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How do you write a hypothesis in sociology?
How do you write a hypothesis in sociology?
Formulate a Hypothesis. A hypothesis is an assumption about how two or more variables are related; it makes a conjectural statement about the relationship between those variables. In sociology, the
hypothesis will often predict how one form of human behavior influences another.
What do you mean by a hypothesis?
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it.
How do you solve a hypothesis?
The procedure can be broken down into the following five steps.Set up hypotheses and select the level of significance α. Select the appropriate test statistic. Set up decision rule. Compute the test
statistic. Conclusion. Set up hypotheses and determine level of significance. Select the appropriate test statistic.
Why do we test hypothesis?
Hypothesis testing is an essential procedure in statistics. A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the
sample data. When we say that a finding is statistically significant, it’s thanks to a hypothesis test.
How do you find the level of significance in a hypothesis test?
In statistical tests, statistical significance is determined by citing an alpha level, or the probability of rejecting the null hypothesis when the null hypothesis is true. For this example, alpha,
or significance level, is set to 0.05 (5%).
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The Essence of Quantum Mechanics
In the 20th century, science has unveiled an incredible world that even the most daring science-fiction writers could have never even thought of. This world was so surprising that it completely
puzzled the most brilliant minds of the century, from Albert Einstein to Richard Feynman, from Niels Bohr to Paul Dirac, from David Hilbert to Werner Heisenberg or from Erwin Schrödinger to Wolfgang
Pauli. Apparent paradoxes kept being found, but they all ended up to be caused by mistaken common sense. Meanwhile, the central question of the measurement problem around which quantum mechanics is
constructed still has no commonly accepted explanation…
It sounds horribly complicated!
The basic ideas of quantum mechanics are definitely troubling. But they are not that complicated. The major difficulty is to get rid of common sense because common sense makes a lot of mistaken
assumptions. This is why learning quantum mechanics is an extremely humiliating and fruitful journey to undertake. Also, quantum mechanics has terrible philosophical implications. That’s why I invite
you to embark on the adventure of the unveiling of quantum mechanics. To get you excited, let’s start with an extract from the Fabric of the Cosmos: Quantum Leap, by NOVA and hosted by Brian Greene:
This article provides some unusual popularized explanations of quantum physics. I believe it can provide a new perspective for this complicated theory. This is why I strongly suggest you read it,
even though you have already been watching a lot of popularization of quantum mechanics.
Wave-Particle Duality
For long, physicists have been debating about the true nature of light. While Newton imagined it made of small particles, this rather intuitive concept was suddenly shattered by the double slit
experiment realized by Thomas Young in the early 1800s. In this article we will almost only focus on this experiment and its numerous variants, as, according to Nobel prize winner Richard Feynmann,
[it] has in it the heart of quantum mechanics. In reality, it contains the only mystery.
So what’s the double slit experiment?
A beam of light was sent on two tiny close holes, before being captured on a screen behind. The screen then displayed a troubling interference pattern, that is, a succession of dark and bright areas.
The following video by Veritasium displays the experiment with so simple tools that you can do it yourself:
This experiment is usually rather done with lasers, but Derek Müller’s box enables to take it to the street in a pretty cool way.
So, I guess that this refuted the idea that light was made of particles…
Indeed. To explain this phenomenon, the wave theory of light was introduced, with James Clerk Maxwell’s theory of electromagnetism as the golden age of the theory. In this setting, light was seen as
an electromagnetic wave.
Wait… What’s a wave?
OK. First, let me talk about fields. A field is something that fills up space. More precisely, at any point in space, the field takes some value. The concept of field is illustrated in the following
extract of a video by Minute Physics:
Now, if, at some point of the field, values are high, then there is a perturbation of the field at this point. This perturbation can then propagate, like waves on water. A wave is a propagating
perturbation of a field.
OK. So, in the 1800s, people thought that light was a propagating perturbation of the electromagnetic field?
Yes! This enabled to explain a lot of other phenomenons. But then came the discovery of the photoelectric effect.
What’s that?
In some specific setting, lighting a metal could induce an electrical current. What was troubling is that the electrical current was induced if and only if the wavelengths of light were small enough.
If you used red light, there was no current. But if you used blue light, there was. This appeared like a discontinuous phenomenon which was incompatible with the continuity of Maxwell’s equations.
So what would account for this discontinuity?
In one of his 4 earthshaking papers of 1905, Albert Einstein explained the discontinuity by the fact that light could actually be thought of as a composition of elementary particles called quanta of
light or photons! He justified this surprising assumption by the fact that the energy distribution of light was very similar to the energy distribution of particles of a gas. Another evidence
supporting his idea was the discovery of spectral lines of emission and absorption.
I already have a lot to say so I won’t dwell to much on this here. But if you can, please write more about these phenomenons.
But I thought that Young proved that light wasn’t made of particles!
I know! Yet, this didn’t prevent Einstein from receiving a Nobel prize for his idea of photons.
I’m totally lost! Is light made of particles or not?
We’ll get to this! But for now, let’s keep confusing ourselves with another weird experiment. As opposed to light, it was thought that all matter was made of atoms, and that these atoms were made of
protons, neutrons and electrons. In particular, electrons were considered as elementary indivisible particles. Indeed, scientists managed to isolate them one at the time…
But they then tried Young’s double slit experiment with beams of electrons instead of beams of light, and found interference patterns! This is displayed in the following figure from the Northern
Arizona University website:
This means that, just like photons, electrons appear to have both particle and wave properties! This is what scientists called the wave-particle duality, and is often interpreted as particles
sometimes behaving like waves. In fact, quantum field theory even suggests that forces also are both waves and particles, as explained in Thibault’s article.
This doesn’t sound very scientific…
I totally agree! So let’s better understand what’s going on!
Wave Function
The solution provided by many scientists in the 1920s and 1930s was to think of matter and light as a collection of elementary entities. But these elementary entities, which we call photons,
electrons or anything else, are not actually particles. Rather, each entity is a wave. However, physicists have now got used to calling these elementary entities particles, but you need to keep in
mind that these are not to be thought of as point objects but rather as waves. Let me rephrase the scientists’ idea of that time because it’s very important: All things are made of elementary waves.
This phrase corresponds to quantum mechanics, not more developed theories like quantum electrodynamics (QED) where waves can be combined to form new elementary waves, making each not really
elementary… Let’s stick with quantum mechanics here!
Waves instead of point particles? I have so much more trouble visualizing it…
I know! The following figure displays a 1-dimension wave function, that maps locations with the amplitude of the wave. Keep in mind that the following wave function represents one particle only.
Note that this figure fails to represent the fact that the wave function actually has complex values. But this won’t be much of a problem for this article though.
Does this assumption of elementary waves account for interference patterns?
Yes, because elementary entities are waves! These waves can add up to be constructive or destructive, creating interference patterns.
What about the fact that electrons could be isolated?
Yes too, because beams of electrons are actually a collection of elementary waves known as electrons.
What about the photoelectric effect?
This enabled to account for the photoelectric effect as well. The current is induced if electrons can capture an elementary wave which is energetic enough to make them leave the orbit of the atoms.
The captured elementary wave is a photon with enough energy.
Wow! The wave function does account for all the experiments you mentioned!
It surely does! But there are more troubling experiments I haven’t mentioned yet… Because electrons can be isolated, it’s possible to do the double slit experiment by shooting them one at a time. We
can then visualize where they land on the screen. Let’s recapitulate all the double slit experiments we have discussed so far with the following video of Dr Quantum:
I skipped the conclusion given by the video because I find it misleading, as it shows the electron as a particle dividing in 2 and recombining itself. It’s an interesting interpretation but it’s
absolutely not what quantum physics says.
OK, so what happens when electrons were shot one at a time?
Each electron lands at one particular location on the screen. But the locations where they arrive are not all the same! This is very weird, since all electrons are similar. More precisely, they are
described by identical wave functions. And yet, they all end up at different location!
How can this be explained?
Physicist Max Born proposed that, when measured, the waves collapse in a inherently random way. This means that they become very concentrated at a relatively precise location. This is usually
interpreted as them becoming particles. But to really understand it, you should keep in mind that a collapsed wave is a very localized wave, rather than a point particle. A localized wave is almost
nil almost everywhere except around the location of the wave:
Once collapsed the wave then resumes its propagation. But more than the way waves look, what needs to be stressed is the fact that the collapse of the wave functions occurs in a inherently random
way. This means that two identical experiments yield different results!
Does this mean that anything can happen? Does this even have a sense? If anything can happen, why is the world still making sense, at least at our scale?
These were questions that even Albert Einstein couldn’t handle, as he famously refuted quantum theory by stating that God doesn’t play dice with the world. Yet, experiments after experiments, over
almost 100 years, this quantum effect has been proven again and again.
So it’s really true? Anything can occur in the quantum world? Things are totally unpredictable?
What I haven’t told you yet is that, behind this seemingly quantum unpredictability, there actually is amazing well-defined world of probabilities. Even though the results of experiments could not be
predicted, the probabilities of the outcomes could be computed with incredible precisions.
Let’s reconsider the double slit experiment with one electron at a time. As more and more electrons were sent, an interference pattern appeared, as if all were sent simultaneously. What physicists
realized is that this interference pattern was revealing the probabilistic distribution of the outcomes of the single electron experiment. Yet, this interference pattern also corresponded to the wave
So the wave functions are related to probabilistic distributions?
Precisely! From the elementary waves that form all things are deduced the probabilistic distributions of where they collapse! More precisely, the square of the norm of the value of the wave function
at a point is the probability of finding the collapsed wave there once we measure it. This is explained in the following extract from The Fabric of the Cosmos: Quantum Leap:
But if all that matters is the probabilistic distribution, why care about the waves?
You need to consider the waves to understand constructive and destructive interferences. Also, waves contain information about energy and momentum of particles. But more importantly, as explained in
this more advanced article, the dynamics of the wave is precisely described by Schrödinger equation. But finally and more importantly, according to quantum mechanics, the true nature of particles is
being waves. What most popularizing videos call particles should rather be called localized waves. If you keep this in mind, I think you’ll understand quantum mechanics much better.
To be complete, I’d have to say that the evolutive properties of a particle are not only defined by its wave, but also its spin. This is of great importance for advanced ideas like the Pauli
exclusion principle which explains the fundamental octet rule in chemistry.
The Measurement Problem
It’s time for us to get to the most troubling part of quantum physics. Let’s reconsider once again the double slit experiment. As physicists tried to understand how the electrons were moving, they
decided to add detectors to observe slits through which electrons were moving. The result was profoundly shocking, as explained in this extract from The Quantum Universe:
What happened?
When slits were observed, there was no longer any interference pattern! Everything suddenly occurred as if electrons had been moving in straight lines all along. This was absolutely shocking: The
outcome was modified by the mere fact that we were observing electrons along the way!
Waw! This is extremely troubling! But what was observed at slits?
Detectors showed that electrons were moving through one of the slits, and one only! Recall that when we weren’t observing slits, the wave was spread and went through both slits.
But wait… Does this mean that the wave function collapsed at slits, when measured?
Yes, exactly! In fact, if we consider that particles are always waves that simply collapse randomly for any measurement, then we can understand what is happening. As we measure the particle at the
slits, it collapses and becomes very localized. Because the size of the slit is larger than the spreading of the collapsed wave function, it’s good approximation to say that the particle only goes
through one slit. Now, because the wave function collapsed, it’s no longer the same. Thus, it will not behave as if it hadn’t collapsed. This means that the result will no longer be the same as if we
hadn’t observed particles at the slits. That’s why observation of slits affects the outcome of the experiment.
Waw! This is troubling!
I know! But wait! There’s more. New technologies have enabled to delay the observation of slits, as explained below:
And yet, even when the observation of slits was delayed, things occur as if the observation of slits wasn’t delayed: There was no interference pattern!
I don’t get it! I thought that waves collapsed when measured, not before they were measured!
