it was the initial impetus behind the division of the Engine into a store (that delivered the instructions) and a mill (that made the
calculations themselves) (Jones, 2016, p. 54). While in his memoirs Babbage ascribes mechanical foresight to both the carry mechanism and his game program, it is clear from this survey of his early
work that the latter far predates the former. This example helps to illustrate the unique affordances of games of skill for the central innovations of the Analytical Engine.
And what of this capacity to learn? There is evidence that this too was inspired by games of skill. In 1819 and again in 1820, Babbage recorded in his journals that he attended a demonstration of a
popular chess automaton, the “Mechanical Turk,” playing and losing to it on the second occasion (Standage, 2002, pp. 137-145). This device, which was purportedly a chess-playing automaton but in
reality concealed a skilled chess player manipulating levers from the inside, was invented by Wolfgang von Kempelen in the eighteenth century and was toured across Europe and even America by Johann
Maelzel throughout the nineteenth century (Ibid., p. 103). Numerous historians have suggested that this device modeled for Babbage the possibility that automata could one day play chess -- a
proposition he later made in his Passages -- and that it perhaps inspired the Analytical Engine itself [11]. While his bout with the automaton happened well into his research into games, it is
significant that he drafted his essay on games of skill a year after this event. Thus, it is plausible that even the memory capacity of the Analytical Engine was inspired by his speculations on the
unique affordances of games of skill.
I want to suggest here a more intimate relationship between games and computation in Babbage’s project than is usually acknowledged: Babbage was concerned not only with the fact that an automaton
could play games, but also with the mathematics necessary to optimize the solution to problems that required the coordination of elements into spatial and temporal relationships. In other words,
games were not simply a convenient topic for the development of the geometry of situation. The precise rules, concepts and mechanics of games were necessary antecedents to the development of an
entire mode of mathematical analysis. For Babbage to develop this theory independently of games, he would have needed to stumble upon a problem that involved a turn-based agonistic encounter between
two or more parties where items were placed in abstract space in a linear sequence for the pursuit of a measurable and finite outcome. For many scholars of games, these properties are the very
definition of a game [12]. The geometry of situation points to a capacity of games that has less to do with the modality of play and more with the structure they impose as model systems.
Babbage turns again to games as models of computing technology in his plans for a tic-tac-toe automaton. As described in his Passages, this project first occurred to him as a means of demonstrating
“the power which I possessed over mechanism through the aid of the Mechanical Notation” (Babbage, 1864, p. 465). After taking to the streets to assess whether it was commonly believed that games of
skill required human reason, Babbage endeavored to demonstrate that “every game of skill is susceptible of being played by an automaton” (Ibid., p. 466). His initial theoretical approach focused on
describing chess strategy using chains of conditional logic: “Is the position of the men […] consistent with the rules of the game? If so, has Automaton himself already lost the game? […] If not, can
he win it at the next move? If so, make that move” (Ibid., pp. 466-7). When reflecting on the reaction his audiences might have to an automaton who mastered a game of skill, he waxes philosophical,
noting it to be “worthy of remark how admirably this illustrates the best definitions of chance by the philosopher and the poet,” that, quoting scientist Pierre-Simon Laplace, “Chance is but the
expression of man’s ignorance” (as cited in Babbage, 1864, pp. 469-70). This philosophical objective reiterates that historically specific game forms like chess and tic-tac-toe modeled computing
technologies before those technologies existed.
Although Babbage and Lovelace’s plans to complete a working Analytical Engine never came to fruition, their concepts and designs cast a long shadow over the later history of computing, inspiring the
work of a wide range of early computer scientists and engineers [13]. Babbage’s reflections on the implications of the mechanization of labor processes for political economy also had a profound
influence on the technological theory of Karl Marx, whom Simon Schaffer calls “Babbage’s most penetrating London reader” (Schaffer, 1994, p. 205). Schaffer clarifies that the Analytical Engine’s
capacities of anticipation and memory “were profound resources for Babbage’s metaphysics and his political economy” (Ibid., p.207). Indeed, some of the most influential neoclassical economists leaned
on Babbage in their work, including Alfred Marshall, William Stanley Jevons, John Stuart Mill and Joseph Schumpeter (Cooke, 2005; Niman, 2008; Rosenberg, 2000). Marshall and Jevons in particular
employed Babbage’s chess automaton as a model of rational economic man through its capacity to anticipate and learn from mistakes (Raffaelli, 1994; Jevons, 1883). The use of games to model symbolic
computation did not end with the Analytical Engine, but intensified throughout the twentieth century (Pizelo, 2024a; Pizelo, 2024b). In our moment, games have become the paradigmatic environment for
the study of machine intelligence (Yannakakis and Togelius, 2018). This, too, is part of the history of games.
Games played computers before computers played games. This simple observation has profound implications for how we tell histories of our moment. It inverts the traditional prioritization of computers
as the condition of possibility for games. It also emphasizes the epistemological capacities of games as models and suggests a future research program to identify other cultural logics influenced by
games (Fuchs et al., 2014; Stark, 2024). If computers were themselves dependent on game logics, then Eric Zimmerman’s influential description of our “Ludic Century” is further intensified by
including the “Information Age” as well [14]. The reversal of the historical relationship between games and computation prompts further questions regarding the construction of game studies as a
discipline: If the historical development of electronic computation belongs to the history of games, how should game scholars configure the relationship between “analog” and “digital” games, or
“board” and “video” games? How can histories of games help to denaturalize the Euro-American project of perfecting “reason”? How might game researchers contextualize the operational logics of
computing technologies differently if we recognize that these same logics were at work in ancient media (Pizelo, 2023)? Does game studies have new perspectives, new concepts and new vocabulary to
offer theorists of computation and digital media seeking to describe the spatial and temporal orders imposed by procedural rules (Galloway, 2006; Wark, 2007; Bogost, 2010)?
The many games Babbage studied -- including chess, solitaire, dice games and tic-tac-toe -- were not found objects, they were built objects. They are technologies with their own histories, cultures
and communities. An appreciation for the influence of the material technologies of games necessitates both an acknowledgement of the things games themselves build and the writing of new histories
emphasizing the entanglements between games and the phenomena they model. Game studies, in its generative interdisciplinarity, is well positioned for this task (Chess and Consalvo, 2022; Gecker,
2021; Deterding, 2017). How will the study of our moment change if we learn new ways of describing the generative logics of games? Can we build new systems? Charles Babbage and Ada Lovelace believed
I would like to thank Ranjodh Singh Dhaliwal in particular for his early provocations and later conversations on the enduring relevance of Babbage and Lovelace. Con Diaz and David Dunning were both
so helpful when I was getting a feel for histories of computing and mathematics. Ryan Wright, Doug Stark and Kate Hayles were early readers that encouraged me to be bolder with my intervention. Colin
Milburn, Stephanie Boluk and Patrick LeMieux all helped this article to take shape. In addition, the members of SIGCIS were helpful commentors on an earlier presentation of my argument. Finally, I
want to acknowledge the formative critique of the anonymous reviewers and the meticulous work of the journal editors.
[1] See in particular; Pias, 2017; Malaby, 2007; Crogan, 2011; Milburn, 2015; Milburn, 2018; Jagoda, 2020. For a theory of scientific modeling as a game of make-believe, see; Toon, 2012. For games as
modeling technologies, see; Möring, 2013; Wardrip-Fruin, 2020. There is also a rich tradition of scholarship on the role of games in education and pedagogy. For an introduction to this topic, see;
Gee, 2003; Brougère, 1999.
[2] “A series of cube numbers might be formed, subject to this condition, that whenever the number 2 occurred in the tens’ place, that and all the succeeding cubes should be increased by ten. In such
a series, of course, the second figure would never be a 2, because the addition of ten would convert it into 3.” Ibid., p. 218.
[3] For a discussion of the earliest solutions to the Knight’s Tour which were used to compose Sanskrit poetry, see; Murthy, 2020. Before his published essay on the Knight’s Tour, Euler mentions the
problem in a 1757 letter to Christian Goldbach. See; Zubkov, 2010.
[4] Leibniz made frequent statements to this effect throughout his lifetime correspondence. See; Leibniz, 1996, pp. lxiv-lxv.
[5] Babbage, Charles. “Scribbling Book Volume XIV,” The Babbage Papers at the Science Museum, London. p. 98. CC BY-NC-SA 4.0 License.
[6] For Ernst Zermelo’s influential search tree notation, see; Schwalbe and Walker, 2001.
[7] Babbage, Charles. “Punched cards for Babbage's Analytical Engine,” The Babbage Papers at the Science Museum, London. CC BY-NC-SA 4.0 License.
[8] Leibniz goes into depth on his taxonomy of games and the mathematical study thereof in a letter to fellow probability theorist Pierre Rémond de Montmort. See; Leibniz, 1989, p. 487.
[9] Image taken from the Gottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbibliothek, Leibniz-Handschriften zur Technica, LH 38, fol. 195 v. http://digitale-sammlungen.gwlb.de/resolve?id=
[10] Hacking, 1990. For the development of the theory of probability through games and its ramifications, see also; Daston, 1995; Campe, 2013.
[11] For a recent discussion of this relationship, see; Dhaliwal, 2022a, pp. 377-409. See also; Schaffer, 1999; Schwartz, 2019.
[12] For a recent survey of definitions of games, see; Consalvo and Paul, 2019. For chess and the history of algorithm, see; Larson, 2018.
[13] Babbage had a clear influence on the later designs of Percy Ludgate, the automata of Leonardo Torres y Quevedo, the analytical engine of Louis Couffignal, as well as a cadre of post-WWII
engineers, including Howard Aiken, William Phillips, Vannevar Bush, Konrad Zuse, and others. See; Randell, 2013, pp. 15-17. I am indebted to Ranjodh Singh Dhaliwal for alerting me to the broader
economic influences of Babbage’s work, especially on Karl Marx.
[14] Zimmerman, 2015. In 1997, historian Colas Duflo dubbed the eighteenth century “le siècle du jeu,” that is, “the century of games and play.” Peter Burke has recently depicted a similarly central
role for play during the Renaissance. Johan Huizinga found play to be an explanatory mechanism for the culture of the late Middle Ages long before he generalized this methodology in Homo Ludens. If
we take a global perspective, our list of “ludic centuries” would quickly expand further. See; Duflo, 1997; Burke, 2021; Huizinga, 1996 [1921].
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Babbage, C. (1997). “Essays on the Philosophy of Analysis,” Correspondence and Scientific Manuscripts from the British Library, London. Reel 21. Adam Matthew Publications.
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Babbage, C. (1989). The Works of Charles Babbage (Vols. 2 and 3, M. Campbell-Kelly Ed.). London, Pickering & Chatto.
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Burke, P. (2021). Play in Renaissance Italy. Polity Press.
Campe, R. (2013). The Game of Probability: Literature and Calculation from Pascal to Kleist. Stanford University Press.
Chess, S. & Consalvo, M. (2022). “The Future of Media Studies is Game Studies,” Critical Studies in Media Communication, 39(3), 159-164.
Consalvo, M. & Paul, C. A. (2019). Real Games: What's Legitimate and What's not in Contemporary Videogames. MIT Press.
Cook, S. (2005). “Minds, Machines and Economic Agents: Cambridge Receptions of Boole and Babbage.” Studies in History and Philosophy of Science Part A, 36(2), 331-350.
Crogan, P. (2011). Gameplay Mode: War, Simulation, and Technoculture. U of Minnesota Press.
Daston, L. (1995). Classical Probability in the Enlightenment. Princeton University Press.
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De Risi, V. (2007). Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space. Springer Science & Business Media.
Deterding, S. (2017). "The Pyrrhic Victory of Game Studies: Assessing the Past, Present, and Future of Interdisciplinary Game Research." Games and Culture 12(6), 521-543.
Dhaliwal, R. S. (2022a). “The Cyber-Homunculus: On Race and Labor in Plans for Computation,” Configurations 30(4), 377-409.
Dhaliwal, R. S. (2022b). “On Addressability, or What Even Is Computation?,” Critical Inquiry 49(1), 1-27.
Diaz, G. C. (2020). “Encoding Music: Perforated Paper, Copyright Law, and the Legibility of Code, 1880-1908.” Case Western Reserve Law Review, 71(2), 627-665.
Dubbey, J. M. (2004). The Mathematical Work of Charles Babbage. Cambridge University Press.
Duflo, C. (1997). Le Jeu: De Pascal à Schiller, Presses Universitaires de France (Paris).
Dunning, D. E. (2020). Writing the Rules of Reason: Notations in Mathematical Logic, 1847-1937 [Doctoral dissertation, Princeton University].
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Euler, L. (1766). “Solution d’une Question Curieuse que ne Paroit Soumise à Aucune Analyse,” Euler Archive - All Works. https://scholarlycommons.pacific.edu/euler-works/309
Fickle, T. (2019). The Race Card: From Gaming Technologies to Model Minorities. New York University Press.
Franklin, J. (2001). The Science of Conjecture: Evidence and Probability before Pascal. Johns Hopkins University Press.
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Galloway, A. R. (2021). Uncomputable: Play and Politics in the Long Digital Age. Verso Books.
Gee, J. P. (2003). “What Video Games Have to Teach Us about Learning and Literacy.” Computers in Entertainment (CIE), 1(1), 20-20.
Gekker, A. (2021). "Against Game Studies." Media and Communication, 9(1), 73-83.
Grattan-Guinness, I. (1992). “Charles Babbage as an Algorithmic Thinker,” IEEE Annals of the History of Computing, 14(3), 34-48.
Hacking, I. (1990). The Taming of Chance. Cambridge University Press.
Hollings, C., Martin, U. & Rice, A. C. (2018). Ada Lovelace: The Making of a Computer Scientist. Bodleian Library.
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Jagoda, P. (2020). Experimental Games: Critique, Play, and Design in the Age of Gamification. University of Chicago Press.
Jaenisch, C. F. (1862). Traité des Applications de Analyse Mathématique au Jeu des Échecs. (Vol. 2). Imperial Academy of Sciences.
Jevons, W. S. (1883). Methods of Social Reform: And Other Papers. Macmillan and Company.
Jones, M. L. (2016). Reckoning with Matter: Calculating Machines, Innovation, and Thinking about Thinking from Pascal to Babbage. University of Chicago Press.
Lambert, K. (2021). Symbols and Things: Material Mathematics in the Eighteenth and Nineteenth Centuries. University of Pittsburgh Press.
Larson, M. (2018). “Optimizing Chess: Philology and Algorithmic Culture,” Diacritics, 46(1), 30-53.
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Leroux, V. (2015). Le jeu dans la pensée de Leibniz [Master's thesis, l'Université Paris sciences et lettres].
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Milburn, C. (2015). Mondo Nano: Fun and Games in the World of Digital Matter. Duke University Press.
Milburn, C. (2018). Respawn: Gamers, Hackers, and Technogenic Life. Duke University Press.
Monnens, D. (2013). “'I commenced an examination of a game called ‘tit-tat-to’': Charles Babbage and the 'First” Computer Game.' Proceedings of DiGRA 2013: DeFragging Game Studies.
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Wardrip-Fruin, N. (2020). How Pac-Man Eats. MIT Press.
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Zubkov, A. M. (2011). Euler and Combinatorial Calculus. Proceedings of the Steklov Institute of Mathematics, 274(Suppl 1), 162-168. | {"url":"https://gamestudies.org/2403/articles/pizelo","timestamp":"2024-11-04T05:05:38Z","content_type":"application/xhtml+xml","content_length":"67226","record_id":"<urn:uuid:4c20a561-d3fc-4835-9cee-94c5ba3b386f>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00451.warc.gz"} |
Convert Minutes to Seconds (min to s) | Examples & Steps
Disclaimer: We've spent hundreds of hours building and testing our calculators and conversion tools. However, we cannot be held liable for any damages or losses (monetary or otherwise) arising out of
or in connection with their use. Full disclaimer.
How to convert minutes to seconds (min to s)
The formula for converting minutes to seconds is: s = min × 60. To calculate the minute value in seconds first substitute the minute value into the preceding formula, and then perform the
calculation. If we wanted to calculate 1 minute in seconds we follow these steps:
s = min × 60
s = 1 × 60
s = 60
In other words, 1 minute is equal to 60 seconds.
Example Conversion
Let's take a look at an example. The step-by-step process to convert 4 minutes to seconds is:
1. Understand the conversion formula: s = min × 60
2. Substitute the required value. In this case we substitute 4 for min so the formula becomes: s = 4 × 60
3. Calculate the result using the provided values. In our example the result is: 4 × 60 = 240 s
In summary, 4 minutes is equal to 240 seconds.
Converting seconds to minutes
In order to convert the other way around i.e. seconds to minutes, you would use the following formula: min = s ÷ 60. To convert seconds to minutes first substitute the second value into the above
formula, and then execute the calculation. If we wanted to calculate 1 second in minutes we follow these steps:
min = s ÷ 60
min = 1 ÷ 60
min = 0.0166666666666666666666666666667
Or in other words, 1 second is equal to 0.0166666666666666666666666666667 minutes.
Conversion Unit Definitions
What is a Minute?
A minute is a unit of time measurement that represents 60 seconds or 1/60th of an hour. It is commonly used to measure short durations and as a unit of time on clocks and watches.
To provide an example of a minute, let's consider boiling an egg. The recommended cooking time for a soft-boiled egg is typically around 4 to 6 minutes. This means that you would place the egg in
boiling water and let it cook for approximately 4 to 6 minutes before removing it.
Another example is in scheduling or time management. When discussing meeting durations, it is common to allocate a certain number of minutes for different activities. For instance, a presentation
might be scheduled for 20 minutes, followed by a question and answer session of 10 minutes.
In sports, a minute is often used to measure game time or periods. For example, in basketball, each quarter consists of 12 minutes of playing time. In soccer, a game is divided into two halves, each
typically lasting 45 minutes.
On digital clocks or watches, the time is displayed in hours and minutes. For example, if the clock reads "9:30," it means that it is 9 hours and 30 minutes past midnight.
In summary, a minute is a unit of time measurement that represents 60 seconds or 1/60th of an hour. The examples of boiling an egg, scheduling activities, and game durations illustrate how minutes
are commonly used to measure short durations, track time, and organize events in various contexts.
What is a Second?
A second (s) is the base unit of time measurement in the International System of Units (SI). It is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition
between two hyperfine levels of the ground state of the cesium-133 atom.
To provide an example of a second, let's consider a simple action like snapping your fingers. The time it takes for the sound of a finger snap to occur is typically on the order of milliseconds,
which is a fraction of a second. However, if we zoom in further, a second can be divided into smaller units such as milliseconds, microseconds, and nanoseconds.
For instance, if we take 1 second and divide it into smaller intervals of 1 millisecond each, we would have 1,000 milliseconds in a second. Each millisecond represents a thousandth of a second. This
level of precision is often used in fields that require accurate time measurement, such as scientific experiments, computing, and telecommunications.
In everyday life, we use seconds as a fundamental unit of time to measure durations, intervals, and clock time. For example, when you count "1...2...3...," each count represents a second. When you
check the time on a clock, it displays the hours, minutes, and seconds elapsed since midnight.
Additionally, seconds are crucial in measuring the speed of events, such as the time it takes for a car to accelerate from 0 to 60 miles per hour or the duration of a short video clip.
In summary, a second (s) is the base unit of time in the SI system. It represents the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels
of the cesium-133 atom. The example of snapping your fingers highlights how seconds are used to measure everyday durations, and they can be further divided into smaller units like milliseconds for
more precise time measurement.
Minutes To Seconds Conversion Table
Below is a lookup table showing common minutes to seconds conversion values.
Minute (min) Second (s)
1 min 60 s
2 min 120 s
3 min 180 s
4 min 240 s
5 min 300 s
6 min 360 s
7 min 420 s
8 min 480 s
9 min 540 s
10 min 600 s
11 min 660 s
12 min 720 s
13 min 780 s
Other Common Minute Conversions
Below is a table of common conversions from minutes to other time units.
Conversion Result
1 minute in days 0.000694444444444444444444444444444 d
Minutes To Seconds Conversion Chart | {"url":"https://www.thecalculatorking.com/converters/time/minute-to-second","timestamp":"2024-11-13T04:20:25Z","content_type":"text/html","content_length":"97362","record_id":"<urn:uuid:6f1fbdb1-2444-4663-ae0d-6c78c06a8c02>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00540.warc.gz"} |
The quantities package provides integration of the ‘units’ and ‘errors’ packages for a complete quantity calculus system for R vectors, matrices and arrays, with automatic propagation, conversion,
derivation and simplification of magnitudes and uncertainties.
Blog posts:
• Edzer Pebesma, Thomas Mailund and James Hiebert (2016). “Measurement Units in R.” The R Journal, 8 (2), 486–494. DOI: 10.32614/RJ-2016-061
• Iñaki Ucar, Edzer Pebesma and Arturo Azcorra (2018). “Measurement Errors in R.” The R Journal, 10 (2), 549–557. DOI: 10.32614/RJ-2018-075
Install the release version from CRAN:
The installation from GitHub requires the remotes package.
# install.packages("remotes")
remotes::install_github(paste("r-quantities", c("units", "errors", "quantities"), sep="/"))
This project gratefully acknowledges financial support from the | {"url":"https://cloud.r-project.org/web/packages/quantities/readme/README.html","timestamp":"2024-11-05T17:00:50Z","content_type":"application/xhtml+xml","content_length":"8498","record_id":"<urn:uuid:34239a11-e384-4357-b992-4f0dbc671d73>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00797.warc.gz"} |
xint – Expandable arbitrary precision floating point and integer operations
The xint bundle main modules are:
utilities of independent interest such as expandable and non-expandable loops,
expandable macros implementing addition, subtraction, multiplication, division, and powers for arbitrarily long integers,
extension of xintcore,
extends the scope of xint to decimal numbers, to numbers using scientific notation and also to (exact) fractions,
provides expandable parsers of numeric expressions using the standard infix notations, parentheses, built-in functions, user definable functions and variables (and more ...) which do either exact
evaluations (also with fractions) or floating point evaluations under a user chosen precision.
Further modules of the bundle are: xintkernel (support macros for all the bundle constituents), xintbinhex (conversion to and from hexadecimal and binary bases), xintgcd (provides gcd() and lcm()
functions to xintexpr), xintseries (evaluates numerically partial sums of series and power series with fractional coefficients), and xintcfrac (dedicated to the computation and display of continued
All computations are compatible with expansion-only context.
The packages may be used with Plain TeX, LaTeX, or (a priori) any other macro format built upon TeX.
Sources /macros/generic/xint
Version 1.4m 2022-06-10
Licenses The LaTeX Project Public License 1.3c
Copyright 2013–2022 Jean-François Burnol
Maintainer Jean-François Burnol
TDS archive xint.tds.zip
Contained in TeXLive as xint
MiKTeX as xint
Topics Arithmetic
Download the contents of this package in one zip archive (3.9M).
Community Comments
Maybe you are interested in the following packages as well. | {"url":"https://www.ctan.org/pkg/xint","timestamp":"2024-11-11T03:43:38Z","content_type":"text/html","content_length":"18354","record_id":"<urn:uuid:7790045e-5f29-42bd-89ad-9563300b44d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00814.warc.gz"} |
van der Waals forces
These are the forces that holds individual
together in
. They are very
which act over a very short range. They were discovered by Johannes Diderik van der Waals (a
) in the mid to late 19th century.
It has recently been proposed that it is actually the Van der Waals forces between microscopic hairs/suckers on their feet and the surface they are on that allows Geckos to climb on any surface. | {"url":"https://m.everything2.com/title/Van+der+Waals+Forces","timestamp":"2024-11-03T09:46:21Z","content_type":"text/html","content_length":"49036","record_id":"<urn:uuid:f4bcc67e-1a99-4aed-a790-809a63705bf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00004.warc.gz"} |
Spectral and variational principles of electromagnetic field excitation in wave guides
Possible variational principles for excitation of an electromagnetic field in a wave guide are discussed. Our emphasis is not on the calculation of the modal shapes, which is common in previous art,
but rather on the calculation of modal amplitude evolution, which are important in electron devices such as free electron lasers and gyrotrons. Variational principles have considerable importance in
theoretical physics and are used among other things to derive numerical solution schemes, conservation laws via the Noether theorem and correct boundary conditions for the derived equations including
the important effects of the backward waves amplitudes.
Dive into the research topics of 'Spectral and variational principles of electromagnetic field excitation in wave guides'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/spectral-and-variational-principles-of-electromagnetic-field-exci-3","timestamp":"2024-11-05T01:14:51Z","content_type":"text/html","content_length":"55702","record_id":"<urn:uuid:daec83ec-cb41-44ef-8d7b-60f080844b82>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00639.warc.gz"} |
Do you need calculus for architecture? - Architecture
Do you need calculus for architecture?
No, you do not need calculus for architecture. However, it can be helpful in understanding certain principles and solving certain problems.
No, you do not need calculus for architecture.
Is calculus used in architecture?
Calculus is a powerful tool that can be used to determine the quantities of materials required for constructing support systems. This is especially important for structures that need to withstand
stress over long periods of time. Even notable monuments such as the Eiffel Tower were constructed using calculus to predict the impact of wind resistance.
Geometry, algebra, and trigonometry are all important in architectural design. Architects use these math forms to plan their blueprints or initial sketch designs. They also calculate the probability
of issues the construction team could run into as they bring the design vision to life in three dimensions.
Is architecture math heavy
I agree that math skills are important for success in architecture, but I don’t think they should be a deciding factor. There are many other important skills and qualities that are essential for
success in this field. For example, creativity, spatial awareness, and the ability to think outside the box.
Calculus is a branch of mathematics that deals with the study of change. It is used in many fields, such as physics, engineering, and architecture. Most students finish their Algebra, Geometry, and
Trigonometry requirements in high school and can begin Calculus classes in college right away.
What jobs require calculus?
Calculus is a mathematical tool used to solve problems in a wide range of fields, including animation, engineering, and software development. By understanding and utilizing calculus, professionals in
these fields are able to create more efficient and accurate solutions to the challenges they face.
There is no easy answer when it comes to the question of how difficult it is to study architecture. While the rewards can be great, the level of difficulty is often cited as one of the reasons why
architecture is such a demanding field. To be successful, students need to be able to devote long hours to their studies and be able to pay close attention to detail.
Can I be an architect if I can’t draw?
There are a lot of misconceptions about what it takes to be an architect. Yes, architects do use 3D modeling software, but that’s not all they do. They also spend a lot of time drawing by hand. And
no, you don’t have to be able to draw well to be an architect. There are a lot of other skills that are just as important, like being able to think creatively and solve problems. So if you’re
interested in becoming an architect, don’t let the idea of having to draw stop you.
Architecture students are some of the hardest working college students, averaging 222 hours of study time each week. This figure inevitably takes a toll, andarchitecture students often find
themselves struggling to keep up with the demands of their coursework. If you’re an architecture student, it’s important to be mindful of your mental and physical health, and to make time for
relaxation and self-care. Otherwise, you run the risk of burning out.
What majors require no math
There are a number of online degrees that don’t require math as a prerequisite. This can be a great option for students who don’t feel comfortable with the subject or who want to focus on other areas
of study. Some of the most popular online degrees that don’t require math include anthropology, communications, criminal justice, culinary arts, education, English, and foreign language. Graphic
design is another popular option for students who want to avoid math.
The rigors of the 5-year architecture course are well-known, and every student is required to put in dedicated long hours of studio work, produce meticulously detailed drawings, and gain rigorous
practical on-site knowledge in order to achieve an architecture degree. However, the rewards of successfully completing this tough course are great, and students who are up to the challenge will find
themselves with a wealth of skills and knowledge that they can use to build amazing structures.
Is architecture more math or art?
The architect is responsible for the design, construction, and maintenance of buildings and other structures. They work with both the public and private sector to ensure that the buildings they
create are both functional and aesthetically pleasing.
Architecture degrees teach students to combine math, the arts, engineering and science to create sustainable designs. This means that graduates are not only knowledgeable in the latest construction
methods and materials, but also in the principles of sustainability and how to incorporate them into their designs.
With the world increasingly focused on green living, there is a growing demand for environmentally friendly buildings. As an architect, you will be at the forefront of this movement, creating
structures that not only look good, but also do good for the planet.
If you’re thinking about a career in architecture, you can expect to earn a good salary. In 2021, the median salary for architects was $80,180. The best-paid 25% of architects made $102,160 that
year, while the lowest-paid 25% made $62,500. With a career in architecture, you can expect to earn a good salary and have a stable job.
Do you need a high IQ to be an architect
This study found that architects have an average IQ that is on par with other professionals such as surgeons, lawyers, and engineers. This IQ range is considered to be “superior” or “very superior”
intelligence. This is good news for those pursuing a career in architecture!
This is an important point to remember when considering a career in architecture. Although drawing skills are important, they are just one part of what makes a great architect. Other skills, such as
analysis, synthesis, creative problem-solving, and sensitivity to people’s needs and wants, are also essential. So you don’t need to be ‘good’ at drawing in order to be a successful architect.
How hard is calculus?
Calculus is a very challenging math class that requires students to think beyond the realms of algebra and geometry. The concepts in calculus are very abstract and require a great deal of
imagination. This can be very difficult for some students, but it is worth the effort to try to understand these concepts.
There are a number of majors at colleges and universities that do not require calculus. Examples include anthropology, art and art history, classics, communication, English, environmental studies,
ethnic studies, history, and more. This is good news for students who may not be strong in math or who may not be interested in taking calculus.
Can I go to college without calculus
Calculus is not a required course for getting into most colleges. This is according to a survey of private institution responses. Fewer than 5 percent of respondents said calculus was a requirement
for all or most majors. This means that the majority of colleges do not have a requirement for calculus.
Calculus is a very important branch of mathematics that is used in many different fields, such as physics and engineering. It can be tough for students to learn, but it is essential for anyone
wishing to pursue a career in these fields. With practice and perseverance, anyone can master calculus!
Final Words
No, you do not need calculus for architecture. However, it may be helpful in some aspects of the field, such as when designing buildings or other structures.
Without a doubt, calculus is a necessary tool for architects. This powerful mathematics discipline allows architects to determine dimensions, calculate slopes and hatchings, analyze functions, and
much more. Overall, calculus provides architects with a deeper understanding of the construction process and how to create buildings that are both safe and aesthetically pleasing.
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bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis - Tech Infobicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis - Tech Info
Posted inBusiness
bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis
When we consider a one-bicycle wheel circle radius nyt wheel circle radius nyt of the maximum important additives that come to thoughts are its wheels. The wheel is an elaborate piece of engineering
that performs a vital position in the overall overall performance and efficiency of the bicycle. An essential concept in information about wheels is the radius of the circle they form. This article
delves into the significance of the wheel circle radius nytel’s radius, its impact on the bicycle’s dynamics, and its broader implications.
The Geometry of bicycle wheel circle radius nytWheels
A bicycle wheel is a circle, and its length can be defined by way of its radius. For general bicycles, wheel sizes are regularly specified in phrases of diameter, but the radius is 1/2 of the
diameter. For instance, a common wheel length for road motorcycles is 700c, which corresponds to a diameter of about 622mm. Thus, the radius is 311mm.
Understanding the radius is important because it directly influences the wheel’s circumference, which determines how ways the bicycle wheel circle radius nyt travels with each rotation of the wheel.
Formula of circumference:
[ C = 2pi r ]
in which ( r ) is the radius. For a wheel with a radius of 311mm, the circumference is set at 1,954mm (or 1.954 meters).
Impact on bicycle wheel circle radius nytDynamics
1. Speed and Efficiency:
The radius of the wheel affects the bicycle’s speed and performance. Larger wheels (extra radius) cover the extra floor in keeping with revolution, which can translate to higher speeds and smoother
rides. This is why road motorcycles frequently have large wheels as compared to mountain bikes. The large radius reduces rolling resistance, making it easier to keep velocity on clean surfaces.
2. Handling and Stability:
The wheel radius also impacts the motorcycle’s handling and balance. Larger wheels offer better balance and may roll over limitations greater without difficulty, which is fine on tough terrain.
However, they can also make the bicycle wheel circle radius nyt less maneuverable. Conversely, smaller wheels enhance maneuverability and acceleration however might not cope with rough terrain as
3. Acceleration and Torque:
Smaller wheels, having a smaller radius, require much less torque to begin transferring, this means that they can boost up faster. This is useful for urban commuting or using stop-and-pass visitors.
However, large wheels preserve better speeds greater efficiently once in motion, reaping benefits for lengthy-distance cyclists.
Real-World Applications
Road Bikes
Road motorcycles, designed for velocity and long-distance travel on paved surfaces, generally feature larger wheels with a radius of around 311mm (700c). The larger radius allows for greater
performance and velocity, making them perfect for racing and long rides. The decreased rolling resistance and capacity to preserve momentum are key blessings.
Mountain Bikes
Mountain bikes, on the other hand, frequently use wheels with a radius of around 279mm (29 inches) or even smaller for some models. The barely smaller radius improves maneuverability and manages on
rugged terrain, that is essential for navigating trails and limitations. The stability between stability and agility is critical for mountain biking.
Urban and Folding Bikes
Urban and folding motorcycles commonly have even smaller wheels, with radii starting from 203mm (sixteen inches) to 254mm (20 inches). These motorcycles prioritize compactness and ease of
acceleration, which can be vital for metropolis commuting and garages. The smaller radius makes these bikes extraordinarily maneuverable in tight spaces and brief to respond to site visitors.
The Physics Behind the Ride
The physics of a bicycle wheel circle radius nyt wheel’s radius involves several key standards:
1. Moment of Inertia:
The second of inertia is the resistance of an item to changes in its rotational movement. For bicycle wheel circle radius nyt wheels, a bigger radius will increase the moment of inertia, making it
tougher to accelerate but simpler to maintain pace. Conversely, smaller wheels have a decreased second of inertia, making them quicker to begin but tougher to sustain high speeds.
2. Gyroscopic Effect:
The gyroscopic impact refers to the steadiness furnished through a rotating wheel. Larger wheels, with their extra mass and radius, have a stronger gyroscopic effect, contributing to the bicycle’s
balance at better speeds. This effect is much less pronounced in smaller wheels, which is why they’re regularly used in situations where agility is greater important than stability.
3. Rolling Resistance:
Rolling resistance is the force resisting the movement of the wheel rolling on a floor. Larger wheels generally have decreased rolling resistance due to the reduced deformation of the tire as it
contacts the floor. This is a vast factor within the performance of avenue motorcycles, in which keeping excessive speeds with minimal attempt is vital.
Technological Innovations
Advancements in the bicycle wheel circle radius nyt era have brought about innovations that optimize the blessings of different wheel radii. For example, tubeless tires, remove the inner tube, reduce
rolling resistance and improve trip nice, especially on large wheels. Carbon fiber rims and spokes reduce weight without compromising strength, reaping rewards for each big and small wheel via
enhancing acceleration and handling.
Furthermore, the improvement of hybrid bikes, which combine features of avenue and mountain motorcycles, often includes wheels with intermediate radii. These motorcycles purpose to offer stability
among velocity, balance, and maneuverability, making them versatile for diverse riding situations.
The radius of a bicycle wheel circle radius nyt wheel is a fundamental characteristic that influences the bicycle’s performance, handling, and suitability for specific riding conditions. From the
speed and performance of road motorcycles with larger radii to the agility and manipulation of urban motorcycles with smaller radii, the selection of wheel size is an essential consideration for
cyclists. Understanding the impact of wheel radius on dynamics and performance helps riders make knowledgeable selections primarily based on their specific desires and using environments.
As the generation continues to conform, we can assume further innovations to refine and beautify the abilities of bicycle wheel circle radius nyt wheels, ensuring that cyclists of all sorts can
experience the most suitable overall performance and better driving enjoyment. Whether you are an expert racer, a mountain trail enthusiast, or an everyday commuter, the right wheel radius could make
all of the difference in your cycling journey.
bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius
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Advanced Mathematics I (Lecture) (Winter Semester 2014/15)
Advanced Mathematics I (Lecture) (Winter Semester 2014/15)
This lecture is mandatory for students enrolled in the International program in Mechanical Engineering at the Carl-Benz School of Engineering.
Lecture: Monday 14:00-15:30 ID SR Raum 203
Wednesday 14:00-15:30 ID SR Raum 203
Problem class: Tuesday 14:00-15:30 ID SR Raum 202
Problem Sheets
A problem sheet will be published each Thursday on this webpage. Moreover printouts are available in the students office on Friday.
Students should solve the problems independently, after that they could discuss the solutions in groups of two and could also submit one solution set per group. Only handwritten solutions will be
Please put your solutions in the box in the students office until Monday, 12:30 (11 days after the worksheet was published).
The solutions will be corrected and graded by the tutor (Dhruv Singhal). Your graded solutions are available during the tutorial.
For each problem sheet one can obtain 50 points (10 points per exercise). Please note the criteria for the testat in the section below!
There is a student tutorial given by Dhruv Singhal, Wednesdays 15:45 - 17:15 in SR 203.
In the tutorial you will practice solving the exercises and there is room for questions.
The Testat
If you successfully work on the home work (problem sheets), a testat will be given to you at the end of the semester. This testat is necessary for you to be admitted to the exam.
Please note: The AM I exam is part of the Orientierungsprüfung, which you must complete by the end of the third semester.
For the testat, you have to satisfy the following criteria:
• Obtain at least 125 points on worksheets 1 to 10. This is 25% of all the total points obtainable (each worksheet is worth 50 points).
• For at least 8 of those worksheets have at least 5 points from each.
There is be a written exam on Saturday, February 21'st, 9 - 11 AM.
In order to take the exam it is necessary to obtain the testat (see above)!
For further information please visit the following website.
There will be lecture notes for this class, distributed by the ID.
Besides the lecture notes, we can recommend the following text books
• T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel: Mathematik. Spektrum Akademischer Verlag, Heidelberg (in German).
• J. Stewart: Calculus, Early Transcendentals. Brooks/Cole Publishing Company.
• K. Burg, H. Haf, F. Wille: Höhere Mathematik für Ingenieure. Volumes I-III. Teubner Verlag, Stuttgart (in German).
• E. Kreyszig: Advanced Engineering Mathematics. John Wiley & Sons.
• E.W. Swokowski, M. Olinick, D. Pence, J.A. Cole: Calculus. PWS Publishing Company. Boston. | {"url":"https://www.math.kit.edu/iag6/edu/am12014w/","timestamp":"2024-11-06T12:17:42Z","content_type":"text/html","content_length":"171130","record_id":"<urn:uuid:029b6f9a-d357-4767-9c0f-d399d92a03f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00701.warc.gz"} |
Kimberling’s point X(24)
Kimberling's point X(24)
Kimberling defined point X(24) as perspector of $\triangle ABC$ and Orthic Triangle of the Orthic Triangle of $\triangle ABC$.
Theorem 1
Denote $T_0$ obtuse or acute $\triangle ABC.$ Let $T_0$ be the base triangle, $T_1 = \triangle DEF$ be Orthic triangle of $T_0, T_2 = \triangle UVW$ be Orthic Triangle of $T_1$. Let $O$ and $H$ be
the circumcenter and orthocenter of $T_0.$
Then $\triangle T_0$ and $\triangle T_2$ are homothetic, the point $P,$ center of this homothety lies on Euler line $OH$ of $T_0.$
The ratio of the homothety is $k = \frac {\vec {PH}}{\vec {OP}}= 4 \cos A \cos B \cos C.$
WLOG, we use case $\angle A = \alpha > 90^\circ.$
Let $B'$ be reflection $H$ in $DE.$ In accordance with Claim, $\angle BVD = \angle HVE \implies B', V,$ and $B$ are collinear.
Similarly, $C, W,$ and $C',$ were $C'$ is reflection $H$ in $DF,$ are collinear.
Denote $\angle ABC = \beta = \angle CHD, \angle ACB = \gamma = \angle BHD \implies$
$\angle HDF = \angle HDE = \angle DHB' = \angle DHC' = 180^\circ - \alpha.$
$B'C' \perp HD, BC \perp HD \implies BC|| B'C'.$$OB = OC, HB' = HC', \angle BOC = \angle B'HC' = 360^\circ - 2 \alpha \implies OB ||HB', OC || HC' \implies$
$\triangle HB'C' \sim \triangle OBC, BB', CC'$ and $HO$ are concurrent at point $P.$
In accordance with Claim, $\angle HUF = \angle AUF \implies$ points $H$ and $P$ are isogonal conjugate with respect $\triangle UVW.$
$\[\angle HDE = \alpha - 90^\circ, \angle HCD = 90^\circ - \beta \implies\]$
$\[HB' = 2 HD \sin (\alpha - 90^\circ) = - 2 CD \tan(90^\circ- \beta) \cos \alpha = - 2 AC \cos \gamma \frac {\cos \beta}{\sin \beta} \cos \alpha = - 4 OB \cos A \cos B \cos C.\]$$\[k = \frac {HB'}
{OB} = \frac {HP}{OP}= - 4 \cos A \cos B \cos C \implies \frac {\vec {PH}}{\vec {OP}}= 4 \cos A \cos B \cos C.\]$
Let $\triangle ABC$ be an acute triangle, and let $AH, BD',$ and $CD$ denote its altitudes. Lines $DD'$ and $BC$ meet at $Q, HS \perp DD'.$ Prove that $\angle BSH = \angle CSH.$
Let $\omega$ be the circle $BCD'D$ centered at $O (O$ is midpoint $BC).$
Let $\omega$ meet $AH$ at $P.$ Let $\Omega$ be the circle centered at $Q$ with radius $QP.$
Let $\Theta$ be the circle with diameter $OQ.$
Well known that $AH$ is the polar of point $Q,$ so $QO \cdot HO = QP^2 \implies QB \cdot QC = (QO – R) \cdot (QO + R) = QP^2$$\[\implies P \in \Theta, \Omega \perp \omega.\]$
Let $I_{\Omega}$ be inversion with respect $\Omega, I_{\Omega}(B) = C, I_{\Omega}(H) = O,I_{\Omega}(D) = D'.$
Denote $I_{\Omega}(S) = S'.$
$\[HS \perp DD' \implies S'O \perp BC \implies BS' = CS' \implies \angle OCS' = \angle OBS'.\]$$\[\angle QSB = \angle QCS' = \angle OCS' = \angle OBS' = \angle CSS'.\]$$\[\angle BSH = 90 ^\circ – \
angle QSB = 90 ^\circ – \angle CSS' =\angle CSH.\]$
Theorem 2
Let $T_0 = \triangle ABC$ be the base triangle, $T_1 = \triangle DEF$ be orthic triangle of $T_0, T_2 = \triangle KLM$ be Kosnita triangle of $T_0.$ Then $\triangle T_1$ and $\triangle T_2$ are
homothetic, the point $P,$ center of this homothety lies on Euler line of $T_0,$ the ratio of the homothety is $k = \frac {\vec PH}{\vec OP} = 4 \cos A \cos B \cos C.$ We recall that vertex of
Kosnita triangle are: $K$ is the circumcenter of $\triangle OBC, L$ is the circumcenter of $\triangle OAB, M$ is the circumcenter of $\triangle OAC,$ where $O$ is circumcenter of $T_0.$
Let $H$ be orthocenter of $T_0, Q$ be the center of Nine-point circle of $T_0, HQO$ is the Euler line of $T_0.$ Well known that $EF$ is antiparallel $BC$ with respect $\angle A.$
$LM$ is the bisector of $AO,$ therefore $LM$ is antiparallel $BC$ with respect $\angle A$$\[\implies LM||EF.\]$ Similarly, $DE||KL, DF||KM \implies \triangle DEF$ and $\triangle KLM$ are homothetic.
Let $P$ be the center of homothety.
$H$ is $D$-excenter of $\triangle DEF, O$ is $K$-excenter of $\triangle KLM \implies$$P \in HO.$
Denote $a = BC, \alpha = \angle A, \beta = \angle B, \gamma = \angle C, R$ circumradius $\triangle ABC.$$\angle EHF = 180^\circ - \alpha, EF = BC |\cos \alpha| = 2R \sin \alpha |\cos \alpha|.$$\[LM =
\frac {R}{2} (\tan \beta + \tan \gamma) = \frac {R \sin (\beta + \gamma)}{2 \cos \beta \cdot \cos \gamma} \implies\]$$\[k = \frac {DE}{KL} = 4\cos \alpha \cdot \cos \beta \cdot \cos \gamma \implies\]
$$\frac {\vec {PH}}{\vec {OP}}= 4 \cos A \cos B \cos C \implies P$ is the point $X(24).$
vladimir.shelomovskii@gmail.com, vvsss
Theorem 3
Let $\triangle ABC$ be the reference triangle (other than a right triangle). Let the altitudes through the vertices $A, B, C$ meet the circumcircle $\Omega$ of triangle $ABC$ at $A_0, B_0,$ and
$C_0,$ respectively. Let $A'B'C'$ be the triangle formed by the tangents at $A, B,$ and $C$ to $\Omega.$ (Let $A'$ be the vertex opposite to the side formed by the tangent at the vertex A). Prove
that the lines through $A_0A', B_0B',$ and $C_0C'$ are concurrent, the point of concurrence $X_{24}$ lies on Euler line of triangle $ABC, X_{24} = O + \frac {2}{J^2 + 1} (H – O), J = \frac {|OH|}
At first one can prove that lines $A_0A', B_0B',$ and $C_0C'$ are concurrent. This follows from the fact that lines $AA_0, BB_0,$ and $CC_0$ are concurrent at point $H$ and Mapping theorem (see
Exeter point $X_{22}).$
Let $A_1, B_1,$ and $C_1$ be the midpoints of $BC, AC,$ and $AB,$ respectively.
Let $A_2, B_2,$ and $C_2$ be the midpoints of $AH, BH,$ and $CH,$ respectively.
Let $A_3, B_3,$ and $C_3$ be the foots of altitudes from $A, B,$ and $C,$ respectively.
The points $A, A_2, H,$ and $A_3$ are collinear. Similarly the points $B, B_2, H, B_3$ and $C, C_2, H, C_3$ are collinear.
Denote $I_{\Omega}$ the inversion with respect $\Omega.$ It is evident that $I_{\Omega}(A') = A_1, I_{\Omega}(B') = B_1, I_{\Omega}(C') = C_1, I_{\Omega}(A_0) = A_0, I_{\Omega}(B_0) = B_0, I_{\Omega}
(C_0) = C_0.$
Denote $\omega_A = I_{\Omega}(A'A_0), \omega_B = I_{\Omega}(B'B_0) \omega_C = I_{\Omega}(C'C_0) \implies$$\[A_0 \in \omega_A, A_1 \in \omega_A, O \in \omega_A, B_0 \in \omega_B, B_1 \in \omega_B, O \
in \omega_B \implies\ O = \omega_A \cap \omega_B \cap \omega_C.\]$
It is known that $A_2O = HA_1 = A_0A_1, A_2O || HA_1 \implies \angle OA_2A_0 = \angle A_1A_0A_2, OA_1 ||A_0A_2 \implies A_2 \in \omega_A.$
Similarly, $B_2 \in \omega_B, C_2 \in \omega_C.$
We use Claim and get that the power of point $H$ with respect each circle $\omega_X$ is $\[HA_2 \cdot HA_0 = HB_2 \cdot HB_0 = HC_2 \cdot HC_0 = \frac {R^2 \cdot (1-J^2)} {2}.\]$
$H = AA_0 \cap BB_0 \cap CC_0 \implies H$ lies on common radical axis of $\omega_A, \omega_B,$ and $\omega_C.$
Therefore second crosspoint of these circles point $E$ lies on line $OH$ which is the Euler line of $\triangle ABC \implies$$X_{24} = I_{\Omega}(E)$ lies on the same Euler line as desired.
Last we will find the length of $OX_{24}.$$\[OH \cdot HE = \frac {R^2 \cdot (1–J^2)} {2}.\]$$\[OE \cdot OX_{24} = (OH + HE)\cdot OX_{24} = R^2.\]$$\frac {OX_{24}}{OH} = \frac {R^2}{OH^2 + OH \cdot
HE} = \frac {1}{J^2 + \frac {1– J^2} {2}} = \frac {2}{1+J^2},$ as desired.
Let $AD, BE,$ and $CF$ be the heights of $\triangle ABC, H = AD \cap BE \cap CF.$
Prove that $AH \cdot HD = \frac {R^2 (1 – J^2)}{2},$
where $R$ and $O$ are circumradius and circumcenter of $\triangle ABC, J = \frac {OH}{R}.$
It is known that $\triangle ABC \sim \triangle AEF.$$k = \frac {AF} {AC} = \cos A \implies AH = 2 R k = 2 R \cos A.$
Similarly $BH = 2 R \cos B, DH = BH \sin \angle CBE = BH \cos C = 2R \cos B \cos C.$
Therefore $AH \cdot HD = 4R^2 \cos A \cos B \cos C.$
$\[J^2 = \frac {HO^2}{R^2} = 1 – 8 \cos A \cos B \cos C \ \implies AH \cdot HD = \frac {R^2 (1 – J^2)}{2}.\]$
vladimir.shelomovskii@gmail.com, vvsss | {"url":"https://artofproblemsolving.com/wiki/index.php/Kimberling%E2%80%99s_point_X(24)","timestamp":"2024-11-02T12:21:58Z","content_type":"text/html","content_length":"75618","record_id":"<urn:uuid:356f8360-0d12-405d-8d08-6c686a0d2a9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00553.warc.gz"} |
NETWORKDAYS function | Community
I struggle to understand how NETWORKDAYS function can be used. I read the help page for the function but in vain.
I want to get number of working days in Jan 2023 (ignoring holidays).
My formulas is:
NETWORKDAYS(date(2023,1,1),date(2023,1,31),'01. Day of week'.Working = true)
where Day of week is a dimension I created (below).
I am getting 150 as the result of the formula which is obviously wrong. Help please :) | {"url":"https://community.pigment.com/questions-conversations-40/networkdays-function-1629?postid=4446","timestamp":"2024-11-10T14:18:49Z","content_type":"text/html","content_length":"210026","record_id":"<urn:uuid:774f1c11-43c1-47c8-aa3f-1ce18c16eca5>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00661.warc.gz"} |
What is 204 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
If you’re wondering what 204 degrees Celsius is in Fahrenheit, you’ve come to the right place. 204 degrees Celsius is equal to 399.2 degrees Fahrenheit. Now, let’s take a closer look at the process
of converting Celsius to Fahrenheit.
The Celsius scale is widely used in the scientific and international communities, while the Fahrenheit scale is primarily used in the United States and a few other countries. When it comes to
converting between the two scales, the process is relatively straightforward, but it does involve a simple mathematical formula.
The formula to convert Celsius to Fahrenheit is as follows: (°C × 9/5) + 32 = °F. In other words, you multiply the temperature in Celsius by 9/5 and then add 32 to the result. This will give you the
temperature in Fahrenheit.
So, applying this formula to 204 degrees Celsius, we get (204 × 9/5) + 32 = 399.2 degrees Fahrenheit. This means that 204 degrees Celsius is equivalent to 399.2 degrees Fahrenheit.
Understanding the relationship between Celsius and Fahrenheit is important, especially if you are traveling to a country that uses a different temperature scale than what you are used to. It’s also
crucial for scientific experiments, cooking, and various other applications where precise temperature measurements are necessary.
In conclusion, 204 degrees Celsius is equal to 399.2 degrees Fahrenheit. Converting between the two temperature scales is easy once you understand the simple formula and the relationship between the
two scales. Whether you’re a student, a traveler, or simply curious about temperature conversions, knowing how to convert Celsius to Fahrenheit (and vice versa) is a valuable skill. | {"url":"https://converttemperatureintocelsius.info/what-is-204celsius-in-fahrenheit/","timestamp":"2024-11-05T22:55:30Z","content_type":"text/html","content_length":"72007","record_id":"<urn:uuid:525b30c6-a22a-4cd4-a37f-b18e2533cb58>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00144.warc.gz"} |
• Time limit: 1.00 s
• Memory limit: 512 MB
There are n applicants and m free apartments. Your task is to distribute the apartments so that as many applicants as possible will get an apartment.
Each applicant has a desired apartment size, and they will accept any apartment whose size is close enough to the desired size.
The first input line has three integers n, m, and k: the number of applicants, the number of apartments, and the maximum allowed difference.
The next line contains n integers a_1, a_2, \ldots, a_n: the desired apartment size of each applicant. If the desired size of an applicant is x, he or she will accept any apartment whose size is
between x-k and x+k.
The last line contains m integers b_1, b_2, \ldots, b_m: the size of each apartment.
Print one integer: the number of applicants who will get an apartment.
• 1 \le n, m \le 2 \cdot 10^5
• 0 \le k \le 10^9
• 1 \le a_i, b_i \le 10^9 | {"url":"https://cses.fi/problemset/task/1084/","timestamp":"2024-11-12T14:03:50Z","content_type":"text/html","content_length":"5534","record_id":"<urn:uuid:c9da1ade-65f0-4da4-b368-598158c274b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00270.warc.gz"} |
A Coding Theory Bound and Zero-Sum Square Matrices
For a code C = C(n, M) the level k code of C, denoted C[k], is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive
integer divisible by 8, and n = γk, M = βk ≥ 2k then there is a codeword in C [k] whose weight is either 0 or at most n/2 - n(1/8γ - 6/(4β-2)^2) + 1. In particular, if γ < (4β - 2)^2/48 then there is
a codeword in C[k] whose weight is n/2 - Θ(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k, p) denote the minimum integer
guaranteeing that in any square matrix over Z[p], of order f(k, p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f
(k, 2)/k < 3.836. For general p we obtain, using a different approach, that f(k, p) ≤ p^(k/ln k) (1+o[k](1)).
ASJC Scopus subject areas
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'A Coding Theory Bound and Zero-Sum Square Matrices'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/a-coding-theory-bound-and-zero-sum-square-matrices","timestamp":"2024-11-11T12:40:51Z","content_type":"text/html","content_length":"51649","record_id":"<urn:uuid:74d6d517-ca02-4259-9f96-aa4f75402a81>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00883.warc.gz"} |
[HELP]Why every explanation about quicksort is somehow different?
I'm trying to implement a basic quicksort algorithm, I've understood the basic concept behind quicksort and now I'm looking online for a step by step guide.
I've followed through about 7/8 guides and what I've found is that every guide has different ways of using the pivot and different ways of using the two pointers/counters.
Some follow these steps for the pivot:
1.Decide a pivot
2.Swap the pivot with the last element of the array
3.Search the element from the left that is greater than pivot and element that is smaller than pivot from right
4.Swap those elements
Others do this:
1.Choose a pivot
2.Put left pointer to 0, a right pointer to the last element of the array
3.Start comparing left pointer to pivot, if left is smaller than pivot move pointer forward
Let's just say that before reading the guides I think I knew more about the quicksort algorithm.
What guide do you recommend?
Edit: there's some kind of problem with text formatting, I'm sorry
Top comments (4)
mattother •
So to be honest, if you're struggling with the implementation then you probably don't understand Quicksort. I could be wrong, but I'm going to give you some resources here. I'll also answer your
question directly. However, for me personally I self learned everything I know about Algorithms and it was not an easy process. And a lot of what made is difficult was not having a correct foundation
and starting in the wrong place.
I get the feeling that what's happening to you. If this isn't the case I apologize, but I figure in the worst case, the resources (and most of this reply) can just be ignored. But having a good
understanding of the class of Algorithm Quicksort fall under will probably help a lot.
There's a lot of background knowledge that if you don't have a strong foundation in will make it really frustrating to start out with Algorithms. I'm sure people have gotten by without, so take this
for what it is, which is just a set of personal recommendations, but hopefully this will save you some time in the long run.