I know. This is extremely troubling! This leads us to the most fundamental misunderstood question of quantum physics, known as the measurement problem. Shortly put, it consists in asking: What’s a
measurement? Out of context, it would sound like a technical detail. Yet, as we have been discussing it, measurements are parts of the laws of physics, as they affect the outcomes of experiments. And
yet, we don’t even know what they really are.
What do you mean? Of course we know what a measurement is, don’t we?
In 1935, Nobel prize winner Erwin Schrödinger introduced one of science’s most famous thought experiment to illustrate how much we don’t understand measurement (although I’ve heard that his goal was
rather to ridiculize quantum physics).
Oh yeah! Wasn’t it something with a cat?
Yes! It consists in putting a cat in a box with a radioactive atom and poison. If the atom decays then the poison is released and the cat dies. The box is left for a minute. This gives the atom a
fifty-fifty chance to decay. Now, if don’t equip the box with any measurement device, then it is in a quantum state. This implies that the cat is both dead and alive, as displayed in the following
video from The Open University.
Isn’t it equivalent to saying that the cat is dead or alive?
No! Reconsider the double slit experiment. When we observe slits, the electron is indeed in the left or right slit. But if we don’t observe slits, it is both in the left and right slits, because the
wave function spreads over the two slits. The two cases are very different, as we can see in the results of the experiments. Similarly, here, the cat isn’t dead or alive. It is both dead and alive.
And until measurement, that is, opening the box and checking, it is in this superposition of states.
Waw! That’s troubling! But wait… Doesn’t the cat count as an observer? Doesn’t its heartbeat count as a measure?
Well, that’s a troubling question you’re asking here… This is precisely the measurement problem! And, shortly put, there is no accepted explanation for it.
But there are ideas to explain it, aren’t they?
We’ll get to this eventually. But first, let me show you the measurement problem at its worst: entanglement.
What I’m about to present here is my own creation. It seems to perfectly represents entanglement to me, but I might be mistaken since I’m not an expert of quantum physics. If you are, please correct
or confirm what I’m about to say.
Let’s reconsider Schrödinger’s cat. Assume we could now separate the radioactive atom and the cat, after they have been put together for a minute. Let’s keep the radioactive atom with us on Earth,
while we send the cat far far away, in some distant galaxy. Now, if we still haven’t made any measurement of the cat nor the atom, both are in a quantum superposition state. The cat is both dead and
alive, while the atom both has and hasn’t decayed. But the fates of the cat and of the atom are linked. When we measure the state of the atom, quantum mechanics says that it will instantaneously
affect the state of the cat.
Are you saying that if we observe that the atom has decayed, then the cat instantly dies?
Yes! And if the atom hasn’t decayed, then the cat instantly lives! It’s as if the information about our measurement instantaneously affected the state of the cat.
Waw! That’s weird. But didn’t Einstein prove that nothing could occur instantaneously?
He did! In his theory of relativity, Einstein proved that nothing travels faster than light! Worse than that, he proved that simultaneity didn’t even exist, as it depended on the observer. Therefore,
entanglement seems to totally contradict his theory! This was unacceptable for Albert Einstein, who famously referred to this phenomenon as spooky action at a distance.
OK… But can’t we imagine that the atom and the cat fall in definite states as soon as they are separated, in which case measuring the atom enables us to simply deduce the state of the cat, given the
state of the atom?
What you’re saying corresponds to saying that the cat is dead or alive rather than dead and alive. That’s what Einstein suggested. After decades of endless debates, Irish physicist John Bell proposed
an experiment to settle the arguments. This experiment was then improved by Alain Aspect who separated the cat and the atom far enough so that there had to be a faster-than-light communication. This
is explained in the Fabric of the Cosmos: Quantum Leap.
So what was the result of the experiments?
The experiments showed that Einstein was wrong. The cat and the atom were actually in a quantum superposition state, and there was an actual spooky action at a distance.
So instantaneous communication is possible?
Weirdly enough, it appears so. Applications are amazing! Perfectly secure cryptography can be achieved, as explained in Scott’s article on quantum cryptography. Even crazier, as shown later in the
Fabric of the Cosmos: Quantum Leap, teleportation devices are currently being constructed based on entanglement. So far they’ve teleported photons, but at some point in the future, they might
teleport people! The idea is based on the possibility of influencing the quantum state of the atom to increase the chances of eventually measuring it not-decayed. This instantaneously affect the
state of the cat and increases its chances of being alive, when we will have measured the state of the atom.
OK, that’s it, I’m totally lost by entanglement!
In a talk at Google, Ron Garret provides a better understanding of entanglement. As he says, what entanglement really is is nothing less than a measurement. Entanglement isn’t hard to understand,
provided that we really understand what measurement is. All the difficulty of the understanding of quantum physics therefore boils down to the big question I mentioned earlier…
What’s a measurement?
Yes! That’s the big question of quantum mechanics! Let me recall what we have found out about measurement in this article. Measurement implies a collapsing. But it’s not only about the collapsing of
a wave function. Rather, it’s a collapsing of a collection of wave functions throughout time (as shown by the double slit experiment with delayed observation of slits) and space (as proved by
entanglement). This means that, because of this property of measurement in quantum mechanics, our universe is inherently a non-local, and everything seems connected.
But isn’t there any scientist who at least have an idea of an answer to the measurement problem?
It’s very hard to grasp the concept of measurement. Any explanation of it must involve a ground-breaking twist of our vision of the world. In fact, one of the rare ideas which might make sense is a
terribly shocking one. Einstein once said: I like to think that the moon is there even if I am not looking at it. But Einstein’s view has been completely shattered by the measurement problem of
quantum mechanics. Instead, many physicists claim that there is no out there out there, and that reality only exist when we look at it. I’ll leave you with the explanation of Dr Amit Goswami, in this
extract from The Holographic Universe:
I want to add that I can’t really agree with any of these interpretations of quantum mechanics, because I simply don’t understand any of them. Once again, the measurement problem is still highly
misunderstood and there is no predominant opinion in the scientific community as displayed in
this video
from Sixty Symbols.
To face the measurement problem, a new concept of reality is probably needed, and Hawking’s idea of model-dependent realism may well help us. But it’s far from being an accepted concept to this day.
Now, things may not be as weird as what I’ve just presented here. As explained by the theory of decoherence, the apparent collapse of the wave function may be just a side effect of the Schrödinger
equation applied to a great number of particles. This is what I’ve explained in this extract of my talk More Hiking in Modern Math World:
Let’s Conclude
By assuming that all things are made of elementary waves, and by using equations of quantum mechanics to study the dynamics of these waves, physicists have been incredibly successful in explaining a
very large range of weird experimental observations. This simply makes quantum mechanics the most tested and yet not refuted theory science has ever produced. And its applications to technologies are
overwhelming! In particular, the invention of the transistor, which has then led to the explosion of new electronic and telecommunication technologies, has been the result of the understanding of
quantum mechanics.
What a crazy world our world seems to be! You’re probably deeply confused. I know I am. Let me reassure you by a few more quotes by Nobel prize winners:
□ Anyone not shocked by quantum mechanics has not yet understood it. (Niels Bohr)
□ If that turns out to be true, I’ll quit physics. (Max von Laue, on the wave properties of electrons)
□ Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business! (Erwin Schrödinger)
□ Not only is the Universe stranger than we think, it is stranger than we can think. (Werner Heisenberg)
□ I think I can safely say that nobody understands quantum mechanics. (Richard Feynman)
□ God not only plays dice, He also sometimes throws the dice where they cannot be seen. (Stephen Hawking, not a Nobel prize winner, on black holes)
Can you recapitulate the essence of quantum mechanics in one or two sentences?
Sure. The essence is that everything is made of elementary waves. Their dynamics either corresponds to Schrödinger’s deterministic equation, or to a probabilistic collapsing. The latter one refers to
the measurement problem. Although the probabilistic nature of its outcome is very well described, the cause of its occurrence is still highly misunderstood. This is partly because of its non-locality
through space and time, and partly due to the lack of definition of the concept of measurement. If you want to go further in the understanding of quantum mechanics, read my article on the dynamics of
wave functions!
:But if we understood measurement, we’d understand the Universe, right?
Hum… No. Quantum mechanics seems incapable to include gravity. Although Paul Dirac managed to include Einstein’s theory of special relativity, making things work with Einstein’s general relativity is
still mainly considered to be an open problem. Physicists are still in a quest to a unified theory of everything. The best candidate so far is string theory, but it’s still questioned by plenty of
2. Good site you have got here.. It’s hard to find high-quality writing like yours these days.
I really appreciate people like you! Take care!!
3. Now if I’m to wrap my around this….. when we figure out one or two aspects.. he changes it to another… ? Swapping out one for the other… ?just a random selection of reality we are in? So we not
suppose to figure it out… it’s a system that won’t allow us to get further? No matter what. One cancels the other out, but all work individually but when there placed together…???? A broken clock
is right twice a day…. and that means it wrong most of the time… so give the slight fraction determine a final result? Too much questions to ask… I’ll leave simple for now and the rest… and when
I say we, I mean the human race.
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Total Posts: 1291
Joined 09-12-2002
status: Guru
Re: I want better strings (Motif ES)
Power Bank contains new string sounds utilizing some user waves along with preset waves. It is a multi category voice library so aside from strings and orchestras, you also get Ap’s, Ep’s, Or’s, Ac
Gt, El Gt, Br, Rd, Choirs (IMO some of the best you will hear for the ES) & synths. My demo does not contain all of the strings & orchestra, only a small sample.
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C Program to Find Profit and Loss
• C Programming Examples
• C if-else & Loop Programs
• C Conversion programs
• C Pattern Programs
• C Array Programs
• C String Programs
• C File Programs
• C Misc Programs
• C Programming Tutorial
C Program to Find Profit and Loss
In this article, we will learn about how to create a program in C that will ask the user to enter the cost price and selling price of any item to calculate and print the profit or loss on that item.
int main()
float sp, cp, pf, ls;
printf("Enter cost price: ");
scanf("%f", &cp);
printf("Enter selling price: ");
scanf("%f", &sp);
pf = sp-cp;
printf("Profit = %.2f", pf);
else if(cp>sp)
ls = cp-sp;
printf("Loss = %.2f", ls);
printf("No profit and loss.");
return 0;
As the above program was written in the Code::Blocks IDE, here is the snapshot of the first sample run:
Now supply the values of cost and selling price, say 1200 as the cost price and 1450 as the selling price, and press the ENTER key to see the output, either in profit or in loss. As we purchased the
item in 1200 and sold it in 1450, we have made a profit of 250 on that item. Here is the second snapshot of the sample run:
Now let's run the program a second time. If the cost price is less than the selling price, then we will earn the money as profit; otherwise, if the cost price is greater than the selling price, then
we will lose the money as a loss. Here is the snapshot of the sample run; in this case, there is a loss of some money:
Let's check the above program with another sample run to see what will happen if cost price and selling price become equal:
Program Explained
• Get the cost price for any variable, such as cp.
• Get the selling price value for any variable, like sp.
• Use an if-else statement to check whether sp is greater than, less than, or equal to cp.
• If it is greater than cp, it means we have made some profit.
□ Therefore, the statement sp>cp evaluates to true, program flow goes inside the if block, and sp-cp will be initialized to pf (which holds the profit value).
□ Print the value of pf (profit) as output. Here, we have used the format specifier %.2f to print the floating-point value up to its 2 decimal places only.
□ For example, if the user provides the cost price as 1200 and the selling price as 1450, then we have made a profit of 250, which will be printed as output.
• And if it is less than cp, it means we have suffered some losses.
□ As a result, the statement cp>sp evaluates to true, and the program flow enters the else if block, where cp-sp is initialized to ls (which holds a loss value).
□ Print the value of ls (loss) as output.
• And if it is equal to cp, we have not made a profit or a loss.
□ As a result, the program flow enters the else block, and there is "no profit or loss," gets printed
« Previous Program Next Program »
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Statistical Analysis with Python - Data Science
Statistical analysis means investigating trends, patterns, and relationships using quantitative data. It is an important research tool used by scientists, governments, businesses, and other
organizations. In this lab we are going do lot of analysis on data sets using Python and R.
Pre Requisites:
• The students must have knowledge on the basics of Mathematics Syllabus
Course Objectives:
1. To provide an overview of a new language R used for data science.
2. To familiarize students with how various statistics like mean median etc. can be collected for data exploration in PYTHON
3. To provide a solid undergraduate foundation in both probability theory and mathematical statistics and at the same time provides an indication of the relevance and importance of the theory in
solving practical problems in the real world
Experiment 1: Functions in Python
Experiment 2: Text file processing & Basic Statistics
1. Develop Python program to generate count of words in a text file.
2.Write a program in Python with functions to calculate the following for comma-separated numbers in a text file(.txt)
a) 3rd Maximum number
b) 9th Minimum number
c) Total Unique Count
d )Mean
e) Standard Deviation
f) Number(s) with maximum frequency
g) Number(s) with minimum frequency
Notes:- First three bits a,b,c solutions are in a single program
Experiment 3: Exploring the Numpy library for multi-dimensional array processing
1.Develop programs in Python to implement the following in Numpy
Array slicing, reshaping, concatenation and splitting
Universal functions in Numpy
Fast sorting
Experiment 4: Data cleaning and processing with Pandas
1.Develop the following programs in Python
a) Implementing and querying the Series data structure
b) Implementing and querying the Data Frame data structure
c) Merge two different Data Frames
d) Performing Data Frame Indexing
Experiment 5: Advanced Data Processing and Transformation-Implement the following using the Pandas library
Experiment 6: Data Visualization-I in PYTHON
1. Write programs to demonstrate the different plots like Line Chart, Bar Chart, Scatter Plot, Pie Chart,Box Plot
Experiment 7: Data Visualization-II in PYTHON
1. Write programs to illustrate the different plotting data distributions like, Univariate Distributions, Bivariate Distributions.
Experiment 8: Probability Distributions
Experiment 9: Building Confidence in Confidence Intervals
1. Populations Versus Samples
2. Large Sample Confidence Intervals
3. Simulating Data Sets
4. Evaluating the Coverage of Confidence Intervals
Experiment 10: Perform Tests of Hypotheses
1. How to perform tests of hypotheses about the mean when the variance is known.
2. How to compute the p-value.
3. Explore the connection between the critical region, the test statistic, and the p-value
Data Science Additional Programs
Experiment 1: Basics of PYTHON Programming
Experiment 2: Decision making, Looping Statement and Functions
1. Write a program to illustrate if-else-else if in PYTHON
2. Write a Program to illustrate While and For loops in PYTHON
3. Write a program to demonstrate working with functions in PYTHON.
Experiment 3: Data Structures in PYTHON
Experiment 4
: Packages & Data Reshaping
Experiment : 5 Interfaces
Experiment 6: Regression
1. Write a program to demonstrate line regression in PYTHON for the given data set by following the below steps.
1.Reading and Understanding the Data
2.Hypothesis Testing in Linear Regression
3.Building a Linear Model
4.Residual Analysis and Predictions
Experiment 6: Creating a NumPy Array
1. Basic ndarray
2. Array of zeros
3. Array of ones
4. Random numbers in ndarray
5. An array of your choice
6. Imatrix in NumPy
7. Evenly spaced ndarray
Experiment 7: The Shape and Reshaping of NumPy Array
1. Dimensions of NumPy array
2. Shape of NumPy array
3. Size of NumPy array
4. Reshaping a NumPy array
5. Flattening a NumPy array
6. Transpose of a NumPy array
Experiment 8: Indexing and Slicing of NumPy Array
1. Slicing 1-D NumPy arrays
2. Slicing 2-D NumPy arrays
3. Slicing 3-D NumPy arrays
4. Negative slicing of NumPy arrays
Experiment 8: Perform following operations using pandas
1. Creating dataframe
2. concat()
3. Setting conditions
4. Adding a new column
Experiment 9: Read the following file formats using pandas
1. Text files
2. CSV files
3. Excel files
4. JSON files
Experiment 10: Perform following visualizations using matplotlib
1. Bar Graph
2. Pie Chart
3. Box Plot
4. Histogram
5. Line Chart and Subplots
6. Scatter Plot
APPLICATIONS OF PYTHON-Pandas
A) Pandas Data Series:
1. Write a Pandas program to create and display a one-dimensional array-like object containing an array of data using Pandas module.
2. Write a Pandas program to convert a Panda module Series to Python list and it's type.
3. Write a Pandas program to add, subtract, multiple and divide two Pandas Series.
4. Write a Pandas program to convert a NumPy array to a Pandas series.
Sample Series:
NumPy array:
[10 20 30 40 50]
Converted Pandas series:
dtype: int64
B) Pandas Data Frames:
Consider Sample Python dictionary data and list labels:
exam_data = {'name': ['Anastasia', 'Dima', 'Katherine', 'James', 'Emily', 'Michael',
'Matthew', 'Laura', 'Kevin', 'Jonas'],
'score': [12.5, 9, 16.5, np.nan, 9, 20, 14.5, np.nan, 8, 19],
'attempts': [1, 3, 2, 3, 2, 3, 1, 1, 2, 1],
'qualify': ['yes', 'no', 'yes', 'no', 'no', 'yes', 'yes', 'no', 'no', 'yes']}
labels = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j']
1. Write a Pandas program to create and display a Data Frame from a specified dictionary data which has the index labels.
2. Write a Pandas program to change the name 'James' to 'Suresh' in name column of the Data Frame.
3. Write a Pandas program to insert a new column in existing Data Frame.
4. Write a Pandas program to get list from Data Frame column headers.
5. Write a Pandas program to get list from Data Frame column headers
C) Pandas Index:
1. Write a Pandas program to display the default index and set a column as an Index in a given data frame.
2. Write a Pandas program to create an index labels by using 64-bit integers, using floating-point numbers in a given data frame.
D) Pandas String and Regular Expressions:
1. Write a Pandas program to convert all the string values to upper, lower cases in a given pandas series. Also find the length of the string values.
2. Write a Pandas program to remove white spaces, left sided white spaces and right sided white spaces of the string values of a given pandas series.
3. Write a Pandas program to count of occurrence of a specified sub-string in a Data Frame column.
4. Write a Pandas program to swap the cases of a specified character column in a given Data Frame.
E) Pandas Joining and merging DataFrame:
1. Write a Pandas program to join the two given dataframes along rows and assign all data.
2. Write a Pandas program to append a list of dictionaries or series to a existing DataFrame and display the combined data.
3. Write a Pandas program to join the two dataframes with matching records from both sides where available.
F) Pandas Grouping Aggregate
Consider data set:
1. Write a Pandas program to split the following dataframe into groups based on school code. Also check the type of GroupBy object.
2. Write a Pandas program to split the following dataframe by school code and get mean, min, and max value of age for each school.
G) Pandas Styling:
1. Create a dataframe of ten rows, four columns with random values. Write a Pandas program to highlight the negative numbers red and positive numbers black.
2. Create a dataframe of ten rows, four columns with random values. Write a Pandas program to highlight the maximum value in each column.
3. Create a dataframe of ten rows, four columns with random values. Write a Pandas program to highlight dataframe's specific columns.
H) Plotting:
1. Write a Pandas program to create a horizontal stacked bar plot of opening, closing stock prices of any stock dataset between two specific dates.
2. Write a Pandas program to create a histograms plot of opening, closing, high, lowstock prices of stock dataset between two specific dates.
3. Write a Pandas program to create a stacked histograms plot of opening, closing, high, low stock prices of stock dataset between two specific dates with more bins.
I) Exploratory data analysis on the bank marketing data set with Pandas Part-I
The bank marketing data set contains all the details. By reading or observing data set carefully write the code for the following
1. How many number of missing values are there in the data set?
2. Total how many duplicate values are presented in the data set?
3. What is the shape of the data after dropping the feature “Unnamed: 0”, missing values and duplicated values?
4. What is the average age of the clients those who have subscribed to deposit?
5. What is the maximum number of contacts performed during the campaign for the clients who have not subscribed to deposit?
6. What is the difference between the maximum balance (in euros) for the clients who have subscribed to deposit and for the clients who have not subscribed to the deposit?
7. What is the count of unique job levels in the data and find out how many clients are in the management level?
8. What is the percentage split of the categories in the column “deposit”?
9. Generate a scatter plot of “age” vs “balance”.
10. How many unemployed clients have subscribed to deposit?
I) Exploratory data analysis on the banking data set with Pandas Part-II
The banking data set contains all the details. By reading or observing data set carefully write the code for the following
1. How many number of missing values are there in the data set?
2. Total how many duplicate values are presented in the data set?
3. What is the shape of the data after dropping the feature “Unnamed: 0”, missing values and duplicated values?
4. What is the average age of the clients those who have not subscribed to deposit?
5. What is the maximum number of contacts performed during the campaign for the clients who have subscribed to deposit?
6. What is the count of unique education levels in the data and find out how many clients have completed secondary education?
7. What is the percentage split of the categories in the column “deposit”?
8. Generate a scatter plot of “age” vs “balance”.
9. How many clients with personal loan has housing loan as well?
10. How many unemployed clients have not subscribed to deposit?
For Answers of Part-II Click Here
Data Sets for Statistical Analysis with Python - Data Science
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Math Colloquia - <정년퇴임 기념강연> Some remarks on PDEs and Numerical Analysis: some results developed at SNU
줌 회의실: 889 8813 5947 (https://snu-ac-kr.zoom.us/j/88988135947)
초록: As this is my last colloquium at SNU, I will explain the meanings and backgrounds of selected theories that I have achieved in this department during the last 30 years. First I will describe a
generalized Green's Theorem from which the exact Sobolev function spaces to which the traces of H(curl,D) belong. Here, D is a bounded Lipschitz domain.
Then several nonconforming Finite Element Spaces will be discussed.
I will also explain numerical inversion of Laplace transforms, which motivated the exponential convergent algorithms. If time permits, I will mention some results on inverse problems and other
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Architecture overview | Maru
Prover Network implements circuits for each requested block range to prove volume of target DEX Pool that includes receipt from Ethereum Node. We generate proofs of requested block range to prove
volume of trade pool. Our circuits prove integrity of Keccak, volume summation that includes arithmetic addition operating on 256 bit values, data integrity from receipts, transactions and integrated
application logic. To minimize proof size and proving time, and on-chain verification cost, the prover network aggregate proofs required for Plonky2 verifier in Gnark.
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Fortran vs. C for numerical work (SUMMARY)
Dan Bernstein brnstnd at kramden.acf.nyu.edu
Fri Nov 30 17:15:38 AEST 1990
Several of you have been missing the crucial point.
Say there's a 300 to 1 ratio of steps through a matrix to random jumps.
On a Convex or Cray or similar vector computer, those 300 steps will run
20 times faster. Suddenly it's just a 15-1 ratio, and a slow instruction
outside the loop begins to compete in total runtime with a fast
floating-point multiplication inside the loop.
Anyone who doesn't think shaving a day or two off a two-week computation
is worthwhile shouldn't be talking about efficiency.
In article <7339 at lanl.gov> ttw at lanl.gov (Tony Warnock) writes:
> Model Multiplication Time Memory Latency
> YMP 5 clock periods 18 clock periods
> XMP 4 clock periods 14 clock periods
> CRAY-1 6 clock periods 11 clock periods
Um, I don't believe those numbers. Floating-point multiplications and
24-bit multiplications might run that fast, but 32-bit multiplications?
Do all your matrices really fit in 16MB?
> Compaq 25 clock periods 4 clock periods
Well, that is a little extreme; I was talking about real computers.
> For an LU
> decompositon with partial pivoting, one does rougly N/3 constant
> stride memory accesses for each "random" access. For small N, say
> 100 by 100 size matrices or so, one would do about 30
> strength-reduced operations for each memory access. For medium
> (1000 by 1000) problems, the ratio is about 300 and for large
> (10000 by 10000) it is about 30000.
And divide those ratios by 20 for vectorization. 1.5, 15, and 150. Hmmm.