So first thing that will make Algorithms a hell of a lot easier to understand is a good foundation in Math. Understanding the Mathematical process and Discrete Mathematics in particular. If you
already do great, if not, then these are resources that helped me out a lot.
What I would personally recommend being most comfortable with is how Mathematical proof works, especially induction. Recursion is such a core aspect of programming and induction is the huge backbone
that most proofs surrounding it rely on.
Understanding these principles makes it a lot of easier to grasp Algorithm correctness which plays a key role usually in their time costs.
In general I've personally always found this an incredible tool to have for programming. One key aspect that I've never really heard mentioned outside of Math + Algorithms is invariants.
Understanding how invariants allow you to guarantee outcomes of recursive Algorithms is incredibly helpful. And applying it in an inductive way makes it much easier for writing Algorithms, basically
induction from n to 0.
f(n) -> f(n-1)
f(0) = [is good]
By proving that an invariant holds across an iteration and results in the desired n-1 state combined with a proven terminating state you've basically shown your Algorithms will complete as expected.
Obviously there always bugs and things you'll run into when programming, but I've found applying this principle can really reduce some of the debugging headaches when first starting out with
It's also useful as a primer for property based testing, which can be very useful for confirming Algorithms. Example for Quicksort a key testing invariant
is that next value is greater than of equal to the current value.
values = [1, 2, 3, 5, 4]
for i, _ in enumerate(values[:-1]):
assert values[i] <= values[i+1]
In the case above the assertion would fail because 5 should be after 4.
Anyways now that I've made an argument for Math here's a bunch of resources:
Personally I recommend the following:
Introduction to Mathematical Thinking (Dr. Keith Devlin) (Coursera)
If you've never really studied formal mathematics this is a great primer and a very gentle introduction. However if you're already comfortable with Mathematical notation, rational vs real, etc. then
this might be a bit too elementary.
Mathematics for Computer Science (MIT Open Courseware)
Great primer on mathematics needed for Algorithms, highly recommend this course. Biggest downside, is it's not a formal course so testing yourself can be a bit tricky.
Introduction to proof in abstract mathematics
This book really helped fill in some practical gaps for creating proofs that I felt I hadn't fully grasped via other resources. So if you find you're still really struggling to create proofs this
might be a good book to reference.
Concrete Mathematics / Discrete Mathematics with Application
Both are just really good resources to have available if you're trying to self-learn Math.
There's other higher level aspects of Algorithms that personally I find really help in understanding a Algorithms better. The class of Algorithms gives you a huge hint into what underlying principle
allows it to work.
In the case of Quicksort knowing that it's a Divide and Conquer Algorithms already tells a lot of what you need to know about the Algorithm. It's also the key that really makes the Algorithm work.
In the case you outlined above for Quicksort there are all kinds of methods for choosing a pivot. But the pivot choices isn't really the key ingredient that makes a Quicksort a Quicksort Algorithms,
it's a detail. And once you grasp that fundamental aspect of a Quicksort it will make it a lot easier to understand each "flavor" of it in turn.
So here's the resources I would recommend:
Python Algorithms: Mastering Basic Algorithms in the Python Language
This is an amazing book for learning Algorithms. I would highly recommend it. It goes beyond implementation details and explains the fundamental principle behind certain Algorithms. I don't remember
if it covers Quicksort explicitly but either way a really good resource for learning Algorithms.
Algorithms Specialization (Coursera, Standford)
Good primer on Algorithms, with the added benfit of weekly excerises etc. to test yourself.
Introduction to Algorithms (MIT Open Courseware)
This tackles a lot of the same things as the Algorithms Specialization above. But last time I checked there's not material for testing yourself, etc. So it's a much more DIY approach.
The Algorithm Design Manual
This one is a classic and worth having around for reference. It has great explanations and tons of example code on Algorithms, but less introductory than some of the other resources above.
Design and Analysis of Algorithms (MIT Open Courseware)
This is more of an intermediate level course, but might still be useful.
Back to the question
So to answer the top question (and as I tackled in Algorithms section) they are different because there are a bunch of different ways you can implement Quicksort, usually it's just the method for
selecting a pivot that changes. The Algorithm Design Manual has a good primer on Quicksort, so that's probably a good place to start.
But the essence of what makes Quicksort work is the Divide and conquer nature of it.
What to learn
Obviously there's a lot of resources here so here's the course of action I would probably take, but obviously feel free to take or not take whatever you want from this reply.
A good starting point is the book Python Algorithms. I've come back to this book again and again, it's really a great resource for learning Algorithms.
If you find yourself struggling with it. I would probably try out the Algorithm Specialization courses next, having a concrete assignment each week that can be verified can really go a long way and
you get the added benefit of being able to seek help from people grappling the same content as you.
If you find you can't grasp the Mathematical principles in the course or just feel like you'd like to fill in more details, then I would turn to Mathematics for Computer Science. If you're lost in
that, then start with Introduction to Mathematical Thinking.
And if you feel like you know all this and it really is just implementation details tripping you up then I'd recommend the Algorithm Design Manual. In general it's just a great resource to have
The other book you could try is Algorithms by Robert Sedgewick. He has a lot of great Algorithm books and videos that could be worth checking out.
Anyways, so that's my list, hopefully it helps and if not I apologize. Again these are just resources that have helped me so I would say find what works for you, but hopefully this give you some
stepping stones.
IMRC21 •
Yeah, I think this defines perfectly my struggle with algorithms, I'm currently in my second year at university but couldn't pass some of the math courses and I'm having issues following the online
lectures with all of this covid stuff.
At this point, I'm asking myself if it makes sense with continuing studying CS, I feel like I could even implement a quicksort (and everything else) without having the mathematical concepts but that
would just make me a code monkey and I don't know if I want that.
I've watched the introduction of the course "introduction to mathematical thinking", looks like it's a really interesting course but I don't think I'll have the time.
Thank you for taking the time to answer me, I really appreciate it, you really made me think about my math gaps.
mattother •
No problem! I will say it mostly likely depends on your end goals.
Personally I have quite a few friends who are very successful in games and web development and do not have strong math or algorithmic skills. So if your concern is lacking knowledge in Algorithms
will somehow make you unemployable that's definitely not the case. While helpful, they are not really necessary in a lot of areas of programming.
If you goal in academic then that could be different, a lot of CS is heavy in math and from what I've seen a lot of graduate CS work does require it. But you're probably best to verify for yourself.
My goal wasn't to dissuade you from exploring Algorithms either, just to try and point out materials that I personally found really helped me understand Algorithms a lot better.
Personally I'm somebody who understands better when I understand the fundamental reasons behind something. I found this a lot with classes like Calculus and Linear Alegbra. I didn't necessarily do
terrible in them, but I didn't really grasp the fundamental ideas behind things like complex numbers, integrals, etc. And I found once I understand things like sets, the types of numbers systems,
etc. it really helped things fall into place. And at least for me this wasn't something really covered in highschool either, so it was something I really ended having to explore myself.
If that is part of your struggles then I do think Introduction to Mathematical Thinking will be really helpful. I'll also point out he has a book too, if you're low on time, but I didn't find the
book as good as the lectures themselves.
I find with Math and Algorithms there's also a lot of eureka moments too. It just takes finding the right angle to really get it. And they are both not easy subject matters, so I wouldn't to get too
down if you're struggling either, pretty much everybody does.
I would give Python Algorithms a try first though. A lot of Algorithm books focus mostly on implementation rather than fundamental concepts, but Python Algorithms does a really good job of explaining
the why it works part.
Also don't be afraid to ask questions to professors, online, etc. I think a lot of people are afraid of looking stupid, but there's nothing stupid about not understanding. These are tough subject
matters and everybody learns in different ways. And there are plenty of people happy to help you learn (and who have probably struggled through the same material as well).
Anyways like I said I would check out Python Algorithms, I think it will probably help.
Also if you need a more in-depth explanation of Quicksort just let me know. I'm happy to try and explain it to you as best I can.
Jen Miller • • Edited on • Edited
hmm, I think I may not be understanding the issue you see. I'll take a shot. Lets call the left pointer "i" and right pointer "j"
There are different ways to select the pivot. Some will select a random element, others index 0 or last or the middle. Regardless, i will keep moving right and j will keep moving left...until they
In the end, you want elements divided into two groups. The dividing point of the two arrays is where index when i and j meet. Algorithms treat the meeting differently. Some will stop the i and j
progression when i==j. Others will stop before. But this is just a implementation detail.
Some algorithms will put the pivot in the middle as a final swap, then recursively quick sort the two halves (not including the pivot in the middle -as this element is in the correct final position).
They key is to understand regardless of the partitioning specifics, the two arrays will be partitioned into two groups. The left group will be less than the pivot and the right side will be larger.
Again, it's an implementation detail if the pivot is included in the group, in the right side or left side.
I can understand why it's complicated though. In particular this video is pretty helpful (and is how I think of quicksort).
Another method is explained in the hackerrank video below, however, the explanation misses the key point of how the meeting of i and j is dealt with...so I can see how it's confusing to follow.
Though it is talked about in the code writeup. A lot of people in the comments are also frustrated too.
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/imrc21/why-every-explanation-about-quicksort-is-somehow-different-1f0","timestamp":"2024-11-04T15:17:20Z","content_type":"text/html","content_length":"128245","record_id":"<urn:uuid:08294057-cec5-413b-ae8a-7e5ebbdfbce7>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00640.warc.gz"} |
Notes and Questions Correlation Class 11 Economics
Students should refer to Worksheets Class 11 Economics Correlation Chapter 7 provided below with important questions and answers. These important questions with solutions for Chapter 7 Correlation
have been prepared by expert teachers for Class 11 Economics based on the expected pattern of questions in the class 11 exams. We have provided Worksheets for Class 11 Economics for all chapters on
our website. You should carefully learn all the important examinations questions provided below as they will help you to get better marks in your class tests and exams.
Correlation Worksheets Class 11 Economics
Important terms and concepts
Correlation studies the relationship between tow variables in which change in the value of one variable causes change in the other variable. It is denoted by letter ‘r’.
Kinds of correlation:-
1. Positive and Negative correlation.
2. Linear and non – linear correlation.
3. Simple and multiple correlations.
Positive correlation: When both variables move in the same direction. If one increases, other also increases and vice-versa.
Negative correlation: – When two variables move in the opposite direction, they are negatively correlated.
Linear Correlation: – When two variables change in a constant proportion.
Non- linear correlation: – When two variables do not change in the same proportion.
Simple correlation – Relationship between two variables are studied.
Multiple Correction – Relationship between three or more than three variables are studied.
Degrees of Correlation:
1. Perfect Correlation – When values of both variables changes at a constant rate Types – (a) Perfect positive correlation – when values of both variables changes at a constant ratio in the same
direction correlation coefficient value (r) is + 1
(b) Perfect negative correlation – When values of both the variables change at a constant ratio in opposite direction. Value of coefficient of correlation is -1
2. Absence of correlation : When there is no relation between the variables r = 0
3. Limited degree correlation : The value of r varies between more than O and less than 1
Types – a) High : r his between ± 0.7 & 0.999
b) Moderate = r lies between ± 0.5 and + 0.699
c) Low: r < ± 0.5
Different methods of finding correlation
a) Karl Pearson’s coefficient method
b) Rank method / Spearman’s coefficient method
c) Scatter Diagram
(A) Karl Pearson’s Method
Where X = X – X, Y = Y – Y
N = number of observations
σ X = Standard deviation of series X
σ Y = Standard deviation of series Y
Actual Mean Method
Merits of Karl Pearson’s Method
1. Helps to find direction of correlation
2. Most widely used method
Demerits of Karl Pearson’s method
1. Based on large number of assumptions
2. Affected by extreme values
(B) Spearmans’s Rank Correlation Method
In case of non repeated ranks :-
r[s] = Spearman’s rank correlation
∑D^2 = Sum of squares of difference of ranks
N = Number of observation
In case of repeated ranks:-
M = number of items with repeated ranks.
Merits of Spearman’s Rank Correlation
1. Simple and easy to calculate
2. Not affected by extreme values
Demerits of Spearman’s Rank Correlation
1. Not Suitable for grouped data
2. Not based on original values of observations.
(C) Scatter Diagram – Given data are plotted on a graph paper. By looking at the scatter of points on the graph, degree and direction of two variables can be found.
Merits of Scatter Diagram
1. Most simplest method.
2. Not affected by size of extreme values.
1. Exact degree of correlation cannot be found. | {"url":"https://worksheetsbag.com/notes-correlation-class-11-economics/","timestamp":"2024-11-11T17:32:09Z","content_type":"text/html","content_length":"113440","record_id":"<urn:uuid:5b91ff04-c0c9-4ea8-97ac-99a703f8b910>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00553.warc.gz"} |
Solution with graphical method for linear equations with two variables
To solve with the graphical method linear equations that have two unknowns we must correctly draw the two graphs on the Cartesian plane and find their point of intersection.
How do we do it?
• We will consider each equation as if it were a function and we will make a table of values of $X$ and $Y$.
• We will place in the $X$ of each equation random values (we recommend to put $0,1,2$), then we will deduce the $Y$ corresponding to each $X$ and we will write it down respectively in the table of
• We will draw a Cartesian plane and mark on it the points of each function.
• After we have marked all the points of a function we will draw a line between them and see how the function looks like.
• Only after this we will start marking the points of the second function to avoid confusion, then we will redraw a line between these last points.
• We will analyze what is the point of intersection between the graphs we drew, this point represents the solution graphically for your system of linear equations.
Note that you might see a case in which the lines are parallel and, therefore, there are no points of intersection between them or a case in which the lines overlap and, thus, there would be infinite
points of intersection.
Which graph corresponds to the following equations?
\( 3(2x-4y)=12 \)\( \)
\( \frac{x+y}{3}-\frac{y}{3}=7 \)
Which graph corresponds to the following equations?
\( 3(2x-4y)=12 \)\( \)
\( \frac{x+y}{3}-\frac{y}{3}=7 \) | {"url":"https://www.tutorela.com/math/solution-with-graphical-method-for-linear-equations-with-two-variables","timestamp":"2024-11-02T12:27:12Z","content_type":"text/html","content_length":"70201","record_id":"<urn:uuid:2c880f67-acf5-4a04-860f-f7677e6fb388>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00100.warc.gz"} |
Aleksey Nogin's Curriculum Vitae
Citizenship: USA
Cornell University Ithaca, NY
Aug 2002 Ph.D. in Computer Science; Minor in Cognitive Studies
May 2000 M.S. in Computer Science
Moscow State University Moscow, Russia
Jun 1997 Diploma with Honors, Faculty of Mathematics and Mechanics, Division of Mathematics
Major: Mathematics, Applied Mathematics (Department of Mathematical Logic and Theory of Algorithms)
Development, application, and theory of systems and tools for computer-aided programming language research and experimentation, computer-aided software engineering, and computer-aided
reasoning and verification. Formal methods for reliable software design process. Development, application, and theory of interactive proof assistants and logical frameworks. Type
theories, higher-order abstract syntax, and their applications.
HRL Laboratories, LLC Malibu, CA
Since 8/ Research Staff Scientist
California Institute of Technology Pasadena, CA
9/2002-8/ Postdoctoral Scholar / Senior Postdoctoral Scholar with Prof. Jason Hickey
Research Projects: Computer-aided and formal software engineering based on the logical frameworks, including building reliable extensible compilers. Improving the software engineering
capabilities of the MetaPRL formal toolkit. Foundations for practical syntax-based reasoning about properties of programming languages and languages with bindings, as well as for formal
reasoning in the area of logical reflection in general. Formalizing the foundations of abstract algebra and number theory. Designing and implementing of the OMake build system. Designing
serializability protocols for distributed filesystems.
Jul-Aug Visitor with Prof. Jason Hickey
MetaPRL Project
Since 1999 Coordinator and a Lead Developer. MetaPRL is a formal methods programming toolkit that can be used as a computer-aided software engineering tool. It is also an interactive tactic-based
theorem prover. It also implements a logical framework that allows its users to specify and work with different logical theories and formalisms. Finally, MetaPRL is a basis for a
programming language research, experimentation and meta-reasoning toolkit that is currently being developed.
Cornell University Ithaca, NY
9/1997-9/ Research Assistant with Prof. Robert Constable
Ph.D. Dissertation: Theory and Implementation of an Efficient Tactic-Based Logical Framework
Research Projects: Increased the logical speed of the MetaPRL system by two orders of magnitude. Came up with several methods of improving expressivity and making type theory
formalization more usable in both automated and user-guided theorem proving.
Mar-Apr Visitor with Prof. Robert Constable
9/1992-6/ Moscow State University Moscow, Russia
9/1995-6/ Laboratory for Logical Problems in Computer Science (headed by Prof. Sergei Artemov)
9/1994-6/ Department of Mathematical Logic and Theory of Algorithms; Advisor: Prof. Alexander Razborov
May 1997 Diploma Thesis: Improving the Efficiency of NuPRL Proofs
Participated in creation of three successful grant applications:
Laboratory for Computational Methods at Moscow State University (approximately 30 men-years worth of funding);
ITR: Reliable Distributed Programming with Speculations”;
Building Interactive Digital Libraries of Formal Algorithmic Knowledge”.
Co-organized a day-long tutorial “Introduction to MetaPRL Theorem Prover” given at the 16th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2003).
California Institute of Technology Pasadena, CA
Fall 2005 Creator and Instructor, undergraduate course Language-Based Security.
June 2004 Creator and Instructor, course Introduction into formal computer-aided reasoning and the MetaPRL theorem prover, North American Summer School in Logic, Language and Information (NASSLI)
2004 at UCLA.
Winter 2004 Creator and Instructor, graduate / advanced undergraduate course Programing Language Semantics.
Spring 2003 Creator and Instructor, graduate / advanced undergraduate MetaPRL-based course Type Theory and Formal Methods.
Cornell University Ithaca, NY
Spring 2002 TA, undergraduate course Structure and Interpretation of Computer Programs.
Fall 2000 Instructor, undergraduate course Introduction to Unix.
Spring 1998 TA, undergraduate course Structure and Interpretation of Computer Programs.
Fall 1997 TA, undergraduate course Introduction to Theory of Computing (Honors).
Mechanized meta-reasoning using a hybrid HOAS/de Bruijn representation and reflection.
In John H. Reppy and Julia L. Lawall, editors, Proceedings of the 11th ACM SIGPLAN International Conference on Functional Programming, ICFP 2006, pages 172–183. ACM, 2006.
OMake: Designing a scalable build process.
In Luciano Baresi and Reiko Heckel, editors, Fundamental Approaches to Software Engineering, 9th International Conference, FASE 2006, volume 3922 of Lecture Notes in Computer Science,
pages 63–78. Springer, 2006.
An extended version is available as California Institute of Technology technical report CaltechCSTR:2006.001.
Formalizing type operations using the “Image” type constructor.
In Proceedings of the 13th Workshop on Logic, Language, Information and Computation (WoLLIC 2006), volume 165 of Electronic Notes in Theoretical Computer Science, pages 121–132.
Elsevier, 2006.
Extended version was submitted to Information and Computation Journal.
Practical reflection for sequent logics.
In Proceedings of the International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP'06), Electronic Notes in Theoretical Computer Science, 2006.
Formal Compiler Construction in a Logical Framework.
Higher-Order and Symbolic Computation, 19(2–3):197–230, September 2006.
A Computational Approach to Reflective Meta-Reasoning about Languages with Bindings.
In MERLIN '05: Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized reasoning about languages with variable binding, pages 2–12. ACM Press, 2005.
An extended version is available as California Institute of Technology technical report CaltechCSTR:2005.003.
Building Extensible Compilers in a Formal Framework. A Formal Framework User's Perspective.
In Konrad Slind, editor, Emerging Trends. Proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004), pages 57–70. University of Utah, 2004.
A Simple Serializability Mechanism for a Distributed Objects System.
In David A. Bader and Ashfaq A. Khokhar, editors, Proceedings of the 17th International Conference on Parallel and Distributed Computing Systems (PDCS-2004). International Society for
Computers and Their Applications (ISCA), 2004.
Extensible Hierarchical Tactic Construction in a Logical Framework.
In Konrad Slind, Annette Bunker, and Ganesh Gopalakrishnan, editors, Proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004), volume 3223
of Lecture Notes in Computer Science, pages 136–151. Springer-Verlag, 2004.
MetaPRL — A Modular Logical Environment.
In David Basin and Burkhart Wolff, editors, Proceedings of the 16th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2003), volume 2758 of Lecture Notes in
Computer Science, pages 287–303. Springer-Verlag, 2003.
Compiler Implementation in a Formal Logical Framework.
In Proceedings of the 2003 workshop on Mechanized reasoning about languages with variable binding, pages 1–13. ACM Press, 2003.
Formalizing Abstract Algebra in Type Theory with Dependent Records.
In David Basin and Burkhart Wolff, editors, 16th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2003). Emerging Trends Proceedings, pages 13–27. Universität
Freiburg, 2003.
Implementing and Automating Basic Number Theory in MetaPRL Proof Assistant.
In David Basin and Burkhart Wolff, editors, 16th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2003). Emerging Trends Proceedings, pages 29–39. Universität
Freiburg, 2003.
Building Reliable Compilers with a Formal Methods Framework.
In Dr. Indrakshi Ray, editor, The 14th International Symposium on Software Reliability Engineering (ISSRE 2003). Supplementary Proceeding, pages 319–320. Chillarege Press, 2003.
Theory and Implementation of an Efficient Tactic-Based Logical Framework.
Ph.D. Thesis, Cornell University. August 2002.
Quotient Types: A Modular Approach.
In Victor A. Carreño, Cézar A. Muñoz, and Sophiène Tahar, editors, Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002), volume 2410
of Lecture Notes in Computer Science, pages 263–280. Springer-Verlag, 2002.
Sequent Schema for Derived Rules.
In Victor A. Carreño, Cézar A. Muñoz, and Sophiène Tahar, editors, Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002), volume 2410
of Lecture Notes in Computer Science, pages 281–297. Springer-Verlag, 2002.
Markov's principle for propositional type theory.
In L. Fribourg, editor, Computer Science Logic, Proceedings of the 10th Annual Conference of the EACSL, volume 2142 of Lecture Notes in Computer Science, pages 570-584. Springer-Verlag,
Jprover: Integrating connection-based theorem proving into interactive proof assistants.
In International Joint Conference on Automated Reasoning, volume 2083 of Lecture Notes in Artificial Intelligence, pages 421-426. Springer-Verlag, 2001.
Fast tactic-based theorem proving.
In J. Harrison and M. Aagaard, editors, Theorem Proving in Higher Order Logics: 13th International Conference, TPHOLs 2000, volume 1869 of Lecture Notes in Computer Science, pages
252-266. Springer-Verlag, 2000.
Writing constructive proofs yielding efficient extracted programs.
In Didier Galmiche, editor, Proceedings of the Workshop on Type-Theoretic Languages: Proof Search and Semantics, volume 37 of Electronic Notes in Theoretical Computer Science. Elsevier
Science Publishers, 2000
Improving the efficiency of NuPRL proofs.
Department of Computer Science TR97-1643, Cornell University, August 1997.
On Horn interpolation in linear logic.
Mathematical Logic and Theoretical Computer Science Prepublication Series 1995-11, Steklov Mathematical Institute, April 1995.
In Russian.
et al.
OMake Home Page.
et al.
MetaPRL Home Page.
A listing of MetaPRL theories.
Introduction to MetaPRL Theorem Prover. Tutorial: Implementing FOL in MetaPRL.
MetaPRL System Description
MetaPRL User Guide.
MetaPRL Developer Guide.
Sep 2005 3rd ACM SIGPLAN workshop on Mechanized reasoning about languages with variable binding, Tallinn, Estonia
Mar 2005 CS Colloquium, Purdue University, West Lafayette, IN
Oct 2004 Microsoft Research, Redmond, WA
Sep 2004 Theorem Proving in Higher Order Logics, 17th International Conference, Park City, UT
Sep 2003 Seminar Logical Problems in Computer Science, Department of Mathematical Logic and Theory of Algorithms, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia
Apr 2003 Microsoft Research, Redmond, WA
Dec 2002 City University of New York Graduate Center, Computer Science Colloquium
Aug 2002 Theorem Proving in Higher Order Logics, 15th International Conference, Hampton, VA, two talks
Sep 2001 Computer Science Logic, 10th Annual Conference of the EACSL, Paris, France
Sep 2001 Logic Seminar Series, Department of Informatics of Saarland University and MPI Institute, Saarbrücken, Germany
Aug 2000 Theorem Proving in Higher Order Logics, 13th International Conference, Portland, OR
Jun 2000 Workshop on Type-Theoretic Languages: Proof Search and Semantics, Pittsburgh, PA
Mar 2000 Seminar Logical Problems in Computer Science, Department of Mathematical Logic and Theory of Algorithms, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia
2000-2006 Helped to set up and administer a 16x2-node cluster for Jason Hickey's Mojave Group at Caltech
Since 1997 Contributor to Open Source projects, including OCaml, Mozilla and Red Hat Linux/Fedora Project
1997-2002 Set up and administered a department-wide CVS server. Administered Linux servers for the Cornell PRL group
1995-1997 Set up and administered network, mail, and dial-up servers for the Youth Scientific Creativity Center | {"url":"https://nogin.org/CV.html","timestamp":"2024-11-02T15:11:29Z","content_type":"text/html","content_length":"35419","record_id":"<urn:uuid:52fbe152-4268-4284-b5ed-d6444ade7af1>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00596.warc.gz"} |
What Sample Group number mean?
I understand that Pendo is assigning random group numbers from 0 to 99 to visitors, but what these numbers mean? e.g if an user has sample group 35 and one has sample group 98, how I can interpret
this data?
9 comments
• Sample Group is an automatic metadata flag assigned to each unique Visitor at random between 0 and 99. The effect of this is that each Sample Group contains, roughly, 1% of your total Visitor
population. You can use this flag to establish Test and Train groups for experimentation and analysis within your product.
• Thank you Greg!
To get a better sense of what this means: for example I can create a segment of sample group 98, and this will guarantee that whatever I plan to test I will get to 1% of the total Visitor
population. the more sample groups I add the more I control the increase of the % reach of the visitors?
• You've got it! I will reiterate there is a roughness to that percentage due to how Visitors are randomly assigned and growing nature of Visitors to your product. But the percentages will be ~1%
per Sample Group.
• Hello! Is the Sample Group numbering applied across a Pendo Subscription or by the applications you can set up in your subscription?
• Is the Sample Group number assigned to each user static or dynamic? I'd like to know which (very product-specific) actions users took based on which guide they saw, and the "action" in question
can't be tracked via Track Event/Feature Click.
I'm assuming we can export the data of who's seen which guide and match it with our internal product analytics data on user/account ID, and want to verify this is possible.
• Sample Group is assigned during a Visitor's first visit and is not touched again. Since Visitor IDs are shared across all apps within a Subscription (a big reason to keep Visitor IDs consistent
across applications), that Sample Group is shared across applications. However, if you have multiple Subscriptions, then a given Visitor may have different Sample Group values.
• Greg Nutt (or someone else), could you give more explanation about the Sample Group values? I read through the discussion here and the Sample Groups section and I'm still confused. Does it
basically just mean that you can have up to 99 Sample Groups?
Maybe it'll help if I explain what I'm trying to do. I would like to A/B test a guide. Should I create these two Segments?
□ Segment A: Sample Group is equal to 1
□ Segment B: Sample Group is equal to 2
Thanks in advance!
• Sample Group is a value that is automatically assigned by Pendo when a Visitor is first seen by the system. The value can be any number between 0 and 99 and is randomly assigned. So, in theory,
if you were were create one segment where Sample Group < 50 and one where Sample Group >= 50, then these two would split your population in half. Similarly, if you were to create a segment where
Sample Group < 20, this would provide you with ~20% of your population.
The caveat to this is that Sample Groups are assigned randomly by Visitor, not by any other piece of metadata. So if you were to try and further segment down by, say, a metadata value called
"Role", the two segments of equal numbers of Sample Groups would no longer hold statistically even populations.