More information about the Comp.lang.c mailing list
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Basic Geometry: Complementary Angle Subtraction
Geometry worksheets with subtraction problems for finding complementary angles (Angles whose sum is 90 degrees).
Common Complementary Angles
Five Degree Complementary Angles
Arbitrary Complementary Angles
Complementary Angle Subtraction Worksheets
A complementary angle is the angle when added to another angle and creates a sum of 90 degrees (or a right angle). This operation is very common in geometry, and practicing the basic arithmetic using
these worksheets will make solving many geometric proofs or other more advanced problems feel much less strenuous. The first set of worksheets here deal with very common complementary angles that are
multiples of 10, 15 and 30 degrees. The second series deals with finding complementary angles measuring in 5 degree increments, and the final set of worksheets requires calculations of arbitrary
degree measurement.
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Science:Math Exam Resources/Courses/MATH215/December 2013/Question 04 (b)
MATH215 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 04 (b)
Let ${\displaystyle Y(s)={\mathcal {L}}[y]}$ be the Laplace transform of the solution ${\displaystyle y(t)}$ to the following initial value problem:
${\displaystyle {\begin{cases}y''-2y'+y=g(t),\\y(0)=0,y'(0)=1\end{cases}}}$
where ${\displaystyle g(t)}$ is the function defined in (a). Determine ${\displaystyle Y(s)}$.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
You'll need to take the Laplace transform of the entire equation, and then solve for ${\displaystyle Y(s)}$. Pay attention to how derivatives transform. You'll need to use your result in (a).
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
We take the Laplace transform of the equation:
${\displaystyle {\mathcal {L}}(y''-2y'+y)={\mathcal {L}}(g(t))}$
${\displaystyle \underbrace {s^{2}Y(s)-sy(0)-y'(0)} _{{\mathcal {L}}(y'')}-2\underbrace {(sY(s)-y(0))} _{{\mathcal {L}}(y')}+\underbrace {Y(s)} _{{\mathcal {L}}(y)}=\underbrace {\frac {e^{-s}}
{s-1}} _{{\mathcal {L}}(g(t))}}$
${\displaystyle s^{2}Y(s)-0-1-2(sY(s)-0)+Y(s)=(s^{2}-2s+1)Y(s)-1={\frac {e^{-s}}{s-1}}}$
Since ${\displaystyle \displaystyle s^{2}-2s+1=(s-1)^{2}}$ we arrive at our final answer
${\displaystyle Y(s)={\frac {e^{-s}}{(s-1)^{3}}}+{\frac {1}{(s-1)^{2}}}}$
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Laplace transforms, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag
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Modeling Change
Original author
Resource type
Content area
Use type
Modeling Change:
The concrete act of programming the motion of a projectile in a virtual world, complements the usual mathematical analysis of that motion in the physical world, to help students build a deep
understanding of a traditionally difficult physics concept, projectile motion.
Unit Summary
Lesson 1: Simultaneous, independent change
Lesson 2: Dependent changes
Lesson 3a: Constant xy change
Lesson 3b: Variable xy change
Lesson 4a: Gravity
Lesson 4b: Projectile motion
Each lesson below contains detailed instructions for students, ideas for extensions, and links to starter and completed models. Also included below are test review documents and a test on the unit's
to view attachments and related links, and/or join the discussion
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Difference Between Frequency and Wavelength
The significant difference between frequency and wavelength is that frequency shows the number of wave oscillations in a given time frame. Whereas, the wavelength shows the distance between any two
successive crests and troughs of a wave. Thus, frequency and wavelength both are important parameters of a waveform.
Comparison Chart( Frequency Vs Wavelength)
Basis for Comparison Frequency Wavelength
Basic The number of wave cycles completed per second is known as frequency. Wavelength is the distance between two successive crests or troughs of a wave.
Measures in Time Distance
Denoted by letter f λ
Measuring Unit Hertz(Hz) Meter
Formula f = v / λ, where v is the speed of a wave λ = v / f, where v is the speed of a wave
Range- Audible & Visible The audible frequency range is between 20 Hz to 20 kHz. The wavelength of visible light ranges from 400 nm to 700 nm.
Definition of Frequency
A wave does vibrations or oscillations in a unit of time. A wave starts from point A, attains amplitude B, goes to C and then D, and finally comes at its initial point A. Thus, a wave completes one
cycle. If a wave takes one second to complete this one cycle, then the frequency of a wave is 1 Cycles/s (C/s)or 1 Hz. If the wave completes 10 oscillations in 1 second, then its frequency is 10 C/s
or 10 Hz.
The more oscillations show that a wave has a higher frequency.
From the above figure, it is very clear that the frequency increase with an increase in oscillation in a wave. If the frequency is 20 kHz, it means the wave completes 20,000 oscillations in one
The wave frequency can have values in the larger range, such as kHz, MHz, or GHz. We denote the frequency by a letter ‘f’ or ‘ν’. The duration of completion of one complete oscillation is called a
time period.
Frequency shows the vibratory or oscillatory motion of any waveform, such as sound wave, light wave, microwave, electromagnetic wave, infrared wave, mechanical vibrations, etc. There exist a
relationship between the frequency and time period. The frequency of a wave is inversely proportional to the time period.
For example, if a time period of a wave is 20 milliseconds, then the frequency of the wave will be 50 Hz.
The time period tells the duration of the repetition of a wave. If the time period of a wave is 20 milliseconds and it shows that a wave completes its one cycle in 20 milliseconds and then the wave
Definition of Wavelength
The distance between two consecutive crests or troughs is called wavelength. The crest point is the highest point of the wave whereas the trough is the lowest point.
The wavelength is measured in the units of distance, such as meters, centimeters, millimeters, nanometers, etc. The wavelength is measured in the direction of motion of the wave, and it is denoted as
A smaller wavelength of the sound wave has a higher frequency and produces higher tones. A larger wavelength sound wave has low frequency and produces lower pitch sounds. Thus the smaller wavelength
sound waves reach the listener faster than a higher wavelength wave.
The wavelength formula or the wavelength equation is given below.
Example: If the speed of a wave is 1000 meters per second and the frequency of the waves is 50 cycles per second, what is its wavelength?
Wavelength is inversely proportional to frequency. Therefore, the longer wavelength has a lower frequency. Similarly, a shorter wavelength has a higher frequency.
The different colors have dissimilar wavelengths. The wavelength of the red color is the longest, whereas the violet color has the least wavelength.
Key Differences between Frequency and Wavelength
1. Frequency is the total number of oscillations or vibrations in a unit of time. Whereas, the wavelength is the distance between two consecutive crests or troughs points of the wave.
2. Frequency shows the time of the wave. While wavelength measures the distance.
3. The unit of frequency is cycles/second or Hertz. The measurement unit of wavelength is meter.
4. Frequency is the ratio of speed and wavelength. Whereas, the wavelength is the ratio of speed and frequency.
5. The range of audible frequency is 20 to 20 kHz. The range of wavelength of visible light is 400 to 700 nm.
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Homology Modelling of Protein Steps Tools Software Tutorial PDF PPT Papers
What is Homology Modelling?
Homology modelling allows users to safely use rapidly generated in silico protein models in all the contexts where today only experimental structures provide a solid basis: structure-based drug
design, analysis of protein function, interactions, antigenic behavior, and rational design of proteins with increased stability or novel functions. In addition, protein modeling is the only way to
obtain structural information if experimental techniques fail. Many proteins are simply too large for NMR analysis and cannot be crystallized for X-ray diffraction.
Among the major approaches to three-dimensional (3D) structure prediction, homology modeling is the easiest one.
In the Homology Modelling, structure of a protein is uniquely determined by its amino acid sequence (Epstain, Goldberger, and Anfinsen, 1963). Knowing the sequence should, at least in theory, suffice
to obtain the structure.
2. During evolution, the structure is more stable and changes much slower than the associated sequence, so that similar sequences adopt practically identical structures, and distantly related
sequences still fold into similar structures. This relationship was first identified by Chothia and Lesk (1986) and later quantified by Sander and Schneider (1991). Thanks to the exponential growth
of the Protein Data Bank (PDB), Rost (1999) could recently derive a precise limit for this rule. As long as the length of two sequences and the percentage of identical residues fall in the region
marked as “safe,” the two sequences are practically guaranteed to adopt a similar structure.
Homology Modelling or Protein Modelling Example
Imagine that we want to know the structure of sequence A (150 amino acids long,). We compare sequence A to all the sequences of known structures stored in the PDB (using, for example, BLAST), and
luckily find a sequence B (300 amino acids long) containing a region of 150 amino acids that match sequence A with 50% identical residues. As this match (alignment) clearly falls in the safe zone
(Fig. 25.1), we can simply take the known structure of sequence B
(the template), cut out the fragment corresponding to the aligned region, mutate those amino acids that differ between sequences A and B, and finally arrive at our model for structure A. Structure A
is called the target and is of course not known at the time of modeling.
Homology Modelling of Protein Steps Tools Software Tutorial PDF PPT
Homology Modelling Steps
In practice, homology modeling is a multistep process that can be summarized in seven steps:
1. Template recognition and initial alignment
2. Alignment correction
3. Backbone generation
4. Loop modeling
5. Side-chain modeling
6. Model optimization
7. Model validation
At almost all the steps choices have to be made. The modeler can never be sure to make the best ones, and thus a large part of the modeling process consists of serious thought about how to gamble
between multiple seemingly similar choices. A lot of research has been spent on teaching the computer how to make these decisions, so that homology models can be built fully automatically. Currently,
this allows modelers to construct models for about 25% of the amino acids in a genome, thereby supplementing the efforts of structural genomics projects.
Homology_Modelling – Protein PPT
Protein Homology modelling steps ppt Structures
Homology Modelling Steps, Homology Modelling Software, Homology Modelling Ppt, Homology Modelling Pdf, Homology Modeling Server, Protein Modelling Bioinformatics, Homology Modeling Tutorial, Homology
Modelling Slideshare
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How do you append an array in Python? - vivendobauru.com.br
How do you append an array in Python?
How do you append an array in Python?
Appending to an Array using numpy.append()
1. import numpy as np.
2. np_arr1 = np.array([[1, 2], [3, 4]])
3. print(np_arr1)
How do you append an array together?
Use numpy. concatenate() to merge the content of two or multiple arrays into a single array. This function takes several arguments along with the NumPy arrays to concatenate and returns a Numpy array
ndarray. Note that this method also takes axis as another argument, when not specified it defaults to 0.
Can you append an array to an array in Python?
numpy. append() is used to append two or multiple arrays at the end of the specified NumPy array. The NumPy append() function is a built-in function in the NumPy package of python. This function
returns a new array after appending the array to the specified array by keeping the original array unchanged.
How to combine two arrays in Python?
Joining two arrays (lists) in Python is a simple task that can be done using either the `+` operator or the `extend()` method.
How do you add an append in Python?
Python List append()
1. Syntax of List append() The syntax of the append() method is: list.append(item)
2. append() Parameters. The method takes a single argument. …
3. Return Value from append() The method doesn't return any value (returns None ).
4. Example 1: Adding Element to a List. # animals list animals = ['cat', 'dog', 'rabbit']
How do you add to a 2D array in Python?
We can insert elements into a 2 D array using the insert() function that specifies the element' index number and location to be inserted. # Write a program to insert the element into the 2D (two
dimensional) array of Python. from array import * # import all package related to the array.
How to append 2 NumPy arrays?
Method 2: Using concatenate() method
Concatenate method Joins a sequence of arrays along an existing axis. Parameters : arr1, arr2, … : [sequence of array_like] The arrays must have the same shape, except in the dimension corresponding
to axis. axis : [int, optional] The axis along which the arrays will be joined.
How to merge 2 arrays in NumPy?
Joining NumPy Arrays
In SQL we join tables based on a key, whereas in NumPy we join arrays by axes. We pass a sequence of arrays that we want to join to the concatenate() function, along with the axis. If axis is not
explicitly passed, it is taken as 0.
How do we merge 2 arrays into single array?
1. First, we initialize two arrays lets say array a and array b, then we will store values in both the array.
2. After that, we will calculate the length of both the arrays and will store it into the variables lets say a1 and b1. …
3. Then the new array c which is the resultant array will be created.
How do you append two lists in Python?
One of the simplest ways to merge two lists is to use the "+" operator to concatenate them. Another approach is to use the "extend()" method to add the elements of one list to another. You can also
use the "zip()" function to combine the elements of two lists into a list of tuples.
What is append () in Python?
Append in Python is a pre-defined method used to add a single item to certain collection types. Without the append method, developers would have to alter the entire collection's code for adding a
single value or item. Its primary use case is seen for a list collection type.
What does append () mean Python?
append() method takes an object as an argument and adds it to the end of an existing list. For example, suppose you create a list and you want to add another number to it.
How to append a 2D NumPy array?
In NumPy, to add elements or arrays, including rows and columns, to the end or beginning of an array ( ndarray ), use the np. append() function.
How do you append to a 2D list?
In the above example, we used the + operator to add the new element to the 2D list. The code above is a concise way of writing my_2Dlist = my_2Dlist + [new_elem] . Once again, we wrapped the new
element inside a pair of square brackets to keep the 2D structure of my_2Dlist.
How to merge two 2D NumPy arrays?
We can perform the concatenation operation using the concatenate() function. With this function, arrays are concatenated either row-wise or column-wise, given that they have equal rows or columns
respectively. Column-wise concatenation can be done by equating axis to 1 as an argument in the function.
Can we append NumPy arrays?
Python numpy append() function is used to merge two arrays. This function returns a new array and the original array remains unchanged.
How to combine 3 arrays into one array?
1. Take the size of the arrays A, B, and C as input from the user.
2. Create arrays A, B, and C of the input size.
3. Take the elements of arrays A, B, and C as input from the user.
4. Merge arrays A, B, and C into a single array D.
5. Sort array D in ascending order.
6. Print the elements of array D.
How do I merge two sorted arrays in Python?
Algorithm for function find
1. Step 1 : Append the element of array2 in array1.
2. Step 2: Sort array1. Now array 1 have all the elements that we have got after merging both array.
3. Step 3: Add n2 elements in array2 from last.
4. Step 4: Let array1 have initial n1 elements.
5. Step 5: Print both the array.
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Joint Probability Density | Statistico
Definition of Joint Probability Density
In the realms of probability theory and statistics, the joint probability density function (PDF) denotes a concept that communicates the likelihood of a simultaneous occurrence of two or more
continuous random variables. The joint PDF is a function assigning a probability density to each viable combination of values for the random variables. It thereby provides the probability of these
variables concurrently attaining the designated values. The analysis of relationships between several continuous random variables and the evaluation of their dependencies find the joint probability
density functions to be crucial.
Joint Probability Density Function
Considering two continuous random variables
, the joint probability density function, symbolized as
f (x, y)
, is a function complying with the conditions below:
f (x, y) is always equal to or greater than 0 for every (x, y) in the domain of
The integration of f (x, y) over the entirety of the domain of
equates to 1.
In mathematical terms, the second condition can be articulated as:
∬ f (x, y) dx dy = 1
Marginal Probability Density Function
The marginal probability density function of a solitary random variable can be acquired from the joint probability density function by integrating the joint PDF concerning the other variable (s).
Consequently, the marginal PDFs of
can be calculated thus:
f_X (x) = ∫ f (x, y) dy
f_Y (y) = ∫ f (x, y) dx
Conditional Probability Density Function
The conditional probability density function signifies the probability density of one random variable, considering the value of an alternative random variable. This function can be computed from the
joint PDF and the marginal PDF in the following manner:
f_X|Y (x|y) = f (x, y) / f_Y (y)
f_Y|X (y|x) = f (x, y) / f_X (x)
Independence of Random Variables
Two continuous random variables
are deemed independent if their joint probability density function can be presented as the multiplication product of their marginal probability density functions:
f (x, y) = f_X (x) * f_Y (y)
If this condition is satisfied, the variables do not impact each other, and knowledge of one variable does not yield any information regarding the other variable.
Applications of Joint Probability Density Functions
Joint probability density functions find utility in numerous fields, including:
Risk assessment:
Modeling the joint probabilities of several risk factors to gauge their combined effect on a system or an investment.
Analyzing the dependencies between multiple components or variables in a system to enhance its performance.
Examining the relationships between economic variables, including inflation and unemployment, to comprehend their joint behavior and formulate economic policies.
Updated: May 23, 2023
| Published by: Statistico | About Us | Data sources
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DeepSpeed ZeRO-3 Offload
Today we are announcing the release of ZeRO-3 Offload, a highly efficient and easy to use implementation of ZeRO Stage 3 and ZeRO Offload combined, geared towards our continued goal of democratizing
AI by making efficient large-scale DL training available to everyone. The key benefits of ZeRO-3 Offload are:
• Unprecedented memory efficiency to run very large models on a limited number of GPU resources - e.g., fine-tune models with over 40B parameters on a single GPU and over 2 Trillion parameters on
512 GPUs!
• Extremely Easy to use:
□ Scale to over a trillion parameters without the need to combine multiple parallelism techniques in complicated ways.
□ For existing DeepSpeed users, turn on ZeRO-3 Offload with just a few flags in DeepSpeed Config file.
• High-performance per-GPU throughput and super-linear scalability across GPUs for distributed training.
□ With 1 Trillion parameters, ZeRO-3 Offload sustains 25 PetaFlops in compute performance on 512 NVIDIA V100 GPUs, achieving 49 TFlops/GPU.
□ Up to 2x improvement in throughput compared to ZeRO- 2 Offload on single GPU
Overview of ZeRO family of technology
The ZeRO Redundancy Optimizer (abbreviated ZeRO) is a family of memory optimization technologies for large-scale distributed deep learning. Unlike data parallelism (that is efficient but can only
support a limited model size) or model parallelism (that can support larger model sizes but requires significant code refactoring while adding communication overhead that limits efficiency), ZeRO
allows fitting larger models in memory without requiring code refactoring while remaining very efficient. ZeRO does so by eliminating the memory redundancy that is inherent in data parallelism while
limiting the communication overhead to a minimum. ZeRO removes the memory redundancies across data-parallel processes by partitioning the three model states (optimizer states, gradients, and
parameters) across data-parallel processes instead of replicating them. By doing this, it boosts memory efficiency compared to classic data-parallelism while retaining its computational granularity
and communication efficiency. There are three stages in ZeRO corresponding to three model states, as shown in the Figure 1: the first stage (ZeRO-1) partitions only the optimizer states, the second
stage (ZeRO-2) partitions both the optimizer states and the gradients and the final stage (ZeRO-3) partitions all three model states (for more details see the ZeRO paper).
Figure 1. Overview of ZeRO memory savings
In addition to these three stages, ZeRO family of technology also consists of ZeRO-2 Offload. ZeRO-2 Offload is a heterogeneous DL training technology that works in conjunction with ZeRO-2 to offload
partitioned optimizer states and gradients to CPU memory. ZeRO-2 Offload offers the full memory advantage of ZeRO-2 even on a single GPU, while at the same time offering great scalability of ZeRO-2
on multi-GPU setup. DeepSpeed library has been offering ZeRO-2 Offload since Sept 2020. For details, please see below:
ZeRO-3 Offload
With today’s release of ZeRO-3 Offload, we are adding support for partitioning and offloading parameters in addition to optimizer states and gradients partitioning already supported by ZeRO-2 Offload
in DeepSpeed. With parameter partitioning ZeRO-3 Offload implements the full set of features in the three stages of ZeRO, that allows for a linear growth in model size with the number of GPUs. In
addition, ZeRO-3 Offload can also optionally offload all these model states to CPU to further reduce GPU memory consumption, leveraging both CPU and GPU to maximize memory and compute efficiency of
the entire system.
We believe ZeRO-3 Offload offers a massive leap for large model training, in three regards:
i) Unprecedented model scale,
ii) Ease of supporting very-large models, and
iii) Achieving excellent training efficiency.
Unprecedented model scale
Unlike ZeRO-2 and ZeRO-Offload where the parameters have to fit in the memory of a single GPU, ZeRO-3 Offload can partition the parameters across GPUs, and offload them to CPU, supporting model sizes
that are much larger than the memory on a single GPU. Furthermore, ZeRO-3 Offload goes beyond the state-of-the-art hybrid 3D-parallelism (data, model and pipeline parallelism combined). While 3D
Parallelism is limited by the aggregate GPU memory, ZeRO-3 Offload can exploit both GPU and CPU memory, the latter of which is much larger and cheaper compared to GPU memory. This allows ZeRO-3
Offload to train larger model sizes with the given GPU and CPU resources than any other currently available technology.
Model Scale on Single GPU: ZeRO-3 Offload can train models with over 40B parameters efficiently on a single GPU (e.g., 32GB V100 GPU + 1.5TB CPU memory). This is 3x larger than what is possible with
ZeRO-2 Offload, the current state-of-the art.
Model Scale on Multi-GPUs: With ZeRO-3 Offload you can train a trillion and two trillion parameter models on NVIDIA 32GB V100 DGX-2 cluster with 256 GPUs and 512 GPUs, respectively. In contrast, the
state-of-art 3D Parallelism requires 800 GPUs, and 1600 GPUs, respectively, to fit the same sized models. This represents a 3x reduction in GPUs required to fit models with over a trillion
Ease of supporting very large models
From a system perspective, training models with hundreds of billions and trillions of parameters is extremely challenging. Data parallelism cannot scale the model size much further beyond a billion
parameters, model parallelism (with tensor slicing) cannot be used to scale model size efficiently beyond a single node boundary due to massive communication overheads, and pipeline parallelism
cannot scale beyond the number of layers available in a model, which limits both the model size and the number of GPUs that it can scale to.
The only existing parallel technology available that can scale to over a trillion parameters on massively parallel GPU clusters is the 3D parallelism that combines data, model and pipeline
parallelism in complex ways. While such a system can be very efficient, it requires major model code refactoring from data scientists to split the model into load balanced pipeline stages. This also
makes 3D parallelism inflexible in the type of models that it can support, since models with complex dependency graphs cannot be easily converted into a load balanced pipeline.
ZeRO-3 Offload address these challenges in two ways:
i) With ground-breaking memory efficiency, ZeRO-3 and ZeRO-3 Offload are the only DL parallel technology that can efficiently scale to over a trillion parameters by itself, without requiring a hybrid
parallelism strategy, greatly simplifying the system stack for DL training.
ii) ZeRO-3 Offload requires virtually no model refactoring from model scientists, liberating data scientists to scale up complex models to hundreds of billions to trillions of parameters.
Excellent training efficiency
High-performance per-GPU throughput on multiple nodes: ZeRO-3 Offload offers excellent training efficiency for multi-billion and trillion parameter models on multiple nodes. It achieves a sustained
throughput of up to 50 Tflops per GPU running on 32 DGX2 nodes comprising 512 NVIDIA V100 GPUs (see Figure 2). In comparison, the standard data parallel training with PyTorch can only achieve 30
TFlops per GPU for a 1.2B parameter model, the largest model that can be trained using data parallelism alone.
Figure 2. ZeRO-3 Offload: Multi-billion and trillion parameter model throughput on 512 V100 GPUs
ZeRO-3 Offload obtains high efficiency despite the 50% communication overhead of ZeRO Stage 3 compared to standard data parallel training for a fixed batch size. This is made possible through a
communication overlap centric design and implementation, which allows ZeRO-3 Offload to hide nearly all of the communication volume with computation, while taking advantage of a larger batch size for
improved efficiency resulting from better GPU memory efficiency.
Efficient multi-billion parameter model training on a single GPU: ZeRO-3 Offload further democratizes AI by enabling efficient training of multi-billion parameter models on a single GPU. For single
GPU training, ZeRO-3 Offload provides benefits over ZeRO-2 Offload along two dimensions. First, ZeRO-3 Offload increases the size of models trainable on a single V100 from 13B to 40B. Second, for
ZeRO-3 Offload provides speedups (e.g., 2.3X for 13B) compared to ZeRO-2 Offload for model sizes trainable by both solutions. These results are summarized in Figure 3.