There is another consideration based on Population size. If your Visitor Count is 100k (on the exaggerated high-end), then each Sample Group will hold ~1k Visitors. However, if your Visitor
Count is only 100, then each Sample Group, in theory, would hold only 1 Visitor. But due to how randomness is applied, this isn't always strictly true, e.g. if you flip a coin 10 times, you may
not end up with a true 50/50 split. But if you did so 10k times, you'll find the odds even out. So the assumption of even distribution works best the larger the population.
I hope this helps!
• Greg Nutt Got it, that makes sense! Thanks for the detailed explanation!
Please sign in to leave a comment. | {"url":"https://support.pendo.io/hc/en-us/community/posts/4413031425563-What-Sample-Group-number-mean-","timestamp":"2024-11-05T08:59:16Z","content_type":"text/html","content_length":"64071","record_id":"<urn:uuid:7e71c726-71a6-4d07-ba24-ca56cc5e6769>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00121.warc.gz"} |
Shortest Subarray to be Removed to Make Array Sorted | CodingDrills
Shortest Subarray to be Removed to Make Array Sorted
Given an array of integers arr, return the length of the shortest subarray that, when removed, makes the array sorted in ascending order.
Ada AI
I want to discuss a solution
What's wrong with my code?
How to use 'for loop' in javascript?
javascript (node 13.12.0) | {"url":"https://www.codingdrills.com/practice/shortest-subarray-to-be-removed-to-make-array-sorted","timestamp":"2024-11-05T18:56:55Z","content_type":"text/html","content_length":"13338","record_id":"<urn:uuid:a7b3c89f-0a3d-48f4-a8d0-1274de5215dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00385.warc.gz"} |
Lesson 5
Equations and Their Graphs
Lesson Narrative
So far in the unit, students have primarily used descriptions, expressions, and equations to represent relationships and constraints. In this lesson, they revisit the idea that graphs can be a useful
way to represent relationships. Students are reminded that each point on a graph is a solution to an equation the graph represents. They analyze points on and off a graph and interpret them in
context. In explaining correspondences between equations, verbal descriptions, and graphs, students hone their skill at making sense of problems (MP1).
In this lesson, students are also introduced to the use of graphing technology to graph equations. This introduction could happen independently as long as it precedes the second activity in the
Learning Goals
Teacher Facing
• Comprehend that the graph of a linear equation in two variables represents all pairs of values that are solutions to the equation.
• Interpret points on a graph of a linear equation to answer questions about the quantities in context.
• Use graphing technology to graph linear equations and identify solutions to the equations.
Student Facing
• Let’s graph equations in two variables.
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (If students typically access the digital version of the materials,
Desmos is always available under Math Tools.)
Student Facing
• I can use graphing technology to graph linear equations and identify solutions to the equations.
• I understand how the coordinates of the points on the graph of a linear equation are related to the equation.
• When given the graph of a linear equation, I can explain the meaning of the points on the graph in terms of the situation it represents.
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/2/5/preparation.html","timestamp":"2024-11-07T22:17:31Z","content_type":"text/html","content_length":"78785","record_id":"<urn:uuid:9594050b-20a5-429e-a9d7-f76d0c2e3be5>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00503.warc.gz"} |
Business finance analysis problems, NPV, company, stock valuation - Superb-Writers
Business finance analysis problems, NPV, company, stock valuation
Part A: Your answers should be brief and to the point. Show your calculations, do not just give an answer.
Question 1
The free cashflows of an all-equity project are set out below. Year Cashflows
0 -15,000
1 -15,000
2 +1,000
3 +1,500
4 +2,000
5 +2,500
6 +3,300
Thereafter the free cashflows grow indefinitely at 2.5% per annum.
a) If the project has a cost of capital of 10%, what is its net present value? Show your workings.
b) The IRR of this project is 11%. If we tripled all the cashflows in the project but kept the other assumptions the same, what would happen to its IRR? What would happen to the NPV?
Question 2
The following table shows the expected return and standard deviation of returns for two stocks, the market, and a risk free bond. The correlation of returns between stock A and the market is 0.816.
Company A: Expected Return Not given, Standard Deviation 14%
Company B: Expected Return 12%, Standard Deviation Not given
Market: Expected Return 12%, Standard Deviation 16%
Risk free bond: Expected Return 4%, Standard Deviation 0
a) What is the beta of stock B?
b) What is the beta of stock A? What is the return of stock A?
c) What is the expected return of a portfolio that consists of 50% of stock A and 50% of the market?
d) What is the standard deviation of a portfolio that consists of 50% of stock A and 50% of the market?
Question 3
Company XYZ is entirely equity financed and has 1 million shares outstanding. The cost of capital for Company XYZ is 9.5%. The corporate tax rate is 30%. Assume that the CAPM holds.
a) Financial projections for XYZ suggest that it will generate an EBIT of $1.0 million next year. Depreciation next year will be $200,000, the net working capital balance will be zero, and there will
be no capital expenditure. Assuming that the FCF will grow at a constant rate of 2% per annum forever after next year, compute the value of the company today and its current share price.
b) XYZ is contemplating a $3 million perpetual bond issue to fund a share repurchase. The bond yield will be 6% on an annually compounded basis. What will be the effect of the bond issue on the value
of the firm’s equity before the share repurchase?
c) What will XYZ’s cost of equity be after the bond issue and before the share repurchase has taken place?
d) At what share price are equity holders indifferent between tendering and not tendering the shares?
e) How many shares will XYZ buy back if it uses all the proceeds of the bond issue to do so?
f) What will be the fair price of the shares after the buy-back has taken place?
Question 4
a) You are long a call with an exercise price of 8 and short a call with an exercise price of 12. Both options are on the same share of stock with the same exercise date. Plot the value of this
combination as a function of the stock price on the exercise date (a payoff diagram). Briefly describe why you would take this position.
b) b) Beyond Tofu’s current stock price is 40 and in one period it may go up 50% or down 50%. The risk free rate for 1 period is 3%. What would the present value of a European put with a strike price
of 35 be in the current period (its expiration date is in one period)?
c) Suppose the market price of a European put option on Beyond Tofu from part (b) is 3. How could you make arbitrage profits?
Part B
The response to any individual part (a, b, c, d, or e) should be fewer than 200 words (excluding calculations).
a) Choose two established companies in the same industry that are publicly listed in the U.S. Provide an economic explanation why the betas found on Yahoo might differ (and if they don’t differ,
explain why it is possible for them to differ).
b) Choose one of the two companies. Without any research into what the firm’s capital structure is, do expect them to have a lot of leverage or a little? Explain why – what factors might determine
why it would have so much (or so little) debt in its capital structure?
c) Consider the same company you chose in part (b). Suppose it has 2 new projects to evaluate. The first is to expand is to produce and sell candy. The second is to produce and sell toothpaste.
Estimate the cost of capital for both of these projects (you can use the equity cost of capital). Discuss briefly any assumptions you needed to make to estimate these.
d) Discuss why the equity cost of capital may be inappropriate for part (c). Discuss what data you would need to estimate the appropriate cost of capital.
e) Consider again the same company you chose in part (b). Assume it were a private company which was thinking about doing an IPO. What method should it use to IPO and why? What are the tradeoffs with
this method? | {"url":"https://www.superb-writers.com/business-finance-analysis-problems-npv-company-stock-valuation/","timestamp":"2024-11-04T05:37:33Z","content_type":"application/xhtml+xml","content_length":"112143","record_id":"<urn:uuid:478892ce-45cf-4b0f-b203-7202fcb8e4cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00795.warc.gz"} |
Building Modern Algebra: Learning objectives
Last time, I wrote about the Modern Algebra course that I'm teaching this semester and how I'll be writing about how it's being built. This is the first post in that series, and it starts where the
course build process starts: with learning objectives.
Back in April 2020, when the Big Pivot was still just a few weeks old and I was thinking about how we might improve our online instruction for the Fall, I wrote that the first step toward excellence
in online teaching (or any teaching) is to write clear, measurable learning objectives for the course at both macro and micro levels. Clear, because learners need to know what the target is in order
to hit it. Measurable, because we have to know what the target is to know if students have hit it. And we have these objectives in the first place because our courses are not about us; they are about
students and what they learn. How do you know if students are learning what they need to learn? You have to start with being clear about what those learning outcomes are.
I won't address the objections that some faculty raise – still, after all this time – to the concept of learning objectives. I've done that before and doing it yet again feels like arguing that
the Earth revolves around the Sun. Â Instead, I want to write about the learning objectives for the Modern Algebra course, because the process worked out much differently than for Calculus.
The approach with Calculus was simple: Go through the course module-by-module and identify the "micro" level objectives students will encounter. These are things that students should be able to do,
but I don't necessarily want to assess every single one of them. I began the course build process by doing this and putting those objectives in a list. Then, from that list of micro-objectives,
distill a smaller set of objectives that address the main categories of things students should do. I called those learning targets and I also put those in a list, at the end of the syllabus. The
Learning Targets are what I actually assessed, through the use of "Checkpoints" (described in the syllabus; here's a sample one) which used the micro-level objectives not as targets to assess but as
raw material for how to assess those targets. I also had some over-arching course-level objectives that described the big ideas of the course. Â
I tried this with Modern Algebra, and it didn't work.
It's because Calculus, while it has many conceptual ideas that are important, is a course that can be assessed on the basis of skills. Compute a derivative; look at a graph and state the value of a
limit; write out the setup for a Riemann sum. And those tasks that students perform are easily categorized: If I want to assess the ability to "determine the intervals of concavity of a function and
find all of its points of inflection" (Learning Target DA.2), then it's simple, I just give them a function and tell them to do exactly that. There is really only one thing students can do to
demonstrate their skill: Take the second derivative, set up a sign chart, etc. and if they do this reasonably well, it's evidence of proficiency.
Modern Algebra is different. Modern Algebra has skills embedded in it but is not primarily about those skills. I want students to be able to find all the units and zero divisors of a ring, but not
because that skill is relevant or interesting in and of itself, because it isn't. The only reason I want students to be able to carry out that task is in service of some bigger idea. And unlike
Calculus where the micro skills map more or less on to just one or at most a small number of big ideas, micro skills in algebra could be used for anything.
Several years ago I taught the second semester of this course, which focuses on group theory. I took the Calculus approach of teasing out every skill that could be important and making sure I
assessed them. I ended up with – I am almost ashamed to say it – 67 learning objectives in all. Here they are in all their God-awful glory. At the time I thought I was doing the right thing: If
you want students to know something, express it as a learning objective and then assess it. But in retrospect, it's painfully obvious that trying to center the course on skills in this way is nothing
but egregious micromanagement, and in the end the students focused laser-like on the micro objectives and missed all the big ideas. And it's not their fault.
So, don't do that.
Here is the approach I am taking this time.
I did go through my course module-by-module (after deciding how the module structure would go, roughly) and wrote down all the micro-level objectives for each module. Here's the list. This process
took me about two hours to complete and I think it will save me far more than two hours' time during the semester, since now I have a map of where everything happens in the course and a list of what
matters and what doesn't matter content-wise. Advice: If you do nothing else for your courses this semester, do this for each of them.
But, I did not distill these into Learning Targets. The class actually has no learning targets as such, like Calculus does. Instead, I went straight to the course-level objectives. That list is:
After successful completion of the course, students will be able to…
1. Write to communicate the topics of abstract algebra using accepted proof writing conventions, explanations, and correct mathematical notation.
2. Identify fundamental structures of abstract algebra including rings, fields, and integral domains.
3. Comprehend abstract definitions and theorem statements by building examples and non-examples of definitions, and drawing conclusions using definitions and theorems given mathematical
4. Demonstrate problem solving skills in the context of abstract algebra topics through consideration of examples, pattern exploration, conjecture, proof construction, and generalization of
5. Analyze similarities and differences between algebraic structures including rings, fields, and integral domains.
This is a combination of the official course objectives mandated by my department and my own ideas. Especially, objective 3 — "comprehending" definitions and theorems — is my own creation.
So, I have two layers of course objectives: The topmost layer (above) and the bottom-most layer (the micro-objectives). Therefore the main difference between this and Calculus is that there is no
"middle" layer where Calculus' Learning Targets resided.
This makes sense, to me at least, because again Modern Algebra is focused on big ideas and goals and not so much (or at all) on "skills". Insofar as I will assess these objectives, I'll be asking
students to do things that provide evidence of proficiency or mastery of the main, course-level objectives. But the focus is not on the things, but rather on the objectives. Students perform tasks in
order to make visible their progress toward the course-level objectives; their performance of those tasks works like a progress meter.
Speaking of assessment: Discussion of the grading system comes later, but it's worthwhile to mention it now. This course uses mastery grading but it's much more along the lines of specifications
grading than standards-based grading. Sometimes we use all three of those terms as synonyms for each other, but there are actually significant differences. Â As I explained above, students will be
doing work that shows their progress toward the course objectives, and that work (as I'll detail in another post) will be graded using simple rubrics that use no point values and allow for lots of
feedback and revision, and the student's course grade is based on "eventual mastery". But the grading system itself does not have discrete learning targets that are checked off one by one. Instead,
students complete "bundles" of tasks, and each bundle maps to a course objective. Doing the work in the course serves to make visible the progress toward mastery of a bundle. But failing to master
micro-objective "X" — possibly ever, in the course — does not necessarily imply lack of progress on course objective "Y".
This all seems very theoretical, but in fact I think Modern Algebra has a lot in common with many non-STEM disciplines. Many such courses also focus more on big ideas than on "learning targets", and
I can see why some faculty in those disciplines have questions about the idea of Learning Targets. But if you're teaching a literature or philosophy or art history course, your course objectives
might not look terribly different than the ones I listed above, and so the interplay between micro-scale and course level objectives might also be similar. I'd love to hear about that in the comments
if you're in that situation.
Next time: A little more about assessments. | {"url":"https://rtalbert.org/building-modern-algebra-learning-objectives/","timestamp":"2024-11-11T03:21:01Z","content_type":"text/html","content_length":"32860","record_id":"<urn:uuid:b452c038-8e3f-414d-be2f-ff9a5accfcfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00443.warc.gz"} |
Philosophy of Physics Seminar (Thursday - Week 7, TT23)
John Stachel, the first editor of the Collected Papers of Albert Einstein and the founder of what is today called Einstein scholarship, divides the creation of the general theory of relativity (GR)
into a drama of three acts. The first act centers around 1907, when Einstein was overwhelmed by the epiphany of the equivalence principle, the idea that the force of gravity and the intertia of
bodies were intimately connected. The second act takes place around 1912, when Einstein entered the promised land and proceeded from scalar theories of gravity to those based on a metric tensor. And
the third act finishes in late November 1915, when Einstein found what we now call the Einstein field equations, the successors of Newton's law of gravity. Stachel further argued that the "missing
link" between the second and the third act was Einstein's so-called rotating disc argument, which allowed him to forge a connection between gravity-inertia and non-Euclidean geometry.
In this talk, I shall argue that instead of being the protagonist in said drama, and in which the rotating disc argument is the one heureka moment that allowed Einstein to transition to a metric
theory of gravity, Einstein, in the summer and autumn of 1912, was instead an adventurer walking on six different paths in parallel, all of which led him to the programme of finding a theory of
gravity based on a metric tensor. And yet, I shall argue, it is Einstein's starting point, his scalar theory of gravity of early 1912, that, together with his equivalence principle, pointed him to
these six paths, and determined the way he eventually saw the metric tensor. In particular, I shall argue that Einstein's work on a scalar theory of gravity, and his multi-path journey from there to
the metric tensor, equipped him with many of the interpretational moves and tools that would influence his later interpretation of GR, and made him resist seeing GR as a "reduction of gravity to
spacetime geometry". I shall decipher how Einstein saw the role of geometry in GR instead, what he himself meant by "geometry", and how his notion of geometry differed from his contemporaries and
successors. I shall outline how all this led him to an interpretation of GR that saw the distinction of matter and spacetime geometry as something to be overcome rather than as something to be
With speaker’s consent, talks will be recorded and published on YouTube. Our channel is:
If you wish to join the dinner proceeding the talk, please email Henrique: gomes.ha@gmail.com.
[Philosophy of Physics Seminar Convenors for HT23: Oliver Pooley, Patrick Duerr and Henrique de Andrade Gomes | Philosophy of Physics Group Website] | {"url":"https://www.philosophy.ox.ac.uk/event/philosophy-of-physics-seminar-thursday-week-7-tt23","timestamp":"2024-11-09T06:14:22Z","content_type":"text/html","content_length":"115781","record_id":"<urn:uuid:1a216ec5-d023-447b-a9fb-b0e7173a535a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00274.warc.gz"} |
The goal of esreg is to simultaneously model the quantile and the Expected Shortfall of a response variable given a set of covariates.
CRAN (stable release)
You can install the released version from CRAN:
GitHub (development)
The latest version of the package is under development at GitHub. You can install the development version using these commands:
If you are using Windows, you need to install the Rtools for compilation of the codes.
# Load the esreg package
# Simulate data from DGP-(2) in the paper
x <- rchisq(1000, df = 1)
y <- -x + (1 + 0.5 * x) * rnorm(1000)
# Estimate the model and the covariance
fit <- esreg(y ~ x, alpha = 0.025)
cov <- vcov(object = fit, sparsity = "nid", cond_var = "scl_sp")
A Joint Quantile and Expected Shortfall Regression Framework | {"url":"https://mirror.ibcp.fr/pub/CRAN/web/packages/esreg/readme/README.html","timestamp":"2024-11-14T18:04:14Z","content_type":"application/xhtml+xml","content_length":"2685","record_id":"<urn:uuid:6bc404f3-fd46-4684-9cad-fb82eb562408>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00043.warc.gz"} |
IE8 and mathjax
Hi -
The MathJax version pulled down with the webwork svn is MathJax 2.0. You can be sure that you are running 2.0 by loading a page which displays math using MathJax and then right-clicking on the math
and selecting "About MathJax." Or, on the server you can do the same with the test files in webwork2/htdocs/mathjax/test/.
Also, here's a post about the IE issue from the mathjax team: | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=2709&parent=5961","timestamp":"2024-11-11T17:09:56Z","content_type":"text/html","content_length":"66976","record_id":"<urn:uuid:333d84e7-a623-4f4d-a68f-157116bc6372>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00593.warc.gz"} |
Introduction to probabilistic programming, an emerging field at the intersection of programming languages, probability theory, and artificial intelligence. Shows how to use probabilistic programs to
implement and integrate models and inference algorithms from multiple paradigms. Modeling approaches include generative models, neural networks, symbolic programs, hierarchical Bayesian models,
causal Bayesian networks, graphics engines, and physics simulators. Inference approaches include Markov chain and sequential Monte Carlo methods, optimization, variational inference, and deep
learning. Hands-on projects teach students the fundamentals of probabilistic programming, as well as how to use probabilistic programming to solve problems in data analysis and computer vision, such
as forecasting time series, exploring and cleaning multivariate data, and real-time visual SLAM using depth cameras. Also shows how to write probabilistic programs that learn the structure and
parameters of probabilistic programs from data, and introduces new probabilistic programming-based AI architectures for expert systems that help people analyze and curate data and for common-sense
scene understanding.
Introduces probabilistic programming, an emerging field at the intersection of programming languages, probability theory, and artificial intelligence. Shows how to integrate modeling and inference
approaches from multiple eras of AI, by defining models and inference algorithms using executable code in new probabilistic programming languages. Also shows how to use technical ideas from
programming languages to formalize and generalize AI techniques. Example modeling formalisms include generative models, neural networks, symbolic programs, hierarchical Bayesian models, and causal
Bayesian networks. Example inference approaches include Monte Carlo, numerical optimization, and neural network techniques. Includes hands-on exercises in probabilistic programming fundamentals, plus
applications to computer vision and data analysis, using two new open-source probabilistic programming platforms recently prototyped at MIT. Graduate students must complete an original research
project for H level credit.
Introduces probabilistic programming, a computational formulation of probability theory. Covers how to formalize key ideas from probabilistic modeling and inference as probabilistic meta-programs,
and provides hands-on probabilistic programming experience with Venture, an open-source research platform. Emphasizes practical AI-based techniques for probabilistic data analysis while also
surveying applications to computer vision, robotics, and the exploration and modeling of complex databases in domains such as public health and neuroscience. Illustrates connections with other
approaches to engineering and reverse-engineering intelligence. | {"url":"http://probcomp.csail.mit.edu/classes/","timestamp":"2024-11-01T22:34:25Z","content_type":"text/html","content_length":"30935","record_id":"<urn:uuid:b7b4002b-e5ed-4b33-92e4-bb0e2500131d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00795.warc.gz"} |
What is the tens digit of the positive integer r?
Question Stats:
73% 27% (01:42) based on 2392 sessions
What is the tens digit of the positive integer r?
Let \(r=abc\), tens digit would be \(b\), so the question is \(b=?\)
(1) The tens digit of r/10 is 3 --> \(\frac{r}{10}=ab.c\) --> tens digit of this number is \(a\), so \(a=3\). No info about \(b\). Not sufficient.
(2) The hundreds digit of 10r is 6 --> \(10r=abc0\) --> hundreds digit of this number is \(b\), so \(b=6\). Sufficient.
Answer: B.
Hi Bunuel
Excellent explanation!! Big fan!!
ive understood statement 2 well how ever I tried to use a 2 digit for statement 1.
and this is how I went about it:-
let r = ab
so r/10 = 3 (or r/10 = 3.0)
in the same way ab/10 = 3 ;
which follows a.b = 3
or a.b = 3.0
That leaves us with no tens digit.
so is this the correct reason that a two digit number does not fit for this example?
Also you mentioned that using an integer or any number makes no difference. Please can you elucidate the same.
Thank You
The tens digit of r/10 is the hundreds digit of r. For example, if r=300, then the hundreds digit of r is 3 and the tens digit of r/10=30 is 3. So, from (1) we can tell that r is at least a 3-digit
number, while from the stem we could imply that r is at least a 2-digit number.
As for an integer part: r not necessary to be an integer, for example, consider r=abc.def, we still need the value of b:
(1) r/10=ab.cdef --> the tens digit of this number is a, so s=3. Not sufficient.
(2)10r=abcd.ef --> the hundreds digit of this number is b, so b=6. Sufficient.
Hope it's clear. | {"url":"https://gmatclub.com/forum/what-is-the-tens-digit-of-the-positive-integer-r-101570.html","timestamp":"2024-11-13T16:12:39Z","content_type":"application/xhtml+xml","content_length":"998300","record_id":"<urn:uuid:94c9fad0-f889-4ab4-bc4b-2155a64d7d66>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00467.warc.gz"} |
For many people the mere mention of fractions elicits a wince, and while these lovely math constructs played a notable role in our early years of math, they are reasonably simple entities. While most
humans revisit fractions over many years and often still fail to grasp the concept, for a computer, the elementary operations with fractions (addition, subtraction, multiplication, division, and the
necessary greatest common factor (GCF) and lowest common multiple (LCM)) can be coded in just a few lines.
While by no means a perfect class (quite possibly a rather poorly coded class), the following provides some basic functions necessary for working with fractions.
n = $num;
$this->d = $den;
public function gcf($n1, $n2){
if ($n2>$n1){
$tmp = $n1;
$rem = $n1 % $n2;
$n1 = $n2;
$n2 = $rem;
return $n1;
public function lcm($n1, $n2){
return $n1*($n2/frac::gcf($n1,$n2));
public function reduce (){
$g = $this->gcf($this->n,$this->d);
$this->n /= $g;
$this->d /= $g;
public function multiply (frac $n1, frac $n2){
$f = new frac($n1->n*$n2->n,$n1->d*$n2->d);
return $f;
public function divide (frac $n1, frac $n2){
return frac::multiply($n1, new frac($n2->d,$n2->n));
public function add (frac $n1, frac $n2){
$g = frac::lcm($n1->d,$n2->d);
$f= new frac($n1->n*($g/$n1->d)+$n2->n*($g/$n2->d),$g);
return $f;
public function subtract (frac $n1, frac $n2){
return frac::add($n1, new frac(-1*$n2->n,$n2->d));
public function display(){
return $this->n . "/" . $this->d;
Examples of use:
1/3 + 1/2:
1/8 * 2/5
The gcf function uses Euclid’s algorithm, and the lcm function (used to find the common denominator) calls the gcf function.
Given the significant disparity between the ease with which a computer can ‘learn’ fractions, and the difficulty encountered by most students, perhaps it is time to consider teaching fractions as a
series of concrete steps – an algorithm – instead of the current method. (Granted, most current methods do provide a method for arriving at an answer, but especially for the determination of the
lowest common denominator (or reducing fractions), a procedural methodology (e.g. prime factoring, Euclid’s method, etc) is rarely given.) | {"url":"https://www.thatsgeeky.com/2010/11/fractions/","timestamp":"2024-11-04T02:13:00Z","content_type":"text/html","content_length":"28694","record_id":"<urn:uuid:b2f2e7c5-a547-45c8-99d8-830be12c26e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00103.warc.gz"} |
A flexible and efficient framework for data-driven stochastic disease spread simulations
The package provides an efficient and very flexible framework to conduct data-driven epidemiological modeling in realistic large scale disease spread simulations. The framework integrates infection
dynamics in subpopulations as continuous-time Markov chains using the Gillespie stochastic simulation algorithm and incorporates available data such as births, deaths and movements as scheduled
events at predefined time-points. Using C code for the numerical solvers and ‘OpenMP’ (if available) to divide work over multiple processors ensures high performance when simulating a sample outcome.
One of our design goals was to make the package extendable and enable usage of the numerical solvers from other R extension packages in order to facilitate complex epidemiological research. The
package contains template models and can be extended with user-defined models.
Getting started
You can use one of the predefined compartment models in SimInf, for example, SEIR. But you can also define a custom model ‘on the fly’ using the model parser method mparse. The method takes a
character vector of transitions in the form of X -> propensity -> Y and automatically generates the C and R code for the model. The left hand side of the first arrow (->) is the initial state, the
right hand side of the last arrow (->) is the final state, and the propensity is written between the two arrows. The flexibility of the mparse approach allows for quick prototyping of new models or
features. To illustrate the mparse functionality, let us consider the SIR model in a closed population i.e., no births or deaths. Let beta denote the transmission rate of spread between a susceptible
individual and an infectious individual and gamma the recovery rate from infection (gamma = 1 / average duration of infection). It is also possible to define variables which can then be used in
calculations of propensities or in calculations of other variables. A variable is defined by the operator <-. Using a variable for the size of the population, the SIR model can be described as:
transitions <- c("S -> beta*S*I/N -> I",
"I -> gamma*I -> R",
"N <- S+I+R")
compartments <- c("S", "I", "R")
The transitions and compartments variables together with the constants beta and gamma can now be used to generate a model with mparse. The model also needs to be initialised with the initial
condition u0 and tspan, a vector of time points where the state of the system is to be returned. Let us create a model that consists of 1000 replicates of a population, denoted a node in SimInf, that
each starts with 99 susceptibles, 5 infected and 0 recovered individuals.
n <- 1000
u0 <- data.frame(S = rep(99, n), I = rep(5, n), R = rep(0, n))
model <- mparse(transitions = transitions,
compartments = compartments,
gdata = c(beta = 0.16, gamma = 0.077),
u0 = u0,
tspan = 1:150)
To generate data from the model and then print some basic information about the outcome, run the following commands:
#> Model: SimInf_model
#> Number of nodes: 1000
#> Number of transitions: 2
#> Number of scheduled events: 0
#> Global data
#> -----------
#> Parameter Value
#> beta 0.160
#> gamma 0.077
#> Compartments
#> ------------
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> S 1.00 19.00 30.00 40.74 60.00 99.00
#> I 0.00 0.00 4.00 6.87 11.00 47.00
#> R 0.00 28.00 67.00 56.39 83.00 103.00
There are several functions in SimInf to facilitate analysis and post-processing of simulated data, for example, trajectory, prevalence and plot. The default plot will display the median count in
each compartment across nodes as a colored line together with the inter-quartile range using the same color, but with transparency.