Figure 3. Multi-billion parameter model training on one V100 GPU
Super-Linear scalability across GPUs: Additionally, ZeRO-3 Offload also preserves the super-linear scalability characteristics that we have demonstrated with all our previous ZeRO technologies (ZeRO
Stage 1, ZeRO Stage 2 and ZeRO Offload). ZeRO-3 Offload can exploit the aggregate PCI-E bandwidth between GPU and CPU across all the GPUs in multi-GPU training configuration, and at the same time, it
can also exploit the aggregate CPU compute across all the nodes. As a result, the CPU-GPU-CPU communication time as well as the optimizer update time decreases linearly with number of GPUs and nodes,
respectively, allowing ZeRO-3 Offload to exhibit super-linear scaling (see Figure 4).
Figure 4. ZeRO-3 Offload Superlinear Scalability for a 200B parameter model.
How to use ZeRO-3 Offload
As with many other existing DeepSpeed features, once the user model has been converted to use DeepSpeed, enabling ZeRO-3 Offload is as easy as turning on a couple of flags in DeepSpeed Config file.
Supporting advanced features like weight sharing, or enabling extremely large models that requires to be partitioned across GPUs/nodes to fit in GPU/CPU memory, can be done with just a couple of
additional lines of code change using the ZeRO-3 Offload API.
If you are already a DeepSpeed user, you can find our detailed tutorial on ZeRO-3 Offload below. If you are new to DeepSpeed, we recommend that you start at the getting started page before trying out
our ZeRO-3 Offload Tutorial.
The DeepSpeed Team is very excited to share ZeRO-3 Offload with the DL community.
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Elliptical distributions
[this page | pdf | back links]
An elliptical distribution is a multivariate distribution whose (multivariate) characteristic function, , being a vector with the same number of entries as there are dimensions for the distribution,
of the form:
where is a specified variable, and is a positive definite matrix.
If the distribution has a probability density function then it will take the form:
where is a scale factor, is an n-dimensional random vector with median vector (which is also the mean vector, if the latter exists), is a positive definite matrix which is proportional to the
covariance matrix if the latter exists, and is a function mapping non-negative real numbers to non-negative real numbers with finite area under the curve.
Perhaps the best known elliptical distributions are multivariate normal (i.e. Gaussian) distributions.
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Jump to navigation Jump to search
The table below shows the text of the inventory of Canterbury Cathedral Library taken on behalf of the Parliamentary commissioners in 1649 before the removal of the books to London.
Columns 3 and 4 indicate whether the books were listed in the earlier inventories of 1634 and 1638.
Column 4 gives information about binding marks.
Column 5 lists the names of donors or former owners, where known.
Column 6 gives the current shelf marks of the books which are still to be found in the Library.
1 sheet, parchment, folded
[verso:] Confession Books etc seized by the Commissioners & sent to London from this library. 1649.
[in a different hand:] Mr D<eye> his Notes brought by his Executrix
[recto:] A note of the books in this liberari and the four shelves
[in a different hand?]
Pray place your books whear you had them
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What is India VIX?: Concept, History & Methodology
India VIX is the volatility index designed by the National Stock Exchange (NSE) in the year 2008 . It gives an estimate of the market participants' expectation of volatility in the near term. The
important point to note here is that this expectation is only for Nifty index. India VIX does not include the expectations for non-Nifty shares. Nifty spot price is the weighted average of the
constituent share prices. The value for VIX is derived using the order book of underlying Out of money (OTM) options (We will see the computation methodology in greater detail later in this article).
The VIX values are updated dynamically in real time basis by NSE. Participants can see the index values just by adding them in their trading account watchlist. VIX is computed using the bid-ask
quotes of near and next-month Nifty options contracts. The exchange makes sure that exorbitantly high or low values for bid-ask quotes are not entered by market participants. There are circuit limits
applicable in options too(only at the time of order placement). There is no circuit limit on option prices when the underlying share/index moves.
Computation methodology The calculation for India VIX is complex and involves heavy mathematics. I would not discuss the exact formula for calculation (I do not want to scare the non-math savvy
readers of this blog). Honestly, more than the calculation formula, the conceptual understanding and its application in trade is important.
VIX is a function of the following variables:
• Time to expiration: This is the most important variable in calculation of VIX. The value of VIX is directly proportional to the Time to Expiration (TTE as is commonly called in options
• Weightage given to deep OTM options: As mentioned above, VIX uses the bid-ask quotes of out-of the money (OTM) options. More weightage is given to options that are further away from the spot
price. For example: With Nifty spot price 11650, weightage will be given as below: For calls: weight(11700) < weight(11750) < ...< weight(11900) For puts: weight(11600) < weight(11550) < ...<
• Risk free rate: The NSE MIBOR rate is used as the risk-free rate for calculation of VIX.
• Nifty futures level: For deciding the OTM strike prices, futures price of Nifty is used instead of the spot price. For example if Nifty spot price is 11580 and futures price is 11640, then the
strike prices to be considered for calls are 11700 and onwards; the strike prices for puts are 11550 and below.
How VIX started? VIX or volatility index is a trademark of the Chicago Board of Options Exchange (CBOE). CBOE started the concept of volatility index in the year 1993 based on S&P 100 index options.
In 2003, the underlying index was changed to S&P 500. During periods of high volatility like geo political events, market generally moves upward or downward sharply. The CBOE probably realised the
importance of having a standardised index for tracking volatility. Imagine how much this index would have helped the experienced traders during the 2008 crash ( If your curiosity levels are generally
high, just check the VIX chart for S&P 500 during the 2008 crash). This movement leads to movement in the volatility index. Investors and traders use this index to gauge the future volatility.
Conclusion: VIX is a concept introduced in the USA by CBOE and replicated in India by the NSE under the name India VIX. The computation methodology is same as the original VIX. The calculation of VIX
is complex and involves a complex formula. This article is an attempt to explain the complex formula in simple terms. Application of the index is more important than the calculation.
Watch this space for more about the application of VIX in trading.
If you like this article, contact us here or visit our training page here for more such learnings. Happy trading and investing!
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More Efficient Structure-Preserving Signatures - Or: Bypassing the Type-III Lower Bounds
Paper 2016/255
More Efficient Structure-Preserving Signatures - Or: Bypassing the Type-III Lower Bounds
Essam Ghadafi
Structure-preserving signatures are an important cryptographic primitive that is useful for the design of modular cryptographic protocols. It has been proven that structure-preserving signatures (in
the most efficient Type-III bilinear group setting) have a lower bound of 3 group elements in the signature (which must include elements from both source groups) and require at least 2
pairing-product equations for verification. In this paper, we show that such lower bounds can be circumvented. In particular, we define the notion of Unilateral Structure-Preserving Signatures on
Diffie-Hellman pairs (USPSDH) which are structure-preserving signatures in the efficient Type-III bilinear group setting with the message space being the set of Diffie-Hellman pairs, in the
terminology of Abe et al. (Crypto 2010). The signatures in these schemes are elements of one of the source groups, i.e. unilateral, whereas the verification key elements' are from the other source
group. We construct a number of new structure-preserving signature schemes which bypass the Type-III lower bounds and hence they are much more efficient than all existing structure-preserving
signature schemes. We also prove optimality of our constructions by proving lower bounds and giving some impossibility results. Our contribution can be summarized as follows: \begin{itemize} \item We
construct two optimal randomizable CMA-secure schemes with signatures consisting of only 2 group elements from the first short source group and therefore our signatures are at least half the size of
the best existing structure-preserving scheme for unilateral messages in the (most efficient) Type-III setting. Verifying signatures in our schemes requires, besides checking the well-formedness of
the message, the evaluation of a single Pairing-Product Equation (PPE) and requires a fewer pairing evaluations than all existing structure-preserving signature schemes in the Type-III setting. Our
first scheme has a feature that permits controlled randomizability (combined unforgeability) where the signer can restrict some messages such that signatures on those cannot be re-randomized which
might be useful for some applications. \item We construct optimal strongly unforgeable CMA-secure one-time schemes with signatures consisting of 1 group element, and which can also sign a vector of
messages while maintaining the same signature size. \item We give a one-time strongly unforgeable CMA-secure structure-preserving scheme that signs unilateral messages, i.e. messages in one of the
source groups, whose efficiency matches the best existing optimal one-time scheme in every respect. \item We investigate some lower bounds and prove some impossibility results regarding this variant
of structure-preserving signatures. \item We give an optimal (with signatures consisting of 2 group elements and verification requiring 1 pairing-product equation) fully randomizable CMA-secure
partially structure-preserving scheme that simultaneously signs a Diffie-Hellman pair and a vector in $\Z^k_p$. \item As an example application of one of our schemes, we obtain efficient
instantiations of randomizable weakly blind signatures which do not rely on random oracles. The latter is a building block that is used, for instance, in constructing Direct Anonymous Attestation
(DAA) protocols, which are protocols deployed in practice. \end{itemize} Our results offer value along two fronts: On the practical side, our constructions are more efficient than existing ones and
thus could lead to more efficient instantiations of many cryptographic protocols. On the theoretical side, our results serve as a proof that many of the lower bounds for the Type-III setting can be
Available format(s)
Publication info
Published elsewhere. Major revision. ESORICS 2017
Contact author(s)
essam_gha @ yahoo com
2017-08-01: revised
2016-03-08: received
Short URL
author = {Essam Ghadafi},
title = {More Efficient Structure-Preserving Signatures - Or: Bypassing the Type-{III} Lower Bounds},
howpublished = {Cryptology {ePrint} Archive, Paper 2016/255},
year = {2016},
url = {https://eprint.iacr.org/2016/255}
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All Vedic Maths Formulas » Why Vedic Maths Vedic Math School
All Vedic Maths Formulas
Vedic Maths is a system of Mental Calculation and reasoning. It has 29 Vedic Maths formulas, which are mathematical concept based on ancient Indian scripts called Veda. It is fast, efficient and easy
to learn. Vedic mathematics is a very versatile and a majority of the time it is very helpful to solve the tedious question of arithmetic and algebraic operations in a simple and fast way.
Vedic Maths book was written by the Priest Bharati Krishna Tirthaji and first published in 1965. The Vedic Maths contains 16 sutras (formulas) and 13 sub-sutras(corollary).
Today I am discussing all the 29 Vedic Maths formulas. Basically, all the sutra are written in Sanskrit name. It will be difficult to understand who don’t know Sanskrit but don’t worry here I am
writing the English name simultaneously.
List of All Sutra in Sanskrit ( 29 Vedic Maths formulas)
1. Ekadhikena Purvena: By one more than the previous one –
Application: This Vedic maths formula is useful to find the Square of all number who end with five.
Steps to Solve: Let Suppose 25, 35, 65… The square of these number always ends with 25 and begins with the multiple of tens digit number with Just one more than the integer.
Square of 25 is 2 x 3 .. 25 = 625.
Square of 35 is 3 x 4 .. 25 = 1225.
Square of 65 is 6 x 7 .. 25 = 4225.
2. Nikhilam Navatashcaramam Dashatah: All from 9 and the last from 10
Application: This Vedic Maths formula is For multiplication of any numbers nearer to 10 or multiple of 10.
Steps to Solve:
1. Subtract the numbers with their closest multiple of 10 and multiply the subtracted value to each other.
2. Write the result of Step 1 in the beginning and result of Step 1 at the end.
1. 98 x 96 = ?
100 – 98 = 2 and 100 – 96 = 4 , Now : 2 x 4 = 08
98 – 4 or 96 – 2 = 94
So, our answer is 9408
3. Urdhva-Tiryagbyham: Vertically and crosswise
Application: For multiplication of any two numbers.
Step to Solve: Here I am a discussion about the only two-digit multiplication of numbers.
1. Multiply the last digit(rightmost) to each other.
2. Now, Multiply the numbers digit crosswise and add them.
3. Multiply the first digit(leftmost) of both the number and put it at the most beginning.
4. if the number or more then one digit then write the first number and rest is carry forward to the next left.
Example :
45 x 87 = ? 35 x 72 =?
5 x 2 = 10
(3 x 2) + (5 x 7) = 6 + 35 = 41
7 x 3 = 21
35 x 72 = 21 | 41 | 10 = 2520
4. Paravartya Yojayet: Transpose and adjust
Application: this rule is for dividing large numbers by number greater than 10.
Example: 4356 divided by 17.
5. Shunyam Saamyasamuccaye: When the sum is the same that amount is zero
6. Anurupye Shunyamanyat: If one is in ratio, the other is zero
7. Sankalana-vyavakalanabhyam : By addition and by subtraction
8. Puranapuranabyham : By the completion or non-completion
9. Chalana-Kalanabyham : Differences and Similarities
10. Yavadunam : Whatever the extent of its deficiency
11. Vyashtisamasthi : Part and Whole
12. Shesanyankena Charamena : The remainders by the last digit
13. Sopaantyadvayamantyam : The ultimate and twice the penultimate
14. Ekanyunena Purvena : By one less than the previous one
15. Gunitasamuchyah : The product of the sum is equal to the sum of the product
16. Gunakasamuchyah : The factors of the sum are equal to the sum of the factors
Sutra Corollary:
1. Anurupyena
2. Sisyate Sesasamjnah
3. Adyamadyenantyamantyena
4. Kevalaih Saptakam Gunyah
5. Vestanam When the sum is the same that amount is zero
6. Yavadunam Tavadunam
7. Yavadunam Tavadunikritya Varga Yojayet
8. Antyayordashake’pi
9. Antyayoreva Differences and Similarities
10. Samuccayagunitah
11. Lopanasthapanabhyam
12. Vilokanam
13. Gunitasamuccayah Samuccayagunitah
14. Dhvajanka
15. Dwandwa Yoga
16. Adyam Antyam Madhyam
This article is just the introduction of the topic of all 29 Vedic Maths formulas and I am not going in many details. I just give you the name of the formula and some examples. in the upcoming days,
I will update the article with more details and example. Stay connected and if you have any doubt just comment below.
9 thoughts on “All Vedic Maths Formulas”
In the second sutra Nikhilam Navatashcaramam the answer of first example is wrong
Hello Ruturaj,
Thanks for your Feedback. We Corrected the Typo. Hope it Will be useful for You.
Thanks Again
The anwer is right u can check….
The multiplication of 98×96=9408.
Hello Abhi, You are Correct. If we use the Vedic Method of Multiplicaiton we can get the result within in second. Just follow the Process.
Sir please give eg:-for all formulaes and mention clearly which formulaes are coverd under basic level and which one will cover under advance level
Thank you
Hello Rajkumari, if you will going to enroll into Vedic Maths Beginner to to Advance Class, then you will going to learn all forumula with there example and application. In Beginner class you
will get basic Idea about the vedic maths.
Shared all formats with us ,it very fast easy friendly
Hello Gaouri Prasad, You can start from Here: https://vedicmathschool.org/vedic-mathematics/
Sir kindly share all examples with tricks
Leave a Comment
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Figure 7: Elementary directional encoding module
3.2 Directional distribution - Pan module
Adapting the system to a given output format simply implies replacing each elementary panning module (along with the diffuse distribution module necessary for the late reverberation). Typically, the
Pan stage should include an individual panning module for each synthetic early reflection. However, in a natural situation, the directions of the early reflections are not perceived individually.
This can be exploited in order to improve the efficiency and modularity of the mixing architecture.
Assigning to all early reflections the same direction as the direct sound would be an excessive simplification, creating the risk of a perceived coloration of the source signal in many cases. On the
other hand, a fixed diffuse distribution of early reflections would imply the loss of valuable directional cues [12]. An intermediate approach consists of selecting a subset of early reflections
coming from directions close to the direct sound and adopting a diffuse incidence approximation for other reflections. The directional group of reflections can be rendered by producing two
independent (left and right) distributions of early reflections to form a "halo" surrounding the direction of the direct sound. This is illustrated in Fig. 8 in the case of a frontal sound reproduced
over a 3/2-stereo loudspeaker arrangement.
This principle can be applied to any 2-D or 3-D encoding format, by panning the left and right early reflection signals according to the direction specified for the direct sound, so that the relative
directions of the three signals are preserved. As shown in Fig. 6, this requires only two panning modules for early reflections, instead of one per reflection. The modularity of the system is also
improved: the Room and Pan stages now appear as two independent modules associated in cascade. A general intermediate transmission format is defined, comprising a center channel (C) conveying the
direct sound, two channels (L and R) conveying directional reflections, and four additional ("surround") channels conveying diffuse reflections and reverberation. The Room module thus appears as a
reverberator directly compatible with the 3/2-stereo format, while the Pan module appears as a format converter simultaneously realizing the directional panning function.
Figure 8: Distribution of direct sound and reverberation signals on a 3/2-stereo setup (for a frontal sound).
3.3 Binaural encoding
In the particular case of binaural encoding, the Room - Pan structure of Fig. 6 still allows individual panning of each early reflection, provided that this be realized in the Room module instead of
the Pan module. Although it implies a loss of modularity, this mode can be kept as a special option for accurate auralization using binaural techniques. However, implementing a binaural panning
module requires about 140 operations per sample period, i. e. about 7 MIPS at 48 kHz (assuming that the two directional filters h[1] and h[2] are both made of a variable delay line cascaded with a
12th-order variable IIR filter) [11]. A drastic improvement in efficiency can be obtained by introducing perceptually-based approximations in the rendering of spectral cues for early reflections [11
]. The general approach consists of reducing the order of the filters h[i] (possibly down to preserving only frequency-independent time and amplitude interaural difference cues), and lumping the
remaining spectral cues in an "average" binaural filter (shown on Fig. 4) for the whole set of early reflections. Similarly, the diffuse reverberation requires a static filter simulating the
"diffuse-field head-related transfer function" as well as a 2x2 mixing matrix for controlling the interaural cross-correlation coefficient vs. frequency.
Adopting diffuse-field normalization for all binaural synthesis filters in the system eliminates the "diffuse" filter, and a further perceptually-based approximation eliminates the "average" filter
too [11]. The stereo spatial processor of Fig. 4 can thus be turned into a binaural processor essentially without modifying the reverberator itself. The only significant increase in complexity
results from replacing, in the direct sound path, the stereo panning module by a binaural one.
3.4 Equalization in listening rooms
For loudspeaker reproduction in anechoic conditions, all necessary corrections can be implemented as inverse equalization filters after mixing, and can be merged with the decoding operation in the
case of transaural or Ambisonic techniques (Out module). This includes time and spectrum alignment of all loudspeaker channels, as well as level and spectrum normalization between different
directional encoding techniques and setups. The only remaining discrepancies between situations then result from the intrinsic performance of the 3-D sound techniques, in terms of localization
accuracy and robustness of the sound image according to the position of the listener. In a practical context, however, the effects of the reverberation of the listening room must also be considered,
and compensated, if necessary, to maintain the desired control over reverberation and distance effects, irrespective of the listening conditions.
Applying the inverse filtering approach to typical rooms is impractical because it involves complex deconvolution filters and strong constraints on the listening position. However, assuming that it
is sufficient to specify the desired room response as a distribution of energy vs. time, frequency and direction, one can handle the equalization of the direct sound by inverse filtering, while the
remaining effects due to the reverberation of the listening room are corrected by deconvolving echograms instead of amplitude responses.
Rather than attempting to cancel listening room reflections, this approach takes them into account to automatically derive optimal settings for the synthetic reverberation parameters, so as to
produce the desired target echogram at the reference listening position. The implementation of this "context compensation module" (shown in Fig. 9) is simplified if the synthetic and target
reverberation -denoted R and T, respectively- are modeled by partitioning their energy distribution in adjacent time sections (Fig. 5) and frequency bands. The listening context is then
characterized, in each frequency band, by a set of energy weights C[ijk] (representing the contribution of section R[j] of the synthetic room effect to section T[k] of the target room effect, via
loudspeaker i). The coefficients C[ijk] are computed off-line from echograms measured for each loudspeaker, with an omnidirectional microphone placed at the reference listening position. The
compensation is computed in each frequency band by solving [T[k]] = [C[jk]] [R[j]] to derive the energies R[j] (the matrix inversion is straightforward since, due to causality, the matrix C is lower
triangular). The matrix C depends on the (azimuth and elevation) panning coordinates and the directional reproduction technique: C[jk] = [i][s[ij] C[ijk]], where s[ij] is the energy contribution of
section R[j] in loudspeaker i.
This technique (which can be extended to process live acoustic sources with close miking) suffers from two fundamental limitations. (1) When the existing room effect his too strong compared to the
target room effect, the inversion yields unrealizable negative energy values for some of the synthetic reverberation parameters. This could be remedied by an improved optimization algorithm including
a positivity constraint and based on minimizing a perceptual dissimilarity criterion. (2) Although the method allows controlling the global intensity of early reflections at the listening position,
their temporal and directional distribution can not be controlled exactly (this would imply extending the inverse filtering approach to early reflections). Despite these limitations, a prototype
real-time implementation (in 3 frequency bands and 4 time sections) tested in a variable acoustic room, has confirmed that this method allows a convincing simulation of a reverberant configuration in
a less reverberant one, without increasing the constraints on the listening position.
Figure 9: General structure of a Spat processor
4 Architecture and applications of a spatial sound processing software
The conventional mixing architecture, where directional localization effects and reverberation effects are rendered by independent processing units, implies strong limitations for interactive and
immersive 3-D audio. These relate to the heterogeneity of the control interface, the adaptation to various reproduction formats, and the control of reverberation and distance effects [15].
Chowning [12] introduced the principle of a higher-level perceptually-based control interface for independent control of the angular localization and the perceived distance r, resulting in a 2-D
graphic interface for simulating moving sources. Angular panning was applied to the direct sound and a fraction of the reverberation, while the distance cue involved simultaneous attenuation in
intensity of the direct sound (1/r^2) and the reverberation signal (1/r) [12]. Moore's system [16] used more sophisticated reverberation algorithms [3] and provided individual control of each early
reflection in time, intensity and direction, according to the geometry and physical characteristics of the walls of the virtual room, the position and directivity of each sound source, and the
geometry of the loudspeaker setup. Both of these designs involve controlling the pattern or distribution of the reflections according to the position of the sound source, which implies that an
independent reverberation processor must be associated to each source channel in the mixing system (or, alternatively, to each output channel [12]).
The Spat processor of Fig. 6 presents an intermediate approach: early reflections are handled separately from the later reverberation, yet not with the exhaustivity of Moore's model. The software
library includes several families of elementary modules (early, cluster, reverb, pan...) which can be combined to build reverberation and mixing systems. The Pan and Out modules (Fig. 9) can be
configured for pairwise intensity panning, B-format, stereo or binaural encoding, with reproduction over headphones or various 2-D or 3-D loudspeaker arrangements. The Room module can be implemented
in several versions differing in the number of internal channels N and the flexibility of the generic model (with or without the early and cluster modules, or sharing the reverb module between
several sources). The heaviest implementations of a complete Spat processor according to Fig. 9 involve a theoretical processing cost of 20 MIPS at 48 kHz, which is within the capacity of e. g. a
Motorola DSP56002.
A higher-level control interface (Fig. 9) provides a parametrization independent from the chosen encoding format. Reverberation settings can be derived from the analysis of measured room responses.
Reverberation and distance effects can be dynamically controlled via perceptual attributes [17, 15] (derived from earlier psycho-experimental research carried out at IRCAM), or via physically-based
statistical models. These models provide efficient alternatives to the geometrical (image source) model [16], which entails a prohibitive complexity for real-time tracking of source or listener
movements, unless restricted to particularly simple room geometries and only the first few reflections.
The Spat software can be used in a wide variety of applications due to its adaptability to various output formats and mixing configurations [15]. It is currently implemented as a collection of
modules in the FTS/Max environment, and runs in real time on Silicon Graphics or NeXT/ISPW workstations. Since its initial release [17], it has been used for musical composition and production of
concerts and installations, and in the post-production of CD recordings using 3-D sound effects. Other current applications include assisted reverberation systems for auditoria and research on
human-computer interfaces, virtual reality, and room acoustics perception.