Most modeling and simulation studies require custom data analysis once the simulation data has been generated. To support this, SimInf provides the trajectory method to obtain a data.frame with the
number of individuals in each compartment at the time points specified in tspan. Below is the first 10 lines of the data.frame with simulated data.
#> node time S I R
#> 1 1 1 98 6 0
#> 2 2 1 98 6 0
#> 3 3 1 98 6 0
#> 4 4 1 99 5 0
#> 5 5 1 97 7 0
#> 6 6 1 98 5 1
#> 7 7 1 99 5 0
#> 8 8 1 99 5 0
#> 9 9 1 97 7 0
#> 10 10 1 97 6 1
Finally, let us use the prevalence method to explore the proportion of infected individuals across all nodes. It takes a model object and a formula specification, where the left hand side of the
formula specifies the compartments representing cases i.e., have an attribute or a disease and the right hand side of the formula specifies the compartments at risk. Below is the first 10 lines of
the data.frame.
#> time prevalence
#> 1 1 0.05196154
#> 2 2 0.05605769
#> 3 3 0.06059615
#> 4 4 0.06516346
#> 5 5 0.06977885
#> 6 6 0.07390385
#> 7 7 0.07856731
#> 8 8 0.08311538
#> 9 9 0.08794231
#> 10 10 0.09321154
Learn more
See the vignette to learn more about special features that the SimInf R package provides, for example, how to:
• use continuous state variables
• use the SimInf framework from another R package
• incorporate available data such as births, deaths and movements as scheduled events at predefined time-points.
You can install the released version of SimInf from CRAN
or use the remotes package to install the development version from GitHub
We refer to section 3.1 in the vignette for detailed installation instructions.
In alphabetical order: Pavol Bauer , Robin Eriksson , Stefan Engblom , and Stefan Widgren (Maintainer)
Any suggestions, bug reports, forks and pull requests are appreciated. Get in touch.
SimInf is research software. To cite SimInf in publications, please use:
• Widgren S, Bauer P, Eriksson R, Engblom S (2019) SimInf: An R Package for Data-Driven Stochastic Disease Spread Simulations. Journal of Statistical Software, 91(12), 1–42. doi: 10.18637/
• Bauer P, Engblom S, Widgren S (2016) Fast event-based epidemiological simulations on national scales. International Journal of High Performance Computing Applications, 30(4), 438–453. doi:
This software has been made possible by support from the Swedish Research Council within the UPMARC Linnaeus center of Excellence (Pavol Bauer, Robin Eriksson, and Stefan Engblom), the Swedish
Research Council Formas (Stefan Engblom and Stefan Widgren), the Swedish Board of Agriculture (Stefan Widgren), the Swedish strategic research program eSSENCE (Stefan Widgren), and in the framework
of the Full Force project, supported by funding from the European Union’s Horizon 2020 Research and Innovation programme under grant agreement No 773830: One Health European Joint Programme (Stefan
The SimInf package uses semantic versioning.
The SimInf package is licensed under the GPLv3. | {"url":"https://cran.rstudio.org/web/packages/SimInf/readme/README.html","timestamp":"2024-11-03T20:30:13Z","content_type":"application/xhtml+xml","content_length":"18738","record_id":"<urn:uuid:6374d40c-85f0-4c47-b272-bee87f7214ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00047.warc.gz"} |
3D motion detection using neural networks - docshare.tips
3D motion detection using neural networks
In video surveillance, video signals from multiple remote locations are displayed on several TV screens which are typically placed together in a control room. In the so-called third generation
surveillance systems (3GSS), all the parts of the surveillance systems will be digital and consequently, digital video will be transmitted and processed. Additionally, in 3GSS some 'intelligence' has
to be introduced to detect relevant events in the video signals in an automatic way. This allows filtering of the irrelevant time segments of the video sequences and the displaying on the TV screen
only those segments that require the attention of the surveillance operator. Motion detection is a basic operation in the selection of significant segments of the video signals. Once motion has been
detected, other features can be considered to decide whether a video signal has to be presented to the surveillance operator. If the motion detection is performed after the transmission of the video
signals from the cameras to the control room, then all the bit streams have to be previously decompressed; this can be a very demanding operation, especially if there are many cameras in the
surveillance system. For this reason, it is interesting to consider the use of motion detection algorithms operating in the compressed (transform) domain. In this thesis we present a motion detection
algorithm in the compressed domain with a low computational cost. In the following Section, we assume that video is compressed by using motion JPEG (MJPEG), i.e. each frame is individually JPEG
compressed. Motion detection from a moving observer has been a very important technique for computer vision applications. Especially in recent years, for autonomous driving systems and driver
supporting systems, vision-based navigation method has received more and more attention worldwide.
One of its most important tasks is to detect the moving obstacles like cars, bicycles or even pedestrians while the vehicle itself is running in a high speed. Methods of image differencing with the
clear background or between adjacent frames are well used for the motion detection. But when the observer is also moving, which leads to the result of continuously changing background scene in the
perspective projection image, it becomes more difficult to detect the real moving objects by differencing methods. To deal with this problem, many approaches have been proposed in recent years.
Previous work in this area has been mainly in two categories: 1) Using the difference of optical flow vectors between background and the moving objects, 2) calibrating the background displacement by
using camera’s 3D motion analysis result. Calculate the optical flow and estimate the flow vector’s reliability between adjacent frames. The major flow vector, which represents the motion of
background, can be used to classify and extract the flow vectors of the real moving objects. However, by reason of its huge calculation cost and its difficulty for determining the accurate flow
vectors, it is still unavailable for real applications. To analysis the camera’s 3D motion and calibrate the background is another main method for moving objects detection. For on-board camera’s
motion analysis, many motion-detecting algorithms have been proposed which always depend on the previous recognition results like road lane-marks and horizon disappointing. These methods show some
good performance in accuracy and efficiency because of their detailed analysis of road structure and measured vehicle locomotion, which is, however, computationally expensive and over-depended upon
road features like lane-marks, and therefore lead to unsatisfied result when lane mark is covered by other vehicles or not exist at all. Compare with these previous works, a new method of moving
objects detection from an on-board camera is presented in this paper. To deal with the background-change problem, our method uses camera’s 3D motion analysis results to calibrate the background
scene. With pure points matching and the introduction of camera’s Focus of Expansion (FOE), our method is able to determine camera’s rotation and translation parameters theoretically by using only
three pairs of matching points between adjacent frames, which make it faster and more efficient for real-time applications.
A neural network, also known as a parallel distributed processing network, is a computing paradigm that is loosely modeled after cortical structures of the brain. It consists of interconnected
processing elements called nodes or neurons that work together to produce an output function. The output of a neural network relies on the cooperation of the individual neurons within the network to
operate. Processing of information by neural networks is characteristically done in parallel rather than in series (or sequentially) as in earlier binary computers or Von Neumann machines. Since it
relies on its member neurons collectively to perform its function, a unique property of a neural network is that it can still perform its overall function even if some of the neurons are not
functioning. In other words it is robust to tolerate error or failure. All neural networks take numeric input and produce numeric output. The transfer function of a unit is typically chosen so that
it can accept input in any range, and produces output in a strictly limited range (it has a squashing effect). An artificial neural network (ANN), also called a simulated neural network (SNN) or
commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to
computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network. There are different topologies of neural
networks that may be employed for time series modeling. In our investigation we used radial basis function networks which have shown considerably better scaling properties, when increasing the number
of hidden units, than networks with sigmoid activation function. RBF networks were introduced into the neural network literature by Broom head/Lowe and Poggio/Girosi in the late 1980s. The RBF
network model is motivated by the locally tuned response observed in biologic neurons, e.g. in the visual or in the auditory system. RBFs have been studied in multivariate approximation theory,
particularly in the field of function interpolation. The RBF neural network model is an alternative to multilayer perceptron which is perhaps the
most often used neural network architecture. A radial basis function network (RBF), therefore, has a hidden layer of radial units, each actually modeling a Gaussian response surface. Since these
functions are nonlinear, it is not actually necessary to have more than one hidden layer to model any shape of function: sufficient radial units will always be enough to model any function. In
surveillance system estimation of motion is of great importance, which enables the various types of operations to be performed on the detected object. When using motion estimation, an assumption is
made that the objects in the scene have only translational motion. This assumption holds as long as there is no camera pan, zoom, changes in luminance, or rotational motion (quite an assumption!).
After the process of estimation, the detected motion has to be extracted. With the obtained boundary, two objects (with background) can then be extracted from two image frames (both current image
frame and previous image frame). Extracting the moving object from its background can be done by the edge enhancement network and the background remover. In algorithm level, complexity, regularity
and precision are main factors that directly affect the power consumed in extracting an algorithm for motion estimation. Concurrency and modularity are the requirements on algorithms that are
intended to execute on low power architecture. This project aims to reduce the power consumption of motion estimation at algorithm level and architectural level by using neural network concept.
The goals for this thesis have been the following. One goal has been to compile an introduction to the motion detection
algorithms. There exist a number of studies but complete reference on real time motion detection is not as common .we have collected materials from journals, papers and conferences and proposed
approach that can be best to implement a real time motion detection. Another goal has been to search for algorithms that can be used to implement the RBF neural network. A third goal is to evaluate
their performance with regard to motion detected. These properties were chosen because they have the greatest impact on the implementation effort. A final goal has been to design and implement an
algorithm including object extraction. This should be done in high level language or matlab. The source code should be easy to understand so that it can serve as a reference on the standard for
designers that need to implement real time motion detection.
Neural network theory is sometimes used to refer to a branch of computational science that uses neural networks as models to simulate or analyze complex phenomena and/or study the principles of
operation of neural networks analytically. It addresses problems similar to artificial intelligence (AI) except that AI uses traditional computational algorithms to solve problems whereas neural
networks use 'networks of agents' (software or hardware entities linked together) as the computational architecture to solve problems. Neural networks are trainable systems that can "learn" to solve
complex problems from a set of exemplars and generalize the "acquired knowledge" to solve unforeseen problems as in stock market and environmental prediction. i.e., they are self-adaptive systems.
Traditionally, the term neural network has been used to refer to a network of biological neurons. In modern usage, the term is often used to refer to artificial neural networks, which are composed of
artificial neurons or nodes. Thus the term 'Neural Network' has two distinct connotations: 1. Biological neural networks are made up of real biological neurons that are connected or
functionally-related in the peripheral nervous system or the central nervous system. In the field of neuroscience, they are often identified as groups of neurons that perform a specific physiological
function in laboratory analysis.
2. Artificial neural networks are made up of interconnecting artificial neurons (usually simplified neurons) designed to model (or mimic) some properties of biological neural networks. Artificial
neural networks can be used to model the modes of operation of biological neural networks, whereas cognitive models are theoretical models that mimic cognitive brain functions without
necessarily using neural networks while artificial intelligence are well-crafted algorithms that solve specific intelligent problems without using neural network as the computational architecture.
2.1 The brain, neural networks and computers
While it is accepted by most scientists that the brain is a type of computer, it is a computer with a vastly different architecture to the computers that most of us are familiar with. The brain is
massively parallel, even more so than advanced multiprocessor computers. This means that simulating the behavior of a brain on traditional computer hardware is necessarily slow and inefficient.
Neural networks, as used in artificial intelligence, have traditionally been viewed as simplified models of neural processing in the brain, even though the relation between this model and brain
biological architecture is very much debated. To answer this question, David Marr has proposed various levels of analysis which provide us with a plausible answer for the role of neural networks in
the understanding of human cognitive functioning. The question of what is the degree of complexity and the properties that individual neural elements should have in order to reproduce something
resembling animal intelligence is a subject of current research in theoretical neuroscience. Historically computers evolved from von Neumann architecture, based on sequential processing and execution
of explicit instructions. On the other hand origins of neural networks are based on efforts to model information processing in biological systems, which may rely largely on parallel processing as
well as implicit instructions based on recognition of patterns of 'sensory' input from external sources. In other words, rather than sequential processing and execution, at their very heart, neural
networks are complex statistic processors.
2.2 Artificial Neural networks
An artificial neural network (ANN), also called a simulated neural network (SNN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or
computational model for information processing based on a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal
information that flows through the network. In more practical terms neural networks are non-linear statistical data modeling tools. They can be used to model complex relationships between inputs and
outputs or to find patterns in data.
2.3 Background
An artificial neural network involves a network of simple processing elements (neurons) which can exhibit complex global behavior, determined by the connections between the processing elements and
element parameters. One classical type of artificial neural network is the Hopfield net. In a neural network model, simple nodes (called variously "neurons", "neurodes", "PEs" ("processing elements")
or "units") are connected together to form a network of nodes — hence the term "neural network". While a neural network does not have to be adaptive per se, its practical use comes with algorithms
designed to alter the strength (weights) of the connections in the network to produce a desired signal flow. In modern software implementations of artificial neural networks the approach inspired by
biology has more or less been abandoned for a more practical approach based on statistics and signal processing. In some of these systems neural networks or parts of neural networks (such as
artificial neurons) are used as components in larger systems that combine both adaptive and non-adaptive elements.
2.4 Models
Neural network models in artificial intelligence are usually referred to as artificial neural networks (ANN); these are essentially simple mathematical models defining a function such functions. .
Each type of ANN model corresponds to a class of
Fig 1 Artificial Neural Network
Fig 2 A complex neural network
2.5 Employing artificial neural networks
Perhaps the greatest advantage of ANN is their ability to be used as an arbitrary function approximation mechanism which 'learns' from observed data. However, using them is not so straightforward and
a relatively good understanding of the underlying theory is essential.
Choice of model: This will depend on the data representation and the application. Overly complex models tend to lead to problems with learning. Learning algorithm: There are numerous tradeoffs
between learning algorithms. Almost any algorithm will work well with the correct hyper parameters for training on a particular fixed dataset. However selecting and tuning an algorithm for training
on unseen data requires a significant amount of experimentation.
Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.
With the correct implementation ANN can be used naturally in online learning and large dataset applications. Their simple implementation and the existence of mostly
local dependencies exhibited in the structure allows for fast, parallel implementations in hardware.
2.6 Types of neural networks 2.6.1 Feed forward neural network
The feed forward neural networks are the first and arguably simplest type of artificial neural networks devised. In this network, the information moves in only one direction, forward, from the input
nodes, through the hidden nodes (if any) and to the output nodes. There are no cycles or loops in the network.
2.6.2 Single-layer perceptron
The earliest kind of neural network is a single-layer perceptron network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. In this
way it can be considered the simplest kind of feed-forward network. The sum of the products of the weights and the inputs is calculated in each node, and if the value is above some threshold
(typically 0) the neuron fires and takes the activated value (typically 1); otherwise it takes the deactivated value (typically -1). Neurons with this kind of activation function are also called
McCulloch-Pitts neurons or threshold neurons. A perceptron can be created using any values for the activated and deactivated states as long as the threshold value lies between the two. Most
perceptrons have outputs of 1 or -1 with a threshold of 0 and there is some evidence that such networks can be trained more quickly than networks created from nodes with different activation and
deactivation values. Perceptrons can be trained by a simple learning algorithm that is usually called the delta rule. It calculates the errors between calculated output and sample output data, and
uses this to create an adjustment to the weights, thus implementing a form of gradient descent.
Single-unit perceptrons are only capable of learning linearly separable patterns; in 1969 in a famous monograph entitled Perceptrons Marvin Minsky and Seymour Papert showed that it was impossible for
a single-layer perceptron network to learn an XOR function.
2.6.3 Multilayer layer perceptron
This class of networks consists of multiple layers of computational units, usually interconnected in a feed-forward way. Each neuron in one layer has directed connections to the neurons of the
subsequent layer. In many applications the units of these networks apply a sigmoid function as an activation function. The universal approximation theorem for neural networks states that every
continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multi-layer perceptron with just one hidden layer. This
result holds only for restricted classes of activation functions, e.g. for the sigmoid functions. Multi-layer networks use a variety of learning techniques, the most popular being back-propagation.
Here the output values are compared with the correct answer to compute the value of some predefined error-function. By various techniques the error is then fed back through the network. Using this
information, the algorithm adjusts the weights of each connection in order to reduce the value of the error function by some small amount. After repeating this process for a sufficiently large number
of training cycles the network will usually converge to some state where the error of the calculations is small. In this case one says that the network has learned a certain target function. To
adjust weights properly one applies a general method for non-linear optimization that is called gradient descent. For this, the derivative of the error function with respect to the network weights is
calculated and the weights are then changed such that the error decreases (thus going downhill on the surface of the error function). For this reason back-propagation can only be applied on networks
with differentiable.
Fig 3 XOR perceptron A three layer Perceptron net capable of calculating XOR. The numbers within the perceptrons represent each perceptrons' explicit threshold. The numbers that annotate arrows
represent the weight of the inputs. This net assumes that if the threshold is not reached, zero (not -1) is output. Note that the bottom layer of inputs is not always considered a real perceptron
2.6.4 Radial basis function (RBF) network
Radial Basis Functions are powerful techniques for interpolation in multidimensional space. A RBF is a function which has built into a distance criterion with respect to a centre. Radial basis
functions have been applied in the area of neural networks where they may be used as a replacement for the sigmoidal hidden layer transfer characteristic in multi-layer perceptrons.
2.6.5 Echo State Network
The Echo State Network (ESN) is a recurrent neural network with a sparsely connected random hidden layer. The weights of output neurons are the only part of the network that can change and be
learned. ESN are good to (re)produce temporal patterns.
2.6.6 Stochastic neural networks
A stochastic neural network differs from a regular neural network in the fact that it introduces random variations into the network. In a probabilistic view of neural networks, such random variations
can be viewed as a form of statistical sampling, such as Monte Carlo sampling.
2.6.7 Neuro-fuzzy networks
A neuro-fuzzy network is a fuzzy inference system in the body of an artificial neural network. Depending on the FIS type, there are several layers that simulate the processes involved in a fuzzy
inference like fuzzification, inference, aggregation and defuzzification. Embedding an FIS in a general structure of an ANN has the benefit of using available ANN training methods to find the
parameters of a fuzzy system.
3.1 Radial Functions
Radial functions are a special class of function. Their characteristic feature is that their response decreases (or increases) monotonically with distance from a central point. The centre, the
distance scale, and the precise shape of the radial function are parameters of the model, all fixed if it is linear. A typical radial function is the Gaussian which, in the case of a scalar input, is
Its parameters are its centre c and its radius r. The figure illustrates a Gaussian RBF with centre c = 0 and radius r = 1. A Gaussian RBF monotonically decreases with distance from the centre. In
contrast, a multiquadric RBF which, in the case of scalar input, is
monotonically increases with distance from the centre. Gaussian-like RBFs are local (give a significant response only in a neighbourhood near the centre) and are more commonly used than
multiquadric-type RBFs which have a global response.
3.2 Radial Networks
A RBF is a function which has built into a distance criterion with respect to a centre. Radial basis functions have been applied in the area of neural networks where they may be used as a replacement
for the sigmoidal hidden layer transfer characteristic in multi-layer perceptrons. RBF networks have 2 layers of processing: In the first, input is mapped onto each RBF in the 'hidden' layer. The RBF
chosen is usually a Gaussian. In regression problems the output layer is then a linear combination of hidden layer values representing mean predicted output. The
interpretation of this output layer value is the same as a regression model in statistics. In classification problems the output layer is typically a sigmoid function of a linear combination of
hidden layer values, representing a posterior probability. Performance in both cases is often improved by shrinkage techniques, known as ridge regression in classical statistics and known to
correspond to a prior belief in small parameter values (and therefore smooth output functions) in a Bayesian framework. RBF networks have the advantage of not suffering from local minima in the same
way as multi-layer perceptrons. This is because the only parameters that are adjusted in the learning process are the linear mapping from hidden layer to output layer. Linearity ensures that the
error surface is quadratic and therefore has a single easily found minimum. In regression problems this can be found in one matrix operation. In classification problems the fixed non-linearity
introduced by the sigmoid output function is most efficiently dealt with using iterated reweighed least squares. RBF networks have the disadvantage of requiring good coverage of the input space by
radial basis functions. RBF centers are determined with reference to the distribution of the input data, but without reference to the prediction task. As a result, representational resources may be
wasted on areas of the input space that are irrelevant to the learning task. A common solution is to associate each data point with its own centre, although this can make the linear system to be
solved in the final layer rather large, and requires shrinkage techniques to avoid over fitting. Associating each input datum with an RBF leads naturally to kernel methods such as Support Vector
Machines and Gaussian Processes (the RBF is the kernel function). All three approaches use a non-linear kernel function to project the input data into a space where the learning problem can be solved
using a linear model. Like Gaussian Processes, and unlike SVMs, RBF networks are typically trained in a Maximum Likelihood framework by maximizing the probability (minimizing the error) of the data
under the model. SVMs take a different approach to avoiding over fitting by maximizing instead a margin. RBF networks are outperformed in most
classification applications by SVMs. In regression applications they can be competitive when the dimensionality of the input space is relatively small.
3.3 RBF Architecture
Artificial networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The output, , of the network is thus
Where N is the number of neurons in the hidden layer, ci is the center vector for neuron i, and ai are the weights of the linear output neuron. In the basic form all input are connected to each
hidden neuron. The norm is typically taken to be the Euclidean distance and the basis function is taken to be Gaussian.
The Gaussian basis functions are local in the sense that far away from the center of that neuron. RBF networks are universal approximates on a compact subset of function with arbitrary precision. The
weights ai, and the data.
Changing parameters of one neuron has only a small effect for input values that are
. This means
that a RBF network with enough hidden neurons can approximate any continuous
, and β are determined in a manner that optimizes the fit between
Fig 4 Architecture of a radial basis function network.
3.4 Training
In a RBF network there are three types of parameters that need to be chosen to adapt the network for a particular task: the center vectors ci, the output weights wi, and the RBF width parameters βi.
In the sequential training of the weights are updated at each time step as data streams in. For some tasks it makes sense to define an objective function and select the parameter values that minimize
its value. The most common objective function is the least squares function
We have explicitly included the dependence on the weights. Minimization of the least squares objective function by optimal choice of weights optimizes accuracy of fit.
3.5 Interpolation
RBF networks can be used to interpolate a function values of that function are known on finite number of when the points:
. Taking the known points xi to be the centers of the radial basis functions and evaluating the values of the basis functions at the same points gij = ρ( | | xj − xi | | ) the weights can be solved
from the equation
It can be shown that the interpolation matrix in the above equation is non-singular, if the point’s x_i are distinct, and thus the weights w can be solved by simple linear algebra:
3.6 Function approximation
If the purpose is not to perform strict interpolation but instead more general function approximation or classification the optimization is somewhat more complex because there is no obvious choice
for the centers. The training is typically done in two phases first fixing the width and centers and then the weights. This can be justified by considering the different nature of the non-linear
hidden neurons versus the linear output neuron.
3.7 Training the basis function centers
Basis function centers can be either randomly sampled among the input instances or found by clustering the samples and choosing the cluster means as the centers. The RBF widths are usually all fixed
to same value which is proportional to the maximum distance between the chosen centers.
3.8 Pseudoinverse solution for the linear weights
After the centers ci have been fixed, the weights that minimize the error at the output are computed with a linear pseudo inverse solution: , Where the entries of G are the values of the radial basis
functions evaluated at the points xi: gji = ρ (| | xj − ci | |). The existence of this linear solution means that unlike Multi-layer perceptron (MLP) networks the RBF networks have a unique local
minimum (when the centers are fixed).
3.9 Advantages/Disadvantages
• • • RBF trains faster than a MLP. Another advantage that is claimed is that the hidden layer is easier to interpret Although the RBF is quick to train, when training is finished and it is being
than the hidden layer in an MLP. used it is slower than a MLP, so where speed is a factor a MLP may be more appropriate.
Given a number of sequential video frames from the same source the goal is to detect the motion in the area observed by the source. When there is no motion all the sequential frames have to be
similar up to noise influence. In the case when motion is present there is some difference between the frames. For sure, each lowcost system has some aspect of noise influence. And in case of no
motion every two sequential frames will not be the identical. This is why the system must be smart enough to distinguish between noise and real motion. When the systems are calibrated and stable
enough the character of noise is that every pixel value may be slightly different from that in other frame. And in first approximation it is possible to define some noise per pixel threshold
parameter (adaptable for any given state) the meaning of which is how the pixel value (of the same oriented pixel in two sequential frames) might differ but actually the indicating value is the same
one. More precisely, if the pixel with coordinates (Xa,Ya) in frame A differs from the pixel with coordinates (Xb,Yb) in frame B less than on TPP (threshold per pixel) value so we will see them as
pixels with equal values. And we can write it by formulae: Pixel (Xa, Ya) equal to Pixel (Xb, Yb) I if {abs (Pixel (Xa,Ya)-Pixel(Xb,Yb)) < TPP } By adapting the TPP value to current system state we
can make the system to be noise-stable. By applying this threshold operation to every pixel pair we may assume that all the preprocessed pixel values are noise-free. The element of noise that is not
cancelled will be significantly small relative to other part. Ok, if so we have to post-process these values to detect the motion if any. As it was memorized above we have manipulate with different
pixels inside two sequential frames to make conclusion about the motion.
Firstly, to make the system sensitive enough we have not to fix the TPP value too big. It mean that keeping the sensitivity of the system high in any two frames there will be some little number (TPP
related) of different pixels. And in this case we have not to see them as noise. It is the first of the reasons to define a TPF (threshold per frame) value (adaptable for any given state) the meaning
of which is how many pixels at least, inside two sequential frames must differ in order to see them as motion. The second reason to deal with TPF is to filter (to drop) small motion. For instance, by
playing with TPF values we can neutralize motion of the small object (bugs etc.) by still detect the motion of people. And we can write the exact meaning of TPF by formulae: Let’s define NDPPP to be
the Number of Different Pre-Processed by TPP Pixels. So, There is a motion i.e. NDPPP > TPF. Both of TPP and TPF values are variable through the UI to get the optimal system sensitivity. Also the TPF
value has its visual equivalent and it is used as following. After the pixels pre-processing (by TPP) lets color all static (which do not include motion) pixels by lets say black color and all the
dynamic (which indicate the motion) pixels will be left with their original color. This will bring the effect of motion extraction. In the other words, all the static parts of the frames will be
black, and only the moving parts will be seen normally. The enabling/disabling of this effect is possible to control through the GUI. The Camera Manager provides routines for acquiring video frames
from CCD cameras. Any process can request a video frame from any video source. The system manages a request queue for each source and executes them cyclically.
This chapter presents the main software design and implementation issues. It starts by describing the general flow chart of the main program that was implemented in MATLAB. It then explains each
component of the flow chart with some details. Finally it shows how the graphical user interface GUI was designed.
5.1 Basic Architecture
Fig 5 A basic architecture of surveillance system The above block diagram shows the surveillance system which consists of a camera system which monitors the particular area, a video daughter card
which transmits the video signal to electrical signal, a network card which helps in connecting to a network and motion detection algorithm (SAD and Correlation) along with RBF network.
5.2 Main Program Flow Chart
The main task of the software was to read the still images recorded from the camera and then process these images to detect motions and take necessary actions accordingly. Figure 6 below shows the
general flow chart of the main program.
Setup & Initializations
What is Flag value?
Image Acquisition
Motion Detection Algorithm
Break & clear
Is image > threshold
Actions on Motion Detection
Data Record
Figure 6 Main Program Flow Diagram
It starts with general initialization of software parameters and objects setup. Then, once the program started, the flag value which indicates whether the stop button was pressed or not is checked.
If the stop button was not pressed it start reading the images then process them using one of the two algorithms as the operator was selected. If a motion is detected it starts a series of actions
and then it go back to read the next images, otherwise it goes directly to read the next images. Whenever the stop button is pressed the flag value will be set to zero and the program is stopped,
memory is cleared and necessary results are recorded. This terminates the program and returns the control for the operator to collect the results. The next sections explain each process of the flow
chart in figure 6 with some details.
5.2.1 Setup and Initializations
Launch GUI
Start button pressed
Read Threshold Value
Read Algorithm Type
Setup Serial Port
Setup Video Object
Figure 7 Setup and Initializations Process
Figure 7 show the flow chart for the setup and initialization process. This process includes the launch of the graphical user interface (GUI) where the type of motion detection algorithm is selected
and threshold value (the amount of sensitivity of the detection) is being initialized. Also, during this stage a setup process for both the serial port and the video object is done. This process
takes approximately 15 seconds to be completed,(depending on the specifications of the PC used) for the serial port it starts by selecting a communication port and reserving the memory addresses for
that port, then the PC connect to the device using the communication setting that was mentioned in the previous chapter. The video object is part of the image acquisition process but it should be
setup at the start of the program.
5.2.2 Image acquisition
Start Read First Frame Convert to Grayscale
Read Second Frame
Convert to Grayscale Stop
Figure 8 Image acquisitions Process
After setup stage the image acquisition starts as shown in figure 8 above. This process reads images from the PC camera and save them in a format suitable for the motion detection algorithm. There
were three possible options from which one is implemented. The first option was by using auto snapshots software that takes images automatically and save them on a hard disk as JPEG format, and then
another program reads these images in the same sequence as they were saved. It was found that the maximum speed that can be attained by this software is one frame per second and this limits the speed
of detection. Also, synchronization was required between both image processing and the auto snapshot software’s where next images need to be available on the hard disk before processing them. The
second option was to display live video on the screen and then start capturing the images from the screen. This is a faster option from the previous approach but again it faced the problem of
synchronization, when the computer monitor goes into a power saving mode where black images are produced all the time during the period of the black screen. The third option was by using the image
acquisition toolbox provided in MATLAB 6.5.1 or higher versions. The image acquisition toolbox is a collection of functions that extend the capability of MATLAB. The toolbox supports a wide range of
image acquisition operations, including acquiring images through many types of image acquisition devices, such as frame grabbers and USB PC cameras, also viewing a preview of the live video displayed
on monitor and reading the image data into the MATLAB workspace directly. For this project video input function was used to initialize a video object that connects to the PC camera directly. Then
preview function was used to display live video on the monitor. Get snapshot function was used to read images from the camera and place them in MATLAB workspace.
The later approach was implemented because it has many advantages over the others. It achieved the fastest capturing speed at a rate of five frames per seconds depending on algorithm complexity and
PC processor speed. Furthermore, the problem of synchronization was solved because both capturing and processing of images were done using the same software. All read images were converted it into a
two dimensional monochrome images. This is because equations in other algorithms in the system were designed with such image format.
5.2.3 Motion Detection Algorithm
A motion detection algorithm was applied on the previously read images. There were two approaches to implement motion detection algorithm. The first one was by using the two dimensional cross
correlation while the second one was by using the sum of absolute difference algorithm. These are explained in details in the next two sub sections.
5.3 Motion Detection Using Sum of Absolute Difference (SAD)
This algorithm is based on image differencing techniques. mathematically represented using the following equation:
D (t ) = 1 N
It is
∑ I (t
) − I (t j )
Where N is the number of pixels in the image used as scaling factor,
I (t i ) is the image I at time i ,
I (t j )
is the image I at time j and
D (t ) is the normalized sum of absolute difference for that time.
In an ideal case when there is no motion
I (t i ) = I (t j )
and D(t ) = 0 . However noise is always presented in images and a better model of the images in the absence of motion will be
I (t i ) = I (t j ) + n( p )
Where n( p ) is a noise signal. The value D (t ) that represents the normalized sum of absolute difference can be used as a reference to be compared with a threshold value as shown in figure 9 below.
The figure also shows a test case that contains a large change in the scene being monitored by the camera this was done by moving the camera. During the time before the camera was moved the SAD value
was around 1.87 and when the camera was moved the SAD value was around 2.2. If the threshold for detection was fixed around the value less than 2.2 it will continuously detect motion after the camera
stop moving.
Figure 9 Direct Thresholds for SAD Values
This approach solve the need for continuously re-estimate the threshold value. Choosing a threshold of 1*10-3 will detect the times when only the camera is moved. This results into a robust motion
detection algorithm that can not be affected by illumination change and camera movements.
5.3.1 Actions on Motion Detection
Before explaining series of actions happen when motion is detected it is worth to mention that the values of variance that was calculated whether it was above or below the threshold will be stored in
an array, where it will be used later to produce a plot of frame number Vs. the variance value. This plot helps in comparing the variance values against the threshold to be able to choose the optimum
threshold value. Whenever the variance value is less than threshold the image will be dropped and only the variance value will be recorded. However when the variance value is greater than threshold
sequence of actions is being started as shown in figure 10 below. Start
Time Date Frame#
Update Log File Display Image
Convert Image to Frame
Trigger Serial Port
Figure 10 Actions on Motion Detection
As the above flow chart show a number of activities happen when motion is detected. First the serial port is being triggered by a pulse from the PC; this pulse is used to activate external circuits
connected to the PC. Also a log file is being created and then appended with information about the time and date of motion also the frame number in which motion occur is being recorded in the log
file. Another process is to display the image that was detected on the monitor. Finally the image that was detected in motion will be converted to a movie frame and will be added to the film
5.3.2 Break and clear Process
After motion detection algorithm applied on the images the program checks if the stop button on GUI was pressed. If it was pressed the flag value will be changed from one to zero and the program will
break and terminate the loop then it will return the control to the GUI. Next both serial port object and video object will be cleared. This process is considered as a cleaning stage where the
devises connected to the PC through those objects will be released and the memory space will be freed.
5.3.3 Data Record
Finally when the program is terminated a data collection process starts where variable and arrays that contain result of data on the memory will be stored on the hard disk. This approach was used to
separate the real time image processing from results processing. This has the advantage of calling back these data whenever it is required. The variables that are being stored from memory to the hard
disk are variance values and the movie structure that contain the entire frames with motion. At this point the control will be returned to the GUI where the operator can callback the results that
where archived while the system was turned on. Next section will explain the design of the GUI highlighting each button results and callbacks.
Fig 11 Flow chart for SAD algoritham
Fig 12 Frame separation
Fig 13 Divide Quadrants
5.3.4 Graphical User Interface Design
The GUI was designed to facilitate interactive system operation. GUI can be used to setup the program, launch it, stop it and display results.
Clear all Previous Work
Variable Initialization & Setup
Launch program
Call Selected main Program
Terminate Program
View Results
Start Again
Figure 14 GUI flow Chart
During setup stage the operator is promoted to choose a motion detection algorithm and select degree of the detection sensitivity Whenever the start/stop toggle button is pressed the system will be
launched and the selected program will be called to perform the calculations until the start/stop button is pressed again which will terminate the calculation and return control to GUI. Results can
be viewed as a log file, movie and plot of frame number vs. variance value. Figure 14 illustrate a flow chart of the steps performed using the GUI.
5.4 Motion detection using Correlation Network
A correlation neural network (CNN) which accounts for velocity sensitive responses of neurons is suitable for analog circuit implementation of motiondetection systems and has been successfully
implemented on CMOS. The CNN utilizes local motion detectors to correlate signals sampled at one location in the image with those sampled after a delay at adjacent locations; however, an
edgedetection process is required in practical motion detection systems with the CNNs. The term correlation can also mean the cross-correlation of two functions or electron correlation in molecular
systems. In probability theory and statistics, correlation, also called correlation coefficient, indicates the strength and direction of a linear relationship between two random variables. In general
statistical usage, correlation or co-relation refers to the departure of two variables from independence, although correlation does not imply causation. In this broad sense there are several
coefficients, measuring the degree of correlation, adapted to the nature of data. A number of different coefficients are used for different situations. The best known is the Pearson product-moment
correlation coefficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations.
5.4.1 Mathematical properties
The correlation ρX, Y between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:
Where E is the expected value of the variable and cov means covariance. Since μX = E(X), σX2 = E(X2) − E2(X) and likewise for Y, we may also write
The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in
absolute value. The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the
degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are independent then the
correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is
uniformly distributed on the interval from −1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are dependent, but their correlation is zero; they are uncorrelated. However, in
the special case when X and Y are jointly normal, independence is equivalent to uncorrelated ness. A correlation between two variables is diluted in the presence of measurement error around estimates
of one or both variables, in which case disattenuation provides a more accurate coefficient.
5.4.2 Geometric Interpretation of correlation
The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables. This method only works with centered data, i.e., data
which have been shifted by the sample mean so as to have an average of zero. Some practitioners prefer an uncentered (non-Pearson-compliant) correlation coefficient. See the example below for a
comparison. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order)
are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18). By the
usual procedure for finding the angle between two vectors, the uncentered correlation coefficient is:
Note that the above data were deliberately chosen to be perfectly correlated: y = 0.10 + 0.01 x. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting x by E
(x) = 3.8 and y by E(y) = 0.138) yields x = (-2.8, -1.8, -0.8, 1.2, 4.2) and y = (-0.028, -0.018, -0.008, 0.012, 0.042), from which
as expected.
5.4.3 Interpretation of the size of a correlation
Several authors have offered guidelines for the interpretation of a correlation coefficient. Cohen (1988), for example, has suggested the following interpretations for correlations in psychological
research, in the table in the bottom. As Cohen himself has observed, however, all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation
of a correlation coefficient depends on the context and purposes. A correlation of 0.9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very
high in the social sciences where there may be a greater contribution from complicating factors
Correlation Small Medium Large
Negative −0.29 to −0.10 −0.49 to −0.30 −1.00 to −0.50 Table 1
Positive 0.10 to 0.29 0.30 to 0.49 0.50 to 1.00
Fig 15 An unit network of two-dimensional CCN.
Decisi on
Fig 16 Flow chart for correlation
Many attempts have been made to extract data from video and film in a form suitable for use by animators and modelers. Such an approach is attractive, since motions and movements for people and
animals may be obtained in this way that would be difficult using mechanical or magnetic motion capture systems. Visual extraction is also appealing since it is non-intrusive and has the potential to
capture, from film, the motion and characteristics of people or animals long dead or extinct. Almost all attempts to perform visual extraction have been based around bespoke computer vision
applications which are difficult for non-experts to use or adapt to their own needs. This paper presents a generic approach to extracting data from video. Whilst our approach allows low-level
information to be extracted we show that higher-level functionality is available also. This functionality can be utilized in a manner that requires little knowledge of the underlying techniques and
principles. Our approach is to approximate an image using principal component analysis, and then to train a multi-layer perceptron to predict the feature required by the user. This requires the user
to hand-label the features of interest in some of the frames of the image sequence. One of the aims of this work is to keep to a minimum the number of frames that need to need labeled by the user.
The trained multi-layer perceptron is then used to predict features for images that have never been labeled by the user. Other attempts to extract useful information from video sequences include the
use of edge-detection and contour or edge tracking, template matching and template tracking. All such systems work well in some circumstances, but fail or require adaptation to meet the requirements
of new users. For instance, in the case of template tracking, the user needs to be aware of the kinds of features that can be tracked well in an image and also choose a suitable template size. This
is not a trivial task for non-specialists.
6.1 Method
The main steps in extraction using our system are detailed below: The user selects the sequence (or set) of images for which they wish data to be extracted from. This may well comprise of several
shorter clips taken from different parts of a film. These images have some pre-processing performed on them (principal components analysis) to reduce each image to a small set of numbers. The user
decides what feature(s) they wish to extract and labels this feature by hand in a fraction of the images chosen at random. The labeling process may involve clicking on a point to be tracked, labeling
a distance or ratio of distances, measuring an angle, making a binary decision (yes/no, near/far etc.) or classifying the feature of interest into one of several classes. Once this ground-truth data
is available, a neural network is trained to predict the feature values in images that have not been labeled by the user.
6.2 Feature Extraction
Principal components analysis (also known as eigenvector analysis) has been used extensively in computer vision for image reconstruction, pattern matching and classification. Given the i th image in
a sequence of images, each of which consists of M pixels, we form the vector x i by concatenating the pixels of the image in raster scan order and removing the mean image of the sequence. The matrix
X is created using the x i's as column vectors. Traditionally, the principal modes, qi, are extracted by computing X T i=iqi (1) Xq Where i's are the Eigen values.
a measure of the amount of variance each of the eigen vectors accounts for. Unfortunately, the matrix XXT is typically too large to manipulate since it is of size M by M. Such computation is wasteful
anyway since only N princi pal modes are meaningful, where N is the number of example images. In all our work N M. Therefore we compute: X X =i ui(2) T u and we can obtain the qi's that we actually
require using: q =X i i u (3)
In practice only the first P modes are used, P30 N. The principal mode extracted from a short film clip is shown in Figure 1 and is used later to help an animator to construct a cartoon version of
the clip. It is tempting to think that such modes could be used directly to predict, say, the rotation of the man's shoulders. However, the second mode also encodes information about shoulder
movement and it is only by combining information from many modes that rotation can be reliably predicted.
[1] ' Special issue on third generation surveillance systems', froc. IEEE, 2001, 89, JAIN, R., KASTURI, R., and SCHUNCK, B.G. This paper gives the detailks about the surveillance systems [2] 'Machine
vision' (McGraw-Hill Inc., 1995) PONS, J., PRADES-NEBOT, J., ALBIOL, A,, and MOLINA, J.his paper provides the details about intelligence. [3] 'Motion video sensor in the compressed domain'. SCS
Euromedia Conf., Valencia, Spain, 2001, This paper provides the details about compressed domain. [4] Y. Song, A perceptual approach to human motion detection and labeling. PhD thesis, California
Institute of Technology, 2003. This paper provides the details about human motion detection [5] N. Howe, M. Leventon, and W. Freeman, “Bayesian reconstruction of 3D human motion from single-camera
video,” Tech. Rep. TR-99-37, Mitsubishi Electric Research Lab, 1999 This paper provides the details about 3d human detection. [6] L. Goncalves, E. D. Bernardo, E. Ursella, and P. Perona, “Monocular
tracking of the human arm in 3D,” in Proc. 5th Int. Conf. Computer Vision, (Cambridge, Mass), pp. 764– 770, 1995.This paper provides the details about 3d human detection. [7] S. Wachter and H.-H.
Nagel, “Tracking persons in monocular image sequences,” Computer Vision and Image Understanding, vol. 74, pp. 174–192, 1999. This paper provides details about motion detection in image sequences.
algorithms in artificient
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In a previous article, we introduced the Jacobi and Gauss-Seidel methods, which are iterative methods for solving linear systems of equation. Specifically, we noted that the Gauss-Seidel method will
in general converge towards a solution much quicker than the Jacobi method. The main issue with the Gauss-Seidel method is that it is non-trivial to make into a parallel algorithm. However, it turns
out that for a certain class of matrices, it is pretty simple to implement a parallel Gauss-Seidel method. | {"url":"https://erkaman.github.io/articles.html","timestamp":"2024-11-10T17:35:52Z","content_type":"text/html","content_length":"17192","record_id":"<urn:uuid:45225435-0900-48ec-a75e-1ada0eb95d76>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00567.warc.gz"} |
8.2 - The 2x2 Table: Test of 2 Independent Proportions | STAT 500
Say we have a study of two categorical variables each with only two levels. One of the response levels is considered the "success" response and the other the "failure" response. A general 2 × 2 table
of the observed counts would be as follows:
Success Failure Total
Group 1 A B A + B
Group 2 C D C + D
The observed counts in this table represent the following proportions:
Success Failure Total
Group 1 \(\hat{p}_1=\frac{A}{A+B}\) \(1-\hat{p}_1\) A + B
Group 2 \(\hat{p}_2=\frac{C}{C+D}\) \(1-\hat{p}_2\) C + D
Recall from our Z-test of two proportions that our null hypothesis is that the two population proportions, \(p_1\) and \(p_2\), were assumed equal while the two-sided alternative hypothesis was that
they were not equal.
This null hypothesis would be analogous to the two groups being independent.
Also, if the two success proportions are equal, then the two failure proportions would also be equal. Note as well that with our Z-test the conditions were that the number of successes and failures
for each group was at least 5. That equates to the Chi-square conditions that all expected cells in a 2 × 2 table be at least 5. (Remember at least 80% of all cells need an expected count of at least
5. With 80% of 4 equal to 3.2 this means all four cells must satisfy the condition).
When we run a Chi-square test of independence on a 2 × 2 table, the resulting Chi-square test statistic would be equal to the square of the Z-test statistic (i.e., \((Z^*)^2\)) from the Z-test of two
independent proportions.
Political Affiliation and Opinion Section
Consider the following example where we form a 2 × 2 for the Political Party and Opinion by only considering the Favor and Opposed responses:
favor oppose Total
democrat 138 64 202
republican 64 84 148
Total 202 148 350
The Chi-square test produces a test statistic of 22.00 with a p-value of 0.00
The Z-test comparing the two sample proportions of \(\hat{p}_d=\frac{138}{202}=0.683\) minus \(\hat{p}_r=\frac{64}{148}=0.432\) results in a Z-test statistic of \(4.69\) with p-value of \(0.000\).
If we square the Z-test statistic, we get \(4.69^2 = 21.99\) or \(22.00\) with rounding error.
The condiments and gender data were condensed to consider gender and either mustard or ketchup. The manager wants to know if the proportion of males that prefer ketchup is the same as the proportion
of females that prefer ketchup. Test the hypothesis two ways (1) using the Chi-square test and (2) using the z-test for independence with a significance level of 10%. Show how the two test statistics
are related and compare the p-values.
Ketchup Mustard Total
Gender Male 15 23 38
Female 25 19 44
Total 40 42 82
Z-test for two proportions
The hypotheses are:
\(H_0\colon p_1-p_2=0\)
\(H_a\colon p_1-p_2\ne 0\)
Let males be denoted as sample one and females as sample two. Using the table, we have:
\(n_1=38\) and \(\hat{p}_1=\frac{15}{38}=0.395\)
\(n_2=44\) and \(\hat{p}_2=\frac{25}{44}=0.568\)
The conditions are satisfied for this test (verify for extra practice).
To calculate the test statistic, we need:
The test statistic is:
The p-value is \(2P(Z<-1.567)=0.1172\).
The p-value is greater than our significance level. Therefore, there is not enough evidence in the data to suggest that the proportion of males that prefer ketchup is different than the proportion of
females that prefer ketchup.
Chi-square Test for independence
The expected count table is:
Ketchup Mustard Total
Gender Male 15 (18.537) 23 (19.463) 38
Female 25 (21.463) 19 (22.537) 44
Total 40 42 82
There are no expected counts less than 5. The test statistic is:
\(\chi^{2*}=\dfrac{(15-18.537)^2}{18.537}+\dfrac{(23-19.463)^2}{19.463}+\dfrac{(25-21.463)^2}{21.463}+\dfrac{(19-22.537)^2}{22.537}=2.46 \)
With 1 degree of freedom, the p-value is 0.1168. The p-value is greater than our significance value. Therefore, there is not enough evidence to suggest that gender and condiments (ketchup or mustard)
are related.
The p-values would be the same without rounding errors (0.1172 vs 0.1168). The z-statistic is -1.567. The square of this value is 2.455 which is what we have (rounded) for the chi-square statistic.
The conclusions are the same. | {"url":"https://online.stat.psu.edu/stat500/lesson/8/8.2","timestamp":"2024-11-11T20:00:07Z","content_type":"text/html","content_length":"108558","record_id":"<urn:uuid:c1acb108-0eaa-43d2-aa60-dfd9b9b561aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00276.warc.gz"} |
Back to Papers Home Back to Papers of School of Astronomy
Paper IPM / Astronomy / 14113
School of Astronomy
Title: VERTICAL STRUCTURE OF ADVECTION-DOMINATED ACCRETION FLOWS
Author(s): 1. F Z. Zeraatgari
2. Sh. Abbassi
Status: Published
Journal: Astrophysical Journal
Year: 2015
Pages: 7
Supported by: ipm IPM
We solve the set of hydrodynamic equations for optically thin advection-dominated accretion flows by assuming a radially self-similar spherical coordinate system (r, q, f). The disk is considered to
be in steady state and axisymmetric. We define the boundary conditions at the pole and the equator of the disk and, to avoid singularity at the rotation axis, the disk is taken to be symmetric with
respect to this axis. Moreover, only the trf component of the viscous stress tensor is assumed, and we have set vq = 0. The main purpose of this study is to investigate the variation of dynamical
quantities of the flow in the vertical direction by finding an analytical solution. As a consequence, we found that the advection parameter, f adv, varies along the à ¸ direction and reaches its
maximum near the rotation axis. Our results also show that, in terms of the no-outflow solution, thermal equilibrium still exists and consequently advection cooling can balance viscous heating.
Download TeX format
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What is the formula for IMA of a lever?
(b) The ideal mechanical advantage equals the length of the effort arm divided by the length of the resistance arm of a lever. In general, the IMA = the resistance force, Fr, divided by the effort
force, Fe. IMA also equals the distance over which the effort is applied, de, divided by the distance the load travels, dr.
What is the equation for AMA?
Measure the output force of the system. The pulley generates 100N of force from the 40 N input force. Finally, calculate the actual mechanical advantage. Using the formula we can find the actual
mechanical advantage is 100 / 40 = 2.5.
How do you calculate levers?
In a class one lever the force of the effort (Fe) multiplied by the distance of the effort from the fulcrum (de) is equal to the force of the resistance (Fr) multiplied by the distance of the
resistance from the fulcrum (dr). The effort and the resistance are on opposite sides of the fulcrum.
What is the IMA of the system?
The mechanical advantage is a number that tells us how many times a simple machine multiplies the effort force. The ideal mechanical advantage, IMA, is the mechanical advantage of a perfect machine
with no loss of useful work caused by friction between moving parts.
How do you find the IMA of a pulley system?
A simple way to determine the ideal mechanical advantage to a pulley system is to count the number of lengths of rope between pulleys that support the load.
What is a class 3 lever examples?
In a Class Three Lever, the Force is between the Load and the Fulcrum. If the Force is closer to the Load, it would be easier to lift and a mechanical advantage. Examples are shovels, fishing rods,
human arms and legs, tweezers, and ice tongs. A fishing rod is an example of a Class Three Lever.
How do you calculate MA of a pulley?
The most accurate way of calculating the mechanical advantage of a belt driven pulley is to divide the inside diameter of the driven pulley wheel by the inside diameter of the drive pulley wheel. You
can also compare the number of rotations of the driven pulley wheel to one rotation of the drive pulley wheel.
How do you find the IMA of a block and tackle?
To calculate the mechanical advantage, we can either divide the weight of the object being lifted by the force required to lift it or we can divide the amount of rope we have to pull by the distance
the object moves.
How do you find the IMA of a wheel and axle?
Wheel and axle. The ideal mechanical advantage (IMA) of a wheel and axle is the ratio of the radii. If the effort is applied to the large radius, the mechanical advantage is R/r which will be more
than one; if the effort is applied to the small radius, the mechanical advantage is still R/r, but it will be less than 1.
Is AMA less than IMA?
In any real machine some of the effort is used to overcome friction. Thus, the ratio of the resistance force to the effort, called the actual mechanical advantage (AMA), is less than the IMA. | {"url":"https://www.blfilm.com/2021/03/03/what-is-the-formula-for-ima-of-a-lever/","timestamp":"2024-11-07T10:54:17Z","content_type":"text/html","content_length":"64666","record_id":"<urn:uuid:ff1321b8-9ab3-471e-9bae-8f5ceb9feada>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00435.warc.gz"} |
What is a Right Rectangular Prism ⭐ Meaning, Definition, Examples
Right Rectangular Prism – Definition With Examples
Updated on January 12, 2024
Geometry, one of the many branches of mathematics, can sometimes be challenging for young learners. At Brighterly, we are committed to turning complex ideas into simple, fun, and engaging learning
experiences. Today, let’s delve into one of the most common three-dimensional geometric figures – the Right Rectangular Prism.
A right rectangular prism is a three-dimensional solid object where all angles are right angles, and all faces are rectangles. These shapes are found abundantly in our surroundings. They make up the
basic structure of many everyday objects like books, crates, and even buildings! By understanding the properties of a right rectangular prism and knowing how to calculate its various attributes such
as surface area, volume, and diagonal, children can enhance their spatial understanding and mathematical skills.
What Is a Right Rectangular Prism?
A right rectangular prism, a commonly occurring shape in our daily lives and an integral part of our learning in geometry, refers to a solid, three-dimensional figure. The name indicates its
properties – it’s a prism because it has the same cross-section along its length, it’s rectangular because the cross-section is a rectangle, and it’s right because the angles between the bases and
the sides are right angles (90 degrees).
Objects such as shoe boxes, books, and bricks are practical examples of right rectangular prisms. If you think about a typical shoebox, it has six faces that meet at right angles. Each pair of
opposite faces are identical rectangles, making it an excellent real-world illustration of a right rectangular prism. By studying such objects, we can enhance our understanding of three-dimensional
shapes and their properties.
Properties of a Right Rectangular Prism:
A right rectangular prism possesses certain identifiable properties that make it unique among other geometric shapes. It has six faces, and all of them are rectangles. Out of these, the opposite
faces are congruent, meaning they have the same size and shape. It also has 12 edges, with all the corners meeting at a right angle.
Furthermore, if you draw a line segment connecting any two opposite corners inside the prism (known as a space diagonal), it will also intersect at a right angle with the prism’s base, hence the
name, right rectangular prism. These properties allow us to solve various problems and apply our knowledge to practical applications, from architecture to computer graphics.
Formulas of a Right Rectangular Prism
To understand the formulas related to a right rectangular prism, let’s denote the length of the prism as ‘l’, width as ‘w’, and height as ‘h’. The surface area, volume, and diagonal can then be
calculated using these dimensions. These formulas allow us to calculate the amount of material needed to construct a prism (surface area), the space it occupies (volume), and the longest distance
within the prism (space diagonal).
Surface Area of a Right Rectangular Prism
The surface area of a right rectangular prism can be calculated by adding the areas of all the faces. Since it has six rectangular faces, the surface area ‘A’ is given by the formula A = 2lw + 2lh +
2wh. This formula indicates that the surface area comprises twice the product of the length and width, twice the product of length and height, and twice the product of width and height.
Volume of Right Rectangular Prism
The volume of a right rectangular prism represents the amount of space it occupies. It can be calculated by multiplying the length, width, and height of the prism together. So, the volume ‘V’ is
given by the formula V = lwh.
Diagonal of a Right Rectangular Prism
The diagonal of a right rectangular prism, also referred to as the space diagonal, is the longest line that can be drawn within the prism. It connects two opposite corners of the prism passing
through its interior. The formula for the space diagonal ‘d’ is derived from the Pythagorean theorem and given by d = √(l² + w² + h²).
As we conclude our journey of exploration into the fascinating world of the right rectangular prism, we want to remind you that at Brighterly, we believe in making math fun, interactive, and easily
understandable for every child. The understanding of a right rectangular prism and its properties is not just crucial for math class but also plays a vital role in many real-world applications. From
constructing a building to packing a box, the concepts learned today will accompany you in many walks of life. Keep practicing, stay curious, and continue exploring the captivating world of
mathematics with Brighterly!
Frequently Asked Questions on Right Rectangular Prism
What is a right rectangular prism?
A right rectangular prism is a three-dimensional geometric shape in which all the six faces are rectangles and all the angles are right angles. This object is termed ‘right’ due to its right angles,
‘rectangular’ for its rectangular faces, and ‘prism’ as it has the same cross-sectional shape throughout its length.
What are the properties of a right rectangular prism?
A right rectangular prism has several unique properties. It has six faces, all of which are rectangles. The opposite faces are congruent to each other. There are 12 edges in total, and all the
corners meet at a right angle. Moreover, a line segment drawn from one corner to the opposite corner inside the prism forms a right angle with the base.
How do you calculate the surface area of a right rectangular prism?
The surface area of a right rectangular prism is calculated by adding the areas of all six faces. If the length, width, and height of the prism are ‘l’, ‘w’, and ‘h’ respectively, then the surface
area ‘A’ is given by the formula A = 2lw + 2lh + 2wh.
How do you find the volume of a right rectangular prism?
The volume of a right rectangular prism is the amount of space it occupies. It can be calculated by multiplying the length, width, and height of the prism. So, if ‘l’, ‘w’, and ‘h’ are the length,
width, and height, respectively, the volume ‘V’ is given by the formula V = lwh.
What is the formula for the diagonal of a right rectangular prism?
The diagonal of a right rectangular prism, also known as the space diagonal, is the longest line that can be drawn within the prism. It connects two opposite corners of the prism passing through its
interior. Using the Pythagorean theorem, if ‘l’, ‘w’, and ‘h’ are the length, width, and height, respectively, the space diagonal ‘d’ is given by the formula d = √(l² + w² + h²).
Information Sources:
1. “Computer Graphics” – Wikipedia
2. “Geometric Solids” – National Council of Teachers of Mathematics
3. “Volume Formulas” – Wolfram MathWorld
Poor Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Mediocre Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Needs Improvement
Start practicing math regularly to avoid your child`s math scores dropping to C or even D.
High Potential
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Tridiagonal systems, determinants & natural cubic splines
Tridiagonal systems, determinants, and natural cubic splines
Tridiagonal matrices
A tridiagonal matrix is a matrix that has nonzero entries only on the main diagonal and on the adjacent off-diagonals. This special structure comes up frequently in applications. For example, the
finite difference numerical solution to the heat equation leads to a tridiagonal system. Another application, the one we’ll look at in detail here, is natural cubic splines. And we’ll mention an
interesting result with Fibonacci numbers in passing.
Because these matrices have a special structure, the corresponding linear systems are quick and easy to solve. Also, we can find a simple recurrence relation for their determinants.
Let’s label the component of a tridiagonal matrix as below
where every component not shown is implicitly zero. We can expand the determinant of the matrix above using minors along the last row. This gives a recursive expression for the determinant
with initial conditions d[0] = 1 and d[−1] = 0.
Note that if all the a‘s and b‘s are 1 and all the c‘s are −1, then you get the recurrence relation that defines the Fibonacci numbers. That is, the Fibonacci numbers are given by the determinant
Natural cubic splines
A cubic spline interpolates a set of data points with piecewise cubic polynomials. There’s a (potentially) different cubic polynomial over each interval between input values, all fitted together so
that the resulting function, its derivative, and its second derivative are all continuous.
Suppose you have input points, called knots in this context, t[0], t[1], … t[n] and output values y[0], y[1], … y[n]. For the spline to interpolate the data, its value at t[i] must be y[i].
A cubic spline then is a set of n cubic polynomials, one for each interval [t[i], t[i+1]]. A cubic polynomial has four coefficients, so we have 4n coefficients in total. At each interior knot, t[1]
through t[n−1], we have four constraints. Both polynomials that meet at t[i] must take on the value y[i] at that point, and the two polynomials must have the same first and second derivative at that
point. That gives us 4(n − 1) equations. The value of the first polynomial is specified on the left end at t[0] the value of the last polynomial is specified at the right end at t[n]. This gives us 4
n − 2 equations in total.
We need two more equations. A clamped cubic spline specifies the derivatives at each end point. The natural cubic spline specifies instead that the second derivatives at each end are zero. What is
natural about a natural cubic spline? In a certain sense it is the smoothest curve interpolating the specified points. With these boundary conditions we now have as many constraints as degrees of
So how would we go about finding the coefficients of each polynomial? Our task will be much easier if we parameterize the polynomials cleverly to start with. Instead of powers of x, we want powers of
(x − t[i]) and (t[i+1] − x) because these expressions are 0 on different ends of the interval [t[i], t[i+1]]. It turns out we parameterize the spline over the ith interval as
where h[i] = t[i+1] − t[i], the length of the ith interval.
This may seem unmotivated, and no doubt it is cleaner than the first thing someone probably tried, but it’s the kind of thing you’re lead to when you try to make the derivation work out smoothly.
The basic form is powers of (x − t[i]) and (t[i+1] − x), each to the first and third powers, for reasons given above. Why the 6’s in the denominators? They’re not strictly necessary, but they cancel
out when we take second derivatives. Let’s look at the second derivative.
Note how when we stick in t[i] the first term is zero and the second is z[i], and when we stick in t[i+1] the first term is z[i+1] and the second is zero.
We can now write down the system of equations for the z‘s. We have z[0] = z[n] from the natural cubic spline condition, and for 1 ≤ i ≤ n − 1 we have
Note that this is a tridiagonal system because the ith equation only involves z‘s with subscripts i − 1, i, and i + 1.
Because of its tridiagonal structure, these equations can be solved simply and efficiently, much more efficiently than a general system of equations.
Related math posts
2 thoughts on “Tridiagonal systems, determinants, and natural cubic splines”
1. Typo! Species -> specifies
2. Thanks! | {"url":"https://www.johndcook.com/blog/2018/04/30/tridiagonal-systems/","timestamp":"2024-11-09T07:06:32Z","content_type":"text/html","content_length":"59182","record_id":"<urn:uuid:d304b343-960a-4252-aa76-58c97b051327>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00354.warc.gz"} |
How to print something that is not set initially
Basically I need to make a program that will print out the animal letters that the user enters into a textbox. I don't know how many letters will be entered so I need to get it after the button has
been pressed to define the animals[]. But this doesn't work, the print dialog box opens and nothing appears. Yet if I specify that animals was
[] animals = new String[5] at the beginning, I will get 5 pages in the print preview. How can I get this to show without specifying it ahead of time? thanks for any help, I've been stuck on this for
[String[] animals; ]
[ ]
[ public int print(Graphics g, PageFormat pf, int page) throw]
[ ]
[ if (page < animals.length) { ]
[ ]
[ Graphics2D g2d = (Graphics2D)g; ]
[ g2d.translate(pf.getImageableX(), pf.getImageableY()); ]
[ ]
[ g.drawString(animals[page], 340, 134); ] | {"url":"https://coderanch.com/t/541283/java/print-set-initially","timestamp":"2024-11-02T23:31:47Z","content_type":"text/html","content_length":"37750","record_id":"<urn:uuid:eadff487-6f5c-48ce-9e9a-34c5276eadaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00237.warc.gz"} |
Paying off Mortgage Early – How bad is it for your FI Date?
When your stocks drop 50% what are you going to be using to buy those "on sale" funds? Taking out a mortgage or a helo?:) I could see a person playing an interest timing game. You stick you money in
short term treasures and expect to lose 3% per year (versus paying off) and hope to make it up over the last 20 years when you can lock your money into treasuries paying 7+% (imagine if inflation
goes up to 5% for example). I must admit to having a real lack of enthusiasm for bonds these days. At 5-6% I could talk myself into them. Right now I pretty much prefer cash (i.e. 6-18 month cds) to
taking on the interest rate risk.
Are you aware of any place that will offer me a 30 year fixed rate margin loan at <4% that will not be called if the asset drops 50%? They aren't remotely the same product.
Yes leverage adds risk but it also adds return. In this case the added return drastically outweighs the risk due to the extended time period. Imagine you shove all the money in to an S&P 500
fund (lets assume no taxes to keep it real easy and this is NOT how I would invest money I didn't need for 30 years) during the worst 30 year period ever. Hmm that's an 8% return. Maybe we
are starting a new worst 30 year period that will only return 2%. Its possible. But I wouldn't want to bet on it.
Here is a link to a good article on the subject. The author mentions the mental accounting often done to justify using mortgages for leverage. But what is even better is that he makes the
case of increasing risk exposure in the remaining portfolio to improve potential returns. The logical conclusion is that one should get rid of bonds and use the proceeds for debt payments
or trade bonds at the same time you are directing new money towards the mortgage. As always, the decision what to do in the individual case has to be, well, individualized...
Rest assured all of you who are paying off your mortgages early that it is much less of an emotional issue than is commonly perceived. Or, in other words, it may be of psychological
benefit but it is supported by risk analysis and may lead to higher overall returns when done as part of a dynamic investment allocation strategy.
There is no question that a young person with little invested and a 30 years low rate mortgage should continue investing in tax advantaged accounts and defer paying off the mortgage. There's
really not much choice in the matter.
There is also no question that borrowing on margin is different because of the possibility of a margin call, although I would look at what happened to many people with underwater mortgages and
job loss a few years ago as somewhat of a margin call equivalent. The fact remains that leveraged investments can turn ugly in a hurry.
What I am having trouble understanding is that many people have not only mortgage debt but also significant investments in bonds. I have never had more than 10% in bonds ( ok, had another 5% in
TREA which I believed at the time to be bond-like, well it isn't but I was able to time it in 2009).
The reason is probably that asset allocation is often looked at as something happening just within the investment portfolio. So one hears that someone has an AA of 30/70 but then learns that the
same person has a mortgage balance as large as their investment portfolio. That makes no sense at all.
In reality, this person is facing volatility risk of way more than a 100% stock allocation (referenced to all investable assets of course) with the bond investments only limiting upside
potential. And upside potential is clearly all what counts for an investor with such a high stock allocation and that is all what a young person with good future earning potential and low net
worth should be worrying about.
I think it is important to note that by considering carrying debt to be similar to holding negative bonds, the AA becomes more realistic in terms of real world consequences of volatility risk -
that is the effect on net asset worth. I really do not care about how my investment portfolio is doing in isolation. I do care about net invested asset worth increase over time.
Stock market investments have the highest return potential but unfortunately also have the greatest dispersion of portfolio end value. The way to deal with this is asset allocation. Look at your
AA with your debts figured in as negative bonds and see what you need to do to get to your desired AA (30/70, 40/60 or whatever).
If you don't look at debts as negative bonds you may end up not knowing the actual risk you are taking as a young person and you may end up way too conservative when you are older.
By the way, I am about one year away from FIRE and my current AA is way over 100% stock market but buying back my mortgage debt will get me to about 15% bonds within the year. This excludes real
estate equity and annuities. | {"url":"http://forum.mrmoneymustache.com/investor-alley/paying-off-mortgage-early-how-bad-is-it-for-your-fi-date/200/","timestamp":"2024-11-07T01:20:00Z","content_type":"application/xhtml+xml","content_length":"234427","record_id":"<urn:uuid:4add6966-454e-478b-a18a-9379b5bafcfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00772.warc.gz"} |
How do you find the vertex of y = 4 - |x+2| ? | HIX Tutor
How do you find the vertex of #y = 4 - |x+2| #?
Answer 1
Set x + 2 equal to 0 and solve for x:
#x = -2#
This means that the absolute value function flips from negative to positive at #x = -2#, therefore, this is the x coordinate of the vertex and the y coordinate is the function evaluated at this x
#y = 4 - |-2 + 2|#
#y = 4#
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Answer 2
To find the vertex of ( y = 4 - |x+2| ), you first need to determine the absolute value's critical point, which is the point where the expression inside the absolute value equals zero. So, ( x + 2 =
0 ). Solving for ( x ), we get ( x = -2 ). Then, substitute ( x = -2 ) into the original equation to find the corresponding ( y )-value. Thus, the vertex is (-2, 4).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Gateway to Research (GtR) - Explore publicly funded research
First encounters with quantum computing: can games teach quantum reasoning?
Quantum computing is expected to have far-reaching benefits as well as potential security concerns for a wide range of industries in coming years. At present, there is little understanding of, or
expertise in, the skills required for effective quantum computational reasoning outside specialists in physics and mathematics. The goal of this work is to conduct a pilot project in collaboration
with participants from across a range of ages and sectors to develop an understanding of how non-specialists develop their understanding of counter-intuitive quantum computational concepts and
whether this can be assisted through the use of a visual game-like interface. Our project partner Quarks Interactive have developed a game which visually, rather than mathematically, represents the
most common model of quantum computing; the gate model which is universal in the sense that it can model all quantum information processing. This project will allow us to develop and test the
effectiveness of this game as a learning tool, as well as to assess its relative performance with different groups. The game consists of a visual Plinko (grid of pins) board like system where
coloured balls travel down the board following different tracks. Players are tasked to solve increasingly complex puzzles using picture tiles to change the path the balls take down the board, or to
introduce actions on the balls. Behind the scenes each picture tile accurately represents a quantum mechanical rule and the player's sequence of tiles is writing a functioning quantum algorithm.
However, crucially, the player does not have to have any knowledge of the complex mathematics behind the scenes to be able to successfully complete the puzzles. The puzzles therefore enable the
player to develop an intuition about quantum mechanics as well as generating genuine quantum computing algorithms, which can be implemented on quantum computers. Developing a more effective way for
quantum non-specialists across diverse industries to develop intuition about quantum mechanics is crucial for businesses to understand the implications of quantum computing for their own context and
to stay ahead of the game as quantum computing advances. Additionally, management and government will increasingly be expected to make decisions related to quantum computing and will have to become
quantum literate to avoid falling victim to misconceptions and hype.
Higher levels of quantum literacy will enable citizens from a wide variety of backgrounds to bring the potential benefits of exponentially faster and more complex data processing to their own
businesses, industries and disciplines, enabling them to re-imagine the possibilities for data analysis and problem solving in their fields. At present, a relatively small number of quantum computing
experts have this knowledge and understanding. The benefit of a broader range of citizens being able to access this understanding than is currently possible when quantum computing must be learned
through maths and physics, is therefore a wider understanding of the potential applications of quantum computing in diverse areas of society, which currently do not benefit from this technological
revolution. Quantum literacy equates to a different way of understanding reality and therefore, a potentially different way of conceiving of problems in a range of diverse fields.
A quantum computing visualization learning tool could also have an important function in engaging the next generation of learners in the building blocks of quantum computing at an earlier age. We aim
to discover whether learning through this puzzle visualization process also potentially increases engagement and motivation in groups who are traditionally under-represented in more advanced study of
computer science, mathematics and physics.
Technical Summary
The quantum puzzle visualization tool is based on an entirely graphical version of the matrix-vector representation of the Hilbert spaces of full systems. Because the translation for matrices to
visual elements is exact, this representation is also exact. From this visualisation tool the players will be able to learn about fundamental principles behind quantum mechanics such as superposition
and interference. Because high level tools for quantum computing are not yet fully developed, understanding the underlying building blocks is crucial, unlike in classical computing where much of the
low level behaviour of the computer can be abstracted away. The dynamics of classically counter-intuitive processes such as phase amplification can be understood in such a way that even if the
mathematics is hard to grasp, they can be intuitively understood by engaging with the visual tool and solving puzzles. The visualization tool is of something real i.e. quantum circuits, which include
non-Clifford gates and are therefore universal for quantum computing. The fact that the game is a full exact representation of quantum mechanics necessarily limits the systems to small sizes (if the
game itself could exactly simulate large quantum computers, we would not need large quantum computers), however these small sized examples can build intuition for larger systems which could not be
represented in the game. This is a gateway to further learning because it presents complex numbers and linear algebra in a much more accessible way than through equations. These representations mean
that the phase interference which is crucial to quantum mechanics can be understood visually and allows the users to develop a visual, rather than mathematical, mental model of these processes.
Testing how effective this model is is one of the aims of our proposal.
Planned Impact
This project has the potential to ensure that end users of quantum computing are able to become quantum literate and engage with the concepts behind quantum computing much earlier, and to a much
greater extent than they would otherwise be able to. A direct result of this is that these end users will be able to use their intimate knowledge of important problems and the current cutting edge
classical techniques to develop highly optimal applications of quantum computing, when compared to what could be developed by quantum physicists who are newcomers to the application domains.
Therefore, quantum machines can be developed more effectively for maximum benefit within individual industrial applications and the transition from being academically funded research to industrially
focussed applications can occur much sooner.
Additionally the methods studied here can help educate future decision makers, such as managers and members of government who have to make decisions related to quantum computing. By increasing
quantum literacy in this group, we can ensure that companies and governments make the right choices going forward, and are less likely to invest in apparently exciting but technically unsound avenues
toward quantum computation, and perhaps more importantly have the ability to recognize areas where there is truly immense potential and shape policy and/or investments to support these areas. By
being quantum literate, these important decision makers can also maintain realistic expectations of time scales and the rate of development of the technologies, such literacy is likely to help reduce
the potential of a "quantum winter", where excessive enthusiasm leads to unrealistic expectations for the field and a corresponding reduction in investment when these expectations are not met.
Understanding the optimal age at which it becomes appropriate to introduce the concepts of quantum computing within the education system also means that there is potential to develop skills in this
area from a much earlier stage. This project has the potential to provide policy makers and teachers with the ability to introduce the concepts of quantum computation within the school computer
science curriculum without the need for complex mathematical knowledge, thus advancing the level of skill in this area before students enter the workforce or Higher Education.
Raising the level of quantum literacy in the general public will also elevate the public discourse on quantum computing in arenas such as popular science publications and will lesson the incentives
for both the quantum computing industry and academics to over-hype results since the public is less likely to be convinced. This can again help reduce the potential for "quantum winter". This higher
level of quantum literacy also means that members of the public will be more likely to go into professions requiring expertise related to quantum mechanics which includes, but is not limited to,
quantum computing.
These related fields include but are not limited to materials science, physics, chemistry, and nanotechnology. Quantum mechanical effects are becoming increasingly important as technology moves to
smaller scales (in the case of nanotechnology) or as research into exotic states of matter such as superconductors becomes more important. Other fields such as physics and chemistry have aspects
which are fundamentally quantum. Individuals who are not only quantum literate, but have been exposed to quantum laws from an early age and have true quantum intuition are likely to be better
equipped to handle difficult quantum problems such as high temperature superconductivity, or understanding the action of complex molecules, even in the absence of quantum computers.
A key finding is that younger children seem to adapt better to playing the games we developed as opposed to adults. We also found that sometimes taking longer on some puzzles indicates
better overall performance suggesting than non-trivial learning is taking place. Interviews with participants in trials suggests that pairing the game with other educational materials
Description works best. This has implications on how a quantum education strategy should be implemented. Another achievement related to this award is the development of an educational video game
about quantum computing which, while technically not developed under this grant did benefit from the research we conducted. The webpage where the game can be downloaded is linked in the
URL box.
Exploitation The insights gained through this grant are likely to guide how games can be used within quantum education, including what audiences to target and how to present the subject.
Sectors Education
URL https://www.quarksinteractive.com/
The results of the trials have been used to develop educational tools produced by Quarks Interactive, a Romanian company which develops quantum video games. They have now produced a game
Description and have partnered with IBM and the American Physics Society to develop further quantum education tools. This includes the "save Schodinger's cat" card game (reported elsewhere) which has
been used in outreach in a large number of US classrooms. The game has been used in teaching within Romanian Universities.
Sector Education
Impact Cultural Societal Economic
Description Qaurks Interactive
Organisation Quarks Interactive
Country Romania
Sector Private
PI Contribution The research project has provided data about how to most effectively integrate educational tools into the video games produced by quarks interactive.
Collaborator Contribution Quarks interactive provided the video game software (and adapted it to collect data) so that this study could be possible.
Impact Video game produced by quarks interactive. This project is multidisciplinary between physics and education
Start Year 2020
Description APS physics quest
Form Of Engagement Participation in an activity, workshop or similar
Part Of Official No
Geographic Reach International
Primary Audience Schools
Results and Impact We were approached by the American Physics Society to produce a card game for their Physics Quest 2021 webpage https://www.aps.org/programs/outreach/physicsquest/pq21.cfm.
Approximately 20,000 kits will have been sent out in the United States to Middle School age students, as well as a printable version being available from the APS webpage.
Year(s) Of
Engagement 2021
URL https://www.aps.org/programs/outreach/physicsquest/pq21.cfm | {"url":"https://gtr.ukri.org/projects?ref=BB%2FT018666%2F1","timestamp":"2024-11-14T07:26:09Z","content_type":"application/xhtml+xml","content_length":"58669","record_id":"<urn:uuid:1c2267e4-db4a-40d0-b2b1-0af422829ab7>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00582.warc.gz"} |
Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials
It’s 2020 now! I am going to pick up paper reading just for fun (better than playing video games and watching TV all night lol)!
This is a paper from 2012 and my friend recommended it to me. It’s a great example of how to bridge different mathematical skills (including MAP, CRF, optimization, sampling theorem) to make a great
model. Although I still cannot fully understand all the details, reading this paper can serve as a good training of how to extract important takeaways and plan the next reading list from a
sophisticated paper. From another point of view, of course, the authors also make it clear enough for someone without a comprehensive mathematical background to read.
The paper focusing on pixel-wise image segmentation. For Conditional Random Field (CRF) methods, we want to build a network of many components. And we can calculate an “Energy” value of the network
and by increasing/decreasing the energy, we can extract the result from the converged result. However, building a pixel-paired network is almost impossible to calculate because the number of
connections (O(#pixel^2)) is so large.
Previous works tried to use some un-supervised method or hierarchy structure to cluster pixels into regions to reduce the number of components in the CRF. However, this paper proposes a way that can
reduce the calculation from (O(#pixel^2) to O(#pixel)). As a result, this method can achieve a much better and detailed segmentation than previous methods.
I will be as abstract as possible about the takeaways because I don’t fully understand the details for now:
• Optimize the original energy P(X) can be approximated by computing another distribution Q(X) that minimize the KL-divergence D(Q||P) such that Q(X) can be expressed by a product of independent
marginals. [10]
• Calculating that Q(X) can be done in an iterative way. In one of the steps, for each pixel, we need to iterate all other pixels, which takes O(#pixel^2)
• We can reformulate this step as a convolution (and a minus operation) in the feature space. The convolution performs a low-pass filter. By sampling theorem, “this function can be reconstructed
from a set of samples whose spacing is proportional to the standard deviation of the filter.” Therefore, we can downsample, convolute and then upsample in constant time. [16]
• With the help of permutohedral lattice, high dimension (d-dim) features can be also applied to this method to achieve O(#pixel*d) time complexity rather than O(#pixel^d) with a cost of feature
whitening. [1]
So based on the takeaways, if you want to develop your own pair-wise dense model and limit its linear computation complexity, you should read [16] and find an appropriate kernel for your problem as
long as it can be reformulated into a special convolution. Hope I can read [16] and share more with you soon. | {"url":"http://shaofanlai.com/post/94","timestamp":"2024-11-14T23:57:32Z","content_type":"text/html","content_length":"12371","record_id":"<urn:uuid:3a8143fa-811f-41de-bef0-3566b8f7fffd>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00699.warc.gz"} |
M is INTEGER
[in] M The number of rows of the matrix B.
M >= 0.
N is INTEGER
The number of columns of the matrix B, and the order of the
[in] N triangular matrix A.
N >= 0.
L is INTEGER
[in] L The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
NB is INTEGER
[in] NB The block size to be used in the blocked QR. N >= NB >= 1.
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
[in,out] A On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.
LDA is INTEGER
[in] LDA The leading dimension of the array A. LDA >= max(1,N).
B is COMPLEX*16 array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first M-L rows
[in,out] B are rectangular, and the last L rows are upper trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB is INTEGER
[in] LDB The leading dimension of the array B. LDB >= max(1,M).
T is COMPLEX*16 array, dimension (LDT,N)
[out] T The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.
LDT is INTEGER
[in] LDT The leading dimension of the array T. LDT >= NB.
[out] WORK WORK is COMPLEX*16 array, dimension (NB*N)
INFO is INTEGER
[out] INFO = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d9/d4d/ztpqrt_8f.html","timestamp":"2024-11-09T10:38:13Z","content_type":"application/xhtml+xml","content_length":"15849","record_id":"<urn:uuid:2a6661f5-2e2a-4da2-ac74-599a67a145dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00628.warc.gz"} |
2D Simulated Annealing
Model was written in NetLogo 5.0.2 • Viewed 1694 times • Downloaded 52 times • Run 0 times
Do you have questions or comments about this model? Ask them here! (You'll first need to log in.)
WHAT IS IT?
This model demonstrates the use of a simulated annealing algorithm on a very simple two-dimensional problem. Simulated annealing is an optimization technique inspired by the natural annealing process
used in metallurgy, whereby a material is carefully heated or cooled to create larger and more uniform crystalline structures. In simulated annealing, a minimum (or maximum) value of some global
"energy" function is sought. This model attempts to find a maximal solution in a two-dimensional grid.
We use such a simple problem in this model in order to highlight the solution technique only.
In this model, the objective function is defined for each patch in our 2D world. The location of each patch (x and y coordinates) can be thought of as the parameter values of the objective function.
The optimization works as follows. The system has a "temperature", which controls how much change is allowed to happen. A random location for an initial solution is defined, and then for each step, a
potential move to a new solution (location) is either accepted or rejected. Changes that result in a greater solution value are always accepted (changes that result in no change of solution value
will also always be accepted if the ACCEPT-EQUAL-CHANGES? switch is turned on). Changes that result in a lower solution value are only accepted with some probability, which is proportional to the
"temperature" of the system using the Boltzmann Distribution. The temperature of the system decreases over time, according to some cooling schedule, which means that initially changes that decrease
solution values will often be accepted, but as time goes on they will be accepted less and less frequently. This is similar to cooling a material slowly in natural annealing, to allow the molecules
to settle into nice crystalline patterns. Eventually the temperature approaches zero, at which point the simulated annealing method is equivalent to a random mutation hill-climber search, where only
beneficial changes are accepted.
HOW TO USE IT
Press the SETUP button to initialize the model and solution space.
Press the STEP button to go from one temperature to the next lower temperature.
Press the GO button to have the algorithm run until a solution has been found.
Adjust the COOLING-RATE slider to change how quickly the temperature drops.
The current temperature is shown in the TEMPERATURE monitor.
The DELTA-MAX slider controls how far a potential movement can be.
If the ACCEPT-EQUAL-CHANGES? switch is ON, then the system will always accept a new solution that yields no change in solution value. If it is OFF, then equal solutions are treated the same as those
decrease the solution value, and thus are only accepted probabilistically based on the system temperature.
The Solution monitors and plot show how the algorithm is performing and the best solution that has been found.
Slower cooling rates lead to higher optimal solutions (on average).
If you turn ACCEPT-EQUAL-CHANGES? to ON, does slow cooling still work better than fast cooling?
Try varying the DELTA-MAX. Does this help the system to reach more optimal configurations?
Currently, the probability of accepting a change that increases the best solution value is always 1, and the probability of accepting a change that decreases the solution value is based on the
temperature of the system and the amount by which the solution has changed. Try extending the model to make more alternative acceptance decision criteria.
Simulated annealing can be used on a wide variety of optimization problems. Experiment with using this technique on different "energy/cost" function, or even entirely different problems.
Particle Swarm Optimization, Simple Genetic Algorithm, Crystallization Basic, Ising
Original papers describing a simulated annealing
S. Kirkpatrick and C. D. Gelatt and M. P. Vecchi, Optimization by Simulated Annealing, Science, Vol 220, Number 4598, pages 671-680, 1983.
V. Cerny, A thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. Journal of Optimization Theory and Applications, 45:41-51, 1985
If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:
• Stonedahl, F. and Wilensky, U. (2009). NetLogo Simulated Annealing model. http://ccl.northwestern.edu/netlogo/models/SimulatedAnnealing. Center for Connected Learning and Computer-Based Modeling,
Northwestern University, Evanston, IL.
• Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
Copyright 2012 Kevin Brewer. All rights reserved. Copyright 2009 Uri Wilensky. All rights reserved.
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a
letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu and Kevin Brewer at kbrewer@olivet.edu
Comments and Questions
turtles-own [
fitness ;; equal to the solution value of the objective function at the turtle's current location
patches-own [
solution ;; the solution (energy or fitness) for the x,y values (may be negative)
globals [
winner ;; not really necessary since there is only one turtle at a time, but it is the turtle
;; that currently has the best solution
total-work ;; amount of work that is has been done. Essentially the number of times a solution value
;; has been calculated
best-fitness ;; the current best solution - again, not really "best" since there is only one current solution
global-best-fitness ;; the best solution that has ever been found during the progress
global-best-loc ;; where the best solution was found
g-x ;; temporary variables that hold the best x and y location
temperature ;; the current temperature of the system. Starts at 100 and the algorithm terminates when it gets below 1.
max-solution ;; the greatest patch value prior to scaling. used during setup.
top-solution ;; the ultimate greatest patch value after scaling.
low-solution ;; the lowest patch value prior to scaling. used during setup.
to setup
;; create the 2D domain of solutions
set max-solution -100
set top-solution 400
set total-work 0 ;; we start off at 0 work performed, obviously.
;; create the turtle that will represent the algorithm's current parameter set (i.e., location).
create-turtles 1 [
setxy random-xcor random-ycor ;; start the turtle at a random solution point
set color cyan
set shape "circle"
ifelse trace-on? [ pen-down ] [ pen-up ]
set total-work (total-work + 1) ;; everytime we calculate a solution, we add 1 to our total work counter.
;; populate the variables with "best" solution information
set winner max-one-of turtles [fitness]
set best-fitness [fitness] of winner
set global-best-fitness best-fitness
set g-x [xcor] of winner
set g-y [ycor] of winner
set global-best-loc (list g-x g-y)
;; set the initial temperature of the system
set temperature 100
to go
;; anneal at this temperature
;; populate the variables with "best" solution information
set winner max-one-of turtles [fitness]
set best-fitness [fitness] of winner
if best-fitness > global-best-fitness [
set global-best-fitness best-fitness
set g-x [xcor] of winner
set g-y [ycor] of winner
set global-best-loc (list g-x g-y)
;; reduce the temperature based on user input
set temperature temperature * (1 - cooling-rate / 100)
;; determine if we stop. If so, put the overall best solution in the monitors.
if (temperature < 1)
[ ask turtles
set color yellow
setxy (item 0 global-best-loc) (item 1 global-best-loc)
stop ]
to create-solutions
;; solutions are stored as the "solution" patch variable
ask patches [
;; for each patch, we initially set the solution to zero, just so something is in the variable at each location.
set solution 0
;; we will now put a bunch of "single spires" in the patch. We will eventually smooth these individual spires to make them "humps".
if random-float 1 < .5
[ ;; controls the number of spires, on a per patch basis - so want a low probability
ifelse random-float 1 < 0.25
[set solution -.25]
[set solution 1 ]
set solution (solution * random (top-solution * 100))
;; smooth the spires to make humps
repeat 100 [ diffuse solution 1.0 ]
;; now we will add a bit of small scale variability to the solution space - i.e., bumps.
ask patches [
set solution ( solution + random (top-solution / 20))
;;Now for the scaling of solution
;; adjust all solutions to a height proportional to the overall solution of the patch
set max-solution max [solution] of patches
if max-solution > 0
[ ask patches
[ set solution ((solution / max-solution ) * top-solution ) ]
set max-solution max [solution] of patches
set low-solution min [solution] of patches
;; now we will color the patches to make a nice visualization
ask patches [
set pcolor scale-color brown solution low-solution top-solution
;; let's highlight the best solution (maximum patch) by coloring it red
let best-patch max-one-of patches [solution]
ask best-patch [ set pcolor red ]
to calculate-fitness
;; a turtle procedure that returns the patch solution where the turtle is
set fitness [solution] of patch-here
to-report accept-change? [ old-energy new-energy ]
;; a reporter that will return true or false to indicate whether the turtle will move (accept)
;; the new solution location, or stay where it is.
report (new-energy > old-energy) ;; always accept new location if better.
or (accept-equal-changes? and new-energy = old-energy) ;; accept new location at equal value if user control says so
;; the following line is the key simulated annealing control. The idea is that as the temperature is reduced, it is less likely
;; to move to a poorer new location. When the temperature is high, the probabily of moving to poorer locations is greater.
;; this follows the Boltzmann Distribution
or ( exp ((old-energy - new-energy) * -1 / (0.1 * temperature)) > random-float 1.0 )
to anneal-turtle
;; figure out what the new potential solution is, and determine whether to move there or stay put.
ask turtles [ ;; there is only one turtle...
;; in this 2D example, a new solution can be found by going a distance in a direction. We will use
;; the built-in turtle moving routines to make this easy to program.
;; pick a random direction
right random 360
;; get a random distance that is limited by the user control
let my-distance max-pxcor * delta-max / 100 * random-float 1.
;; figure out what the solution is for the new distance and compare to current solution
if ( can-move? my-distance )
[ let o-energy fitness
let n-energy [solution] of patch-ahead my-distance
set total-work (total-work + 1)
if (accept-change? o-energy n-energy )
;; we have determined to move to the new solution, so do it!
move-to patch-ahead my-distance
;; we don't increment the work counter, since we really already calculated it and accounted for it just previously.
; Portions COpyright 2012 Kevin Brewer. All rights reserved.
; Portions Copyright 2008 Uri Wilensky. All rights reserved.
; The full copyright notice is in the Information tab.
There is only one version of this model, created over 10 years ago by Kevin Brewer.
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Multilinear form
Jump to navigation Jump to search
In abstract algebra and multilinear algebra, a multilinear form on ${\displaystyle V}$ is a map of the type
${\displaystyle f:V^{k}\to K}$,
where ${\displaystyle V}$ is a vector space over the field ${\displaystyle K}$ (or more generally, a module over a commutative ring), that is separately K-linear in each of its ${\displaystyle k}$
arguments.^[1] (The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.)
A multilinear k-form on ${\displaystyle V}$ over ${\displaystyle \mathbf {R} }$ is called a (covariant) k-tensor, and the vector space of such forms is usually denoted ${\displaystyle {\mathcal {T}}^
{k}(V)}$ or ${\displaystyle {\mathcal {L}}^{k}(V)}$.^[2]
Tensor product[edit]
Given k-tensor ${\displaystyle f\in {\mathcal {T}}^{k}(V)}$ and ℓ-tensor ${\displaystyle g\in {\mathcal {T}}^{\ell }(V)}$, a product ${\displaystyle f\otimes g\in {\mathcal {T}}^{k+\ell }(V)}$, known
as the tensor product, can be defined by the property
${\displaystyle (f\otimes g)(v_{1},\ldots ,v_{k},v_{k+1},\ldots ,v_{k+\ell })=f(v_{1},\ldots ,v_{k})g(v_{k+1},\ldots ,v_{k+\ell })}$,
for all ${\displaystyle v_{1},\ldots ,v_{k+\ell }\in V}$. The tensor product of multilinear forms is not commutative; however it is bilinear and associative:
${\displaystyle f\otimes (ag_{1}+bg_{2})=a(f\otimes g_{1})+b(f\otimes g_{2})}$, ${\displaystyle (af_{1}+bf_{2})\otimes g=a(f_{1}\otimes g)+b(f_{2}\otimes g)}$, and
${\displaystyle (f\otimes g)\otimes h=f\otimes (g\otimes h)}$.
If ${\displaystyle (v_{1},\ldots ,v_{n})}$ forms a basis for n-dimensional vector space ${\displaystyle V}$ and ${\displaystyle (\phi ^{1},\ldots ,\phi ^{n})}$ is the corresponding dual basis for the
dual space ${\displaystyle V^{*}={\mathcal {T}}^{1}(V)}$, then the products ${\displaystyle \phi ^{i_{1}}\otimes \cdots \otimes \phi ^{i_{k}}}$, with ${\displaystyle 1\leq i_{1},\ldots ,i_{k}\leq n}$
form a basis for ${\displaystyle {\mathcal {T}}^{k}(V)}$. Consequently, ${\displaystyle {\mathcal {T}}^{k}(V)}$ has dimensionality ${\displaystyle n^{k}}$.
Bilinear forms[edit]
Main article: Bilinear forms
If ${\displaystyle k=2}$, ${\displaystyle f:V\times V\to K}$ is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot
product) of vectors.
Alternating multilinear forms[edit]
Main article: Alternating multilinear maps
An important class of multilinear forms are the alternating multilinear forms, which have the additional property that^[3]
${\displaystyle f(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\mathrm {sgn} (\sigma )f(x_{1},\ldots ,x_{k})}$,
where ${\displaystyle \sigma :\mathbf {N} _{k}\to \mathbf {N} _{k}}$ is a permutation and ${\displaystyle \mathrm {sgn} (\sigma )}$ denotes its sign (+1 if even, –1 if odd). As a consequence,
alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., ${\displaystyle \sigma (p)=q,\sigma (q)=p}$ and ${\displaystyle \sigma (i)=i,1\leq i\leq k,ieq
${\displaystyle f(x_{1},\ldots ,x_{p},\ldots ,x_{q},\ldots ,x_{k})=-f(x_{1},\ldots ,x_{q},\ldots ,x_{p},\ldots ,x_{k})}$.
With the additional hypothesis that the characteristic of the field ${\displaystyle K}$ is not 2, setting ${\displaystyle x_{p}=x_{q}=x}$ implies as a corollary that ${\displaystyle f(x_{1},\ldots
,x,\ldots ,x,\ldots ,x_{k})=0}$; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors^[4] use this last condition as the defining property of
alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when ${\displaystyle \mathrm {char} (K)eq 2}$.
An alternating multilinear k-form on ${\displaystyle V}$ over ${\displaystyle \mathbf {R} }$ is called a multicovector of degree k or k-covector, and the vector space of such alternating forms, a
subspace of ${\displaystyle {\mathcal {T}}^{k}(V)}$, is generally denoted ${\displaystyle {\mathcal {A}}^{k}(V)}$, or, using the notation for the isomorphic kth exterior power of ${\displaystyle V^
{*}}$(the dual space of ${\displaystyle V}$), ${\textstyle \bigwedge ^{k}V^{*}}$.^[5] Note that linear functionals (multilinear 1-forms over ${\displaystyle \mathbf {R} }$) are trivially alternating,
so that ${\displaystyle {\mathcal {A}}^{1}(V)={\mathcal {T}}^{1}(V)=V^{*}}$, while, by convention, 0-forms are defined to be scalars: ${\displaystyle {\mathcal {A}}^{0}(V)={\mathcal {T}}^{0}(V)=\
mathbf {R} }$.
The determinant on ${\displaystyle n\times n}$ matrices, viewed as an ${\displaystyle n}$ argument function of the column vectors, is an important example of an alternating multilinear form.
Wedge product[edit]
The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each
term, the wedge product (${\displaystyle \wedge }$) of multicovectors can be defined, so that if ${\displaystyle f\in {\mathcal {A}}^{k}(V)}$ and ${\displaystyle g\in {\mathcal {A}}^{\ell }(V)}$,
then ${\displaystyle f\wedge g\in {\mathcal {A}}^{k+\ell }(V)}$:
${\displaystyle (f\wedge g)(v_{1},\ldots ,v_{k+\ell })={\frac {1}{k!\ell !}}\sum _{\sigma \in S_{k+\ell }}(\mathrm {sgn} (\sigma ))f(v_{\sigma (1)},\ldots ,v_{\sigma (k)})g(v_{\sigma (k+1)},\
ldots ,v_{\sigma (k+\ell )})}$,
where the sum is taken over the set of all permutations over ${\displaystyle k+\ell }$ elements, ${\displaystyle S_{k+\ell }}$. The wedge product is bilinear, associative, and anticommutative: if ${\
displaystyle f\in {\mathcal {A}}^{k}(V)}$ and ${\displaystyle g\in {\mathcal {A}}^{\ell }(V)}$ then ${\displaystyle f\wedge g=(-1)^{k\ell }g\wedge f}$.
Given a basis ${\displaystyle (v_{1},\ldots ,v_{n})}$ for ${\displaystyle V}$ and dual basis ${\displaystyle (\phi ^{1},\ldots ,\phi ^{n})}$ for ${\displaystyle V^{*}={\mathcal {A}}^{1}(V)}$, the
wedge products ${\displaystyle \phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}}$, with ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ form a basis for ${\displaystyle {\mathcal {A}}^{k}(V)}$.
Hence, the dimensionality of ${\displaystyle {\mathcal {A}}^{k}(V)}$ for n-dimensional ${\displaystyle V}$ is ${\textstyle {\tbinom {n}{k}}={\frac {n!}{(n-k)!\,k!}}}$.
Differential forms[edit]
Main article: Differential forms
Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense. Though conceptually and
computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus. Differential forms provide a mathematically rigorous
and precise framework to modernize this long-standing idea. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation
properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds). One far-reaching application is the modern statement of Stokes'
theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.
The synopsis below is primarily based on Spivak (1965)^[6] and Tu (2011).^[3]
Definition and construction of differential 1-forms[edit]
To define differential forms on open subsets ${\displaystyle U\subset \mathbf {R} ^{n}}$, we first need the notion of the tangent space of ${\displaystyle \mathbf {R} ^{n}}$at ${\displaystyle p}$,
usually denoted ${\displaystyle T_{p}\mathbf {R} ^{n}}$ or ${\displaystyle \mathbf {R} _{p}^{n}}$. The vector space ${\displaystyle \mathbf {R} _{p}^{n}}$ can be defined most conveniently as the set
of elements ${\displaystyle v_{p}}$ (${\displaystyle v\in \mathbf {R} ^{n}}$, with ${\displaystyle p\in \mathbf {R} ^{n}}$ fixed) with vector addition and scalar multiplication defined by ${\
displaystyle v_{p}+w_{p}:=(v+w)_{p}}$ and ${\displaystyle a\cdot (v_{p}):=(a\cdot v)_{p}}$, respectively. Moreover, if ${\displaystyle (e_{1},\ldots ,e_{n})}$ is the standard basis for ${\
displaystyle \mathbf {R} ^{n}}$, then ${\displaystyle ((e_{1})_{p},\ldots ,(e_{n})_{p})}$ is the analogous standard basis for ${\displaystyle \mathbf {R} _{p}^{n}}$. In other words, each tangent
space ${\displaystyle \mathbf {R} _{p}^{n}}$ can simply be regarded as a copy of ${\displaystyle \mathbf {R} ^{n}}$ (a set of tangent vectors) based at the point ${\displaystyle p}$. The collection
(disjoint union) of tangent spaces of ${\displaystyle \mathbf {R} ^{n}}$ at all ${\displaystyle p\in \mathbf {R} ^{n}}$ is known as the tangent bundle of ${\displaystyle \mathbf {R} ^{n}}$ and is
usually denoted ${\textstyle T\mathbf {R} ^{n}:=\bigcup _{p\in \mathbf {R} ^{n}}\mathbf {R} _{p}^{n}}$. While the definition given here provides a simple description of the tangent space of ${\
displaystyle \mathbf {R} ^{n}}$, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the article on tangent
spaces for details).
A differential k-form on ${\displaystyle U\subset \mathbf {R} ^{n}}$ is defined as a function ${\displaystyle \omega }$ that assigns to every ${\displaystyle p\in U}$ a k-covector on the tangent
space of ${\displaystyle \mathbf {R} ^{n}}$at ${\displaystyle p}$, usually denoted ${\displaystyle \omega _{p}:=\omega (p)\in {\mathcal {A}}^{k}(\mathbf {R} _{p}^{n})}$. In brief, a differential k-
form is a k-covector field. The space of k-forms on ${\displaystyle U}$ is usually denoted ${\displaystyle \Omega ^{k}(U)}$; thus if ${\displaystyle \omega }$ is a differential k-form, we write ${\
displaystyle \omega \in \Omega ^{k}(U)}$. By convention, a continuous function on ${\displaystyle U}$ is a differential 0-form: ${\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)}$.
We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from
smooth (${\displaystyle C^{\infty }}$) functions. Let ${\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} }$ be a smooth function. We define the 1-form ${\displaystyle df}$ on ${\displaystyle U}$ for $
{\displaystyle p\in U}$ and ${\displaystyle v_{p}\in \mathbf {R} _{p}^{n}}$ by ${\displaystyle (df)_{p}(v_{p}):=Df|_{p}(v)}$, where ${\displaystyle Df|_{p}:\mathbf {R} ^{n}\to \mathbf {R} }$ is the
total derivative of ${\displaystyle f}$ at ${\displaystyle p}$. (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate
functions) ${\displaystyle \pi ^{i}:\mathbf {R} ^{n}\to \mathbf {R} }$, defined by ${\displaystyle x\mapsto x^{i}}$, where ${\displaystyle x^{i}}$ is the ith standard coordinate of ${\displaystyle x\
in \mathbf {R} ^{n}}$. The 1-forms ${\displaystyle d\pi ^{i}}$ are known as the basic 1-forms; they are conventionally denoted ${\displaystyle dx^{i}}$. If the standard coordinates of ${\displaystyle
v_{p}\in \mathbf {R} _{p}^{n}}$ are ${\displaystyle (v^{1},\ldots ,v^{n})}$, then application of the definition of ${\displaystyle df}$ yields ${\displaystyle dx_{p}^{i}(v_{p})=v^{i}}$, so that ${\
displaystyle dx_{p}^{i}((e_{j})_{p})=\delta _{j}^{i}}$, where ${\displaystyle \delta _{j}^{i}}$ is the Kronecker delta.^[7] Thus, as the dual of the standard basis for ${\displaystyle \mathbf {R} _
{p}^{n}}$, ${\displaystyle (dx_{p}^{1},\ldots ,dx_{p}^{n})}$ forms a basis for ${\displaystyle {\mathcal {A}}^{1}(\mathbf {R} _{p}^{n})=(\mathbf {R} _{p}^{n})^{*}}$. As a consequence, if ${\
displaystyle \omega }$ is a 1-form on ${\displaystyle U}$, then ${\displaystyle \omega }$ can be written as ${\textstyle \sum a_{i}\,dx^{i}}$ for smooth functions ${\displaystyle a_{i}:U\to \mathbf
{R} }$. Furthermore, we can derive an expression for ${\displaystyle df}$ that coincides with the classical expression for a total differential:
${\displaystyle df=\sum _{i=1}^{n}D_{i}f\;dx^{i}={\partial f \over \partial x^{1}}dx^{1}+\cdots +{\partial f \over \partial x^{n}}dx^{n}}$.
[Comments on notation: In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices,
respectively. Since differential forms are multicovector fields, upper indices are employed to index them.^[3] The opposite rule applies to the components of multivectors and multicovectors, which
instead are written with upper and lower indices, respectively. For instance, we represent the standard coordinates of vector ${\displaystyle v\in \mathbf {R} ^{n}}$ as ${\displaystyle (v^{1},\ldots
,v^{n})}$, so that ${\textstyle v=\sum _{i=1}^{n}v^{i}e_{i}}$ in terms of the standard basis ${\displaystyle (e_{1},\ldots ,e_{n})}$. In addition, superscripts appearing in the denominator of an
expression (as in ${\textstyle {\frac {\partial f}{\partial x^{i}}}}$) are treated as lower indices in this convention. When indices are applied and interpreted in this manner, the number of upper
indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps
pinpoint errors made during manual computation.]
Basic operations on differential k-forms[edit]
The wedge product (${\displaystyle \wedge }$) and exterior differentiation (${\displaystyle d}$) are two fundamental operations on differential forms. The wedge product of a k-form and an ℓ-form is a
${\displaystyle (k+\ell )}$-form, while the exterior derivative of a k-form is a ${\displaystyle (k+1)}$-form. Thus, both operations generate differential forms of higher degree from those of lower
The wedge product ${\displaystyle \wedge :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)}$ of differential forms is a special case of the wedge product of multicovectors in general (
see above). As is true in general for the wedge product, the wedge product of differential forms is bilinear, associative, and anticommutative.
More concretely, if ${\displaystyle \omega =a_{i_{1}\ldots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}}$ and ${\displaystyle \eta =a_{j_{1}\ldots i_{\ell }}dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\
ell }}}$, then
${\displaystyle \omega \wedge \eta =a_{i_{1}\ldots i_{k}}a_{j_{1}\ldots j_{\ell }}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\wedge dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}}$.
Furthermore, for any set of indices ${\displaystyle \{\alpha _{1}\ldots ,\alpha _{m}\}}$,
${\displaystyle dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _{p}}\cdots \wedge \cdots dx^{\alpha _{q}}\wedge \cdots \wedge dx^{\alpha _{m}}=-dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _
{q}}\cdots \wedge \cdots dx^{\alpha _{p}}\wedge \cdots \wedge dx^{\alpha _{m}}}$.
If ${\displaystyle I=\{i_{1},\ldots ,i_{k}\}}$, ${\displaystyle J=\{j_{1},\ldots ,j_{\ell }\}}$, and ${\displaystyle I\cap J=\emptyset }$, then the indices of ${\displaystyle \omega \wedge \eta }$
can be arranged in ascending order by a (finite) sequence of such swaps. Since ${\displaystyle dx^{\alpha }\wedge dx^{\alpha }=0}$, ${\displaystyle I\cap Jeq \emptyset }$ implies that ${\displaystyle
\omega \wedge \eta =0}$. Finally, as a consequence of bilinearity, if ${\displaystyle \omega }$ and ${\displaystyle \eta }$ are the sums of several terms, their wedge product obeys distributivity
with respect to each of these terms. The collection of the wedge products of basic 1-forms ${\displaystyle \{dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}|1\leq i_{1}<\cdots <i_{k}\leq n\}}$ constitutes
a basis for the space of differential k-forms. Thus, any ${\displaystyle \omega \in \Omega ^{k}(U)}$ can be written in the form
${\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}a_{i_{1}\ldots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}}$ (*),
where ${\displaystyle a_{i_{1}\ldots i_{k}}:U\to \mathbf {R} }$ are smooth functions. With each set of indices ${\displaystyle \{i_{1},\ldots ,i_{k}\}}$ placed in ascending order, (*) is said to be
the standard presentation of ${\displaystyle \omega }$.
In the previous section, the 1-form ${\displaystyle df}$ was defined by taking the exterior derivative of the 0-form (continuous function) ${\displaystyle f}$. We now extend this by defining the
exterior derivative operator ${\displaystyle d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)}$ for ${\displaystyle k\geq 1}$. If the standard presentation of k-form ${\displaystyle \omega }$ is given by (*),
the ${\displaystyle (k+1)}$-form ${\displaystyle d\omega }$ is defined by
${\displaystyle d\omega :=\sum _{i_{1}<\ldots <i_{k}}da_{i_{1}\ldots i_{k}}\wedge dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}}$.
A property of ${\displaystyle d}$ that holds for all smooth forms is that the second exterior derivative of any ${\displaystyle \omega }$ vanishes identically: ${\displaystyle d^{2}\omega =d(d\omega
)\equiv 0}$. This can be established directly from the definition of ${\displaystyle d}$ and the equality of mixed second-order partial derivatives of ${\displaystyle C^{2}}$ functions (see the
article on closed and exact forms for details).
Integration of differential forms and Stokes' theorem for chains[edit]
To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated,
applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates.
Given a differentiable function ${\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}$ and k-form ${\displaystyle \eta \in \Omega ^{k}(\mathbf {R} ^{m})}$, we call ${\displaystyle f^{*}\eta \in \
Omega ^{k}(\mathbf {R} ^{n})}$ the pullback of ${\displaystyle \eta }$ by ${\displaystyle f}$ and define it as the k-form such that
${\displaystyle (f^{*}\eta )_{p}(v_{1p},\ldots ,v_{kp}):=\eta _{f(p)}(f_{*}(v_{1p}),\ldots ,f_{*}(v_{kp}))}$,
for ${\displaystyle v_{1p},\ldots ,v_{kp}\in \mathbf {R} _{p}^{n}}$, where ${\displaystyle f_{*}:\mathbf {R} _{p}^{n}\to \mathbf {R} _{f(p)}^{m}}$ is the map ${\displaystyle v_{p}\mapsto (Df|_{p}(v))
If ${\displaystyle \omega =f\,dx^{1}\wedge \cdots \wedge dx^{n}}$ is an n-form on ${\displaystyle \mathbf {R} ^{n}}$ (i.e., ${\displaystyle \omega \in \Omega ^{n}(\mathbf {R} ^{n})}$), we define its
integral over the unit n-cell as the iterated Riemann integral of ${\displaystyle f}$:
${\displaystyle \int _{[0,1]^{n}}\omega =\int _{[0,1]^{n}}f\,dx^{1}\wedge \cdots \wedge dx^{n}:=\int _{0}^{1}\cdots \int _{0}^{1}f\,dx^{1}\cdots dx^{n}}$.
Next, we consider a domain of integration parameterized by a differentiable function ${\displaystyle c:[0,1]^{n}\to A\subset \mathbf {R} ^{m}}$, known as an n-cube. To define the integral of ${\
displaystyle \omega \in \Omega ^{n}(A)}$ over ${\displaystyle c}$, we "pull back" from ${\displaystyle A}$ to the unit n-cell:
${\displaystyle \int _{c}\omega :=\int _{[0,1]^{n}}c^{*}\omega }$.
To integrate over more general domains, we define an n-chain ${\textstyle C=\sum _{i}n_{i}c_{i}}$ as the formal sum of n-cubes and set
${\displaystyle \int _{C}\omega :=\sum _{i}n_{i}\int _{c_{i}}\omega }$.
An appropriate definition of the ${\displaystyle (n-1)}$-chain ${\displaystyle \partial C}$, known as the boundary of ${\displaystyle C}$,^[8] allows us to state the celebrated Stokes' theorem
(Stokes–Cartan theorem) for chains in a subset of ${\displaystyle \mathbf {R} ^{m}}$:
If ${\displaystyle \omega }$ is a smooth ${\displaystyle (n-1)}$-form on an open set ${\displaystyle A\subset \mathbf {R} ^{m}}$ and ${\displaystyle C}$ is a smooth ${\displaystyle n}$-chain in $
{\displaystyle A}$, then${\displaystyle \int _{C}d\omega =\int _{\partial C}\omega }$.
Using more sophisticated machinery (e.g., germs and derivations), the tangent space ${\displaystyle T_{p}M}$ of any smooth manifold ${\displaystyle M}$ (not necessarily embedded in ${\displaystyle \
mathbf {R} ^{m}}$) can be defined. Analogously, a differential form ${\displaystyle \omega \in \Omega ^{k}(M)}$ on a general smooth manifold is a map ${\displaystyle \omega :p\in M\mapsto \omega _{p}
\in {\mathcal {A}}^{k}(T_{p}M)}$. Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem for
See also[edit]
1. ^ Many authors use the opposite convention, writing ${\displaystyle {\mathcal {T}}^{k}(V)}$ to denote the contravariant k-tensors on ${\displaystyle V}$ and ${\displaystyle {\mathcal {T}}_{k}(V)}
$ to denote the covariant k-tensors on ${\displaystyle V}$.
2. ^ ^a ^b ^c Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). New York: Springer. pp. 22–23. ISBN 978-1-4419-7399-3.
3. ^ Halmos, Paul R. (1958). Finite-Dimensional Vector Spaces (2nd ed.). New York: Van Nostrand. p. 50. ISBN 0-387-90093-4.
4. ^ Spivak uses ${\displaystyle \Omega ^{k}(V)}$ for the space of k-covectors on ${\displaystyle V}$. However, this notation is more commonly reserved for the space of differential k-forms on ${\
displaystyle V}$. In this article, we use ${\displaystyle \Omega ^{k}(V)}$ to mean the latter.
5. ^ Spivak, Michael (1965). Calculus on Manifolds. New York: W. A. Benjamin, Inc. pp. 75–146. ISBN 0805390219.
6. ^ The Kronecker delta is usually denoted by ${\displaystyle \delta _{ij}=\delta (i,j)}$ and defined as ${\textstyle \delta :X\times X\to \{0,1\},\ (i,j)\mapsto {\begin{cases}1,&i=j\\0,&ieq j\end
{cases}}}$. Here, the notation ${\displaystyle \delta _{j}^{i}}$ is used to conform to the tensor calculus convention on the use of upper and lower indices.
7. ^ The formal definition of the boundary of a chain is somewhat involved and is omitted here (see Spivak (1965), pp. 98-99 for a discussion). Intuitively, if ${\displaystyle C}$ maps to a square,
then ${\displaystyle \partial C}$ is a linear combination of functions that maps to its edges in a counterclockwise manner. It should be noted that the boundary of a chain is distinct from the
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