5 Acknowledgments
The author would like to acknowledge the contribution of O. Warusfel through discussions, collaboration in the experimental work, and, with J.-P. Jullien, G. Bloch and E. Kahle, their input on
perceptual and musical aspects in the initial stages of the Spatialisateur project. This project is a collaboration between IRCAM and Espaces Nouveaux, with the support of France Telecom (CNET).
Several aspects of this work are covered by issued or pending patents.
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Kaplan-Meier Estimator - What It Is, Example, Formula, Pros, Cons
Kaplan-Meier Estimator
Last Updated :
21 Aug, 2024
What Is Kaplan-Meier Estimator?
The Kaplan-Meier estimator is a statistical technique used to estimate the probability of survival over a specific period. It is applicable in the case of time-to-event data. The method aids in
calculating the time until a particular event occurs, especially in domains like medical and life sciences.
The technique also accommodates incomplete or censored data, where not all individuals or subjects have experienced the event by the end of the study or may have dropped out. The estimator calculates
the probability of survival at specific time points by considering the observed survival times and status. Additionally, it offers valuable insights into survival patterns and enables comparisons
between different groups or treatments.
• The Kaplan-Meier estimator is a statistical method used in survival analysis to evaluate the probability of an event occurring during a particular time frame, especially when applying
time-to-event data. It is frequently used in disciplines like engineering, social sciences, and medical research.
• The estimator is adept at working with censored data. This approach makes efficient use of all the data available and generates reliable survival probabilities even in cases where some data is
• However, this approach focuses on one kind of event. It cannot handle analyses in which several types of events are involved.
Kaplan-Meier Estimator Explained
The Kaplan-Meier estimator is a statistical tool employed in survival analysis to estimate the probability of an event occurring over a specific period, especially in the case of time-to-event data.
It is extensively used in domains like medical research, social sciences, and engineering, where assessing the time until an event occurs is crucial.
The Kaplan-Meier estimator possesses the ability to handle censored data where not all individuals have experienced the event by the conclusion of the study. It often occurs due to subjects being
lost to follow-up or the study ending before all participants encounter the event. The estimator uses the observed survival times and status information to calculate the probability of survival at
various time intervals throughout the study.
Some assumptions of Kaplan-Meier Estimator include the following:
• Censoring Uniformity: This assumption suggests that the probability of being censored at any specific time should be consistent across all subjects. It implies that the data is censored randomly
and uniformly over time.
• Non-informative Censoring: This assumption states that the reason for censoring is unrelated to the survival prospects of the subjects.
• No Interactions Between Subjects: The estimator assumes that the survival or censoring times of one individual do not impact or relate to the survival or censoring times of others. Each subject's
survival time is independent of the others in the study.
The Kaplan-Meier estimator formula is as follows:
In the formula,
• S(t) is the survival probability at any particular time
• n1 is the number of subjects living at the start
• n2 is the number of subjects who died
Let us study the following examples to understand this method:
Example #1
Suppose Jenny, an analyst at Good Health Hospital, recorded the survival time of a small group of patients after a treatment. The data was as follows:
• Ryan: 10 months (died)
• Jim: 15 months (censored)
• Jake: 18 months (censored)
• Amy: 20 months (died)
To calculate the estimator, Jenny first arranged the observed times: 10,15,18, 20. Then, she calculated the survival probabilities at each observed time. Then, Jenny applied the Kaplan-Meier
estimator formula where S(t)= (n1-n2)/n1. At the start (t=0), all patients were alive, so the survival probability was (4-0)/4 = 4/4 = 1. When t=10, Ryan died, and three patients were remaining.
Thus, the survival probability was (4-1)/4 = 3/4. At t=15, Jim censored, maintaining the same survival probability of 3/4. At t=18, Jake censored, which kept the survival probability the same. At=
20, Amy died, resulting in 1 patient remaining. It made the survival probability 1/2.
Finally, Jenny plotted the probabilities results in the Kaplan-Meier survival curve. It visually represented the estimated survival probabilities over time in this small patient group after
treatment. This is a Kaplan-Meier estimator example.
Example #2
Let us assume David was monitoring the performance of loans in a portfolio to track the default occurrences. The loan of Apex Ltd. defaulted after six months, while that of Legend Software was
censored at nine months. Finally, the loan of Creative Salon defaulted after twelve months. Using the estimator, David calculated the survival probabilities at observed times. At the start, all loans
were performing, resulting in a survival probability of 1.
When the loan of Apex Ltd. defaulted at six months, two loans remained, creating a survival probability of 2 out of 3. At nine months, the loan of Legend Software was censored, maintaining the same
survival probability of 2 out of 3. At twelve months, the second default occurs, leaving one active loan and resulting in a survival probability of 1 out of 1.
Pros And Cons
The pros of the Kaplan-Meier Estimator are as follows:
• The estimator is proficient at handling censored data. It is a common occurrence in studies where not all subjects experience the event in focus or the study ends before all events occur. This
method effectively utilizes all available data and provides reliable estimates of survival probabilities even when some information is incomplete.
• The method is specifically designed for estimating survival probabilities over time. It generates a survival curve that visually illustrates the probability of an event not occurring up to a
specific time. This enables researchers to understand and compare survival experiences between different groups or treatments.
• This estimator does not assume any specific data distribution. It is robust and does not require assumptions about the shape of the survival function. This makes it highly versatile in various
research or practical scenarios.
• It allows for meaningful comparisons between different groups. The technique helps to assess if there are significant differences in survival experiences and provides valuable insights into the
effectiveness of treatments or interventions.
The cons of Kaplan-Meier Estimator are:
• The estimator faces limitations in handling time-dependent variables or changing risk factors over time. It assumes that the risk of an event remains constant over time, which might not always be
the case in real-life scenarios.
• This method primarily focuses on a single type of event. It does not accommodate analyses where there are multiple types of events occurring.
Kaplan-Meier Estimator vs Nelson-Aalen Estimator
The differences between the two are as follows:
Kaplan-Meier Estimator
• This estimator aids in estimating and visualizing survival probabilities in the presence of censored data, especially in medical and life sciences.
• The estimator is valuable for comparing survival experiences between different groups or treatments.
• It assumes non-informative censoring, implying that the reason for censoring is unrelated to the possibility of experiencing the event. Additionally, it doesn't require any assumptions about the
underlying distribution of data.
Nelson-Aalen Estimator
• The Nelson-Aalen estimator helps estimate the cumulative hazard function. It provides the cumulative sum of the hazard rates up to a particular time point, indicating the expected number of
events at that time.
• It is advantageous for time-dependent analysis and can handle changes in hazard rates over time.
• It does not assume constant hazard rates and is more flexible in accommodating varying risks.
Frequently Asked Questions (FAQs)
1. What is the minimum sample size for Kaplan-Meier?
There is no fixed minimum sample size requirement for using the Kaplan-Meier estimator. However, a larger sample size usually results in more reliable and accurate estimations. With smaller sample
sizes, the estimates may be less precise and more sensitive to individual data points or outliers.
2. What is the difference between Kaplan-Meier and hazard ratio?
The Kaplan-Meier estimator calculates the probability of survival over specific time intervals in survival analysis. It shows how long subjects survive without an event occurring. However, the hazard
ratio is a statistical measure derived from Cox proportional hazards regression. It compares the hazard or immediate rate of an event happening between two groups.
3. Can the Kaplan-Meier curve cross?
The Kaplan-Meier estimator curves may continuously decline or remain flat but never intersect or cross. They are stepwise curves that represent the probability of survival over time. Each step in the
curve indicates an event occurrence that leads to a decrease in the estimated survival probability at that specific time point. When new events occur, the survival probability cannot rise, and the
curve remains a non-decreasing function. Thus, these curves, after establishing, maintain a consistent pattern of decline or remain constant but do not cross one another.
Recommended Articles
This article has been a guide to what is Kaplan-Meier Estimator. We explain its examples, formula, assumptions, pros, cons, & comparison with Nelson-Aalen estimator. You may also find some useful
articles here -
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Mastering Data Analysis in Pandas: Mean Median and Mode - Adventures in Machine Learning
Data analysis is an essential skill for anyone working with data. In particular, analyzing data in Pandas can be an efficient way to manage and manipulate large datasets.
One important aspect of data analysis is calculating the mean, median, and mode of numerical data. In this article, we’ll look at the functions available in Pandas for calculating these statistical
measures and how they are applied in the context of basketball player data.
Data Analysis in Pandas
Pandas is a Python library specifically developed for data manipulation and analysis. It provides features for handling different types of data such as DataFrame, Series and Panel, and is
particularly useful in handling large datasets.
Pandas provides functions for many statistical measures, including mean, median, and mode. The mean of a set of numerical data is the average value.
It is calculated by adding up all the values in the dataset and dividing by the number of observations. Pandas provides the function “mean()” that calculates the mean of each column in a DataFrame.
This function can be used to quickly calculate the average value of specific numerical data. The median is the middle value in a sorted dataset, with an equal number of values above and below it.
While not as commonly used as the mean, it can be useful in certain cases. The function “median()” in Pandas calculates the median for each column in a DataFrame.
The mode is the value that appears most frequently in a dataset. The mode can be useful for examining the most common occurrence of a value within a dataset.
Pandas provides the “mode()” function to calculate the mode for each column in a DataFrame. Example of calculating mean, median, and mode for basketball player data
Let’s apply these statistical functions to basketball player data.
We will use a dataset that includes information for basketball players for a number of games. The data includes each player’s points per game, rebounds per game, and minutes per game.
To calculate the mean, median, and mode of the dataset, we can use the following syntax:
import pandas as pd
data = pd.read_csv("basketball_players.csv")
# calculate mean
mean_scores = data.mean()
print("Mean scores: n", mean_scores)
# calculate median
median_scores = data.median()
print("Median scores: n", median_scores)
# calculate mode
mode_scores = data.mode()
print("Mode scores: n", mode_scores)
The output will display the calculated mean, median, and mode for each column in the dataset, as shown below:
Mean scores:
PPG 10.80
RPG 4.14
MPG 17.04
dtype: float64
Median scores:
PPG 10.5
RPG 3.7
MPG 17.5
dtype: float64
Mode scores:
PPG RPG MPG
0 8.0 3.5 16.0
From the output, we can see that the mean number of points per game is 10.80, the median number of points per game is 10.5, and the mode number of points per game is 8.
Mean Calculation in Pandas
The “mean()” function in Pandas calculates the mean of each column in a DataFrame. However, it is important to note that this function will only work on columns with numerical data.
It will ignore any strings or non-numerical data. Here is an example of calculating the mean of specific columns in a dataset using Pandas:
import pandas as pd
data = pd.read_csv("basketball_players.csv")
# calculate mean of points per game
mean_points = data['PPG'].mean()
print("Mean points per game: ", mean_points)
# calculate mean of minutes per game
mean_minutes = data['MPG'].mean()
print("Mean minutes per game: ", mean_minutes)
The output will display the mean value for each specific column, as shown below:
Mean points per game: 10.8
Mean minutes per game: 17.04
Output examples for mean value calculation
In addition to displaying the calculated mean result, we can use the functions “describe()” and “info()” to provide additional information about the data. The “describe()” function provides
statistical information on each column, such as the count, mean, standard deviation, minimum value, and maximum value.
Here is an example of using Pandas to calculate the mean of each column and provide a statistical summary of the data:
import pandas as pd
data = pd.read_csv("basketball_players.csv")
# calculate mean
mean_scores = data.mean()
print("Mean scores: n", mean_scores)
# provide additional information
print("nSummary statistics:")
The output will display the mean value for each column as well as statistical information on each column, as shown below:
Mean scores:
PPG 10.80
RPG 4.14
MPG 17.04
dtype: float64
Summary statistics:
PPG RPG MPG
count 10.000000 10.000000 10.000000
mean 10.800000 4.140000 17.040000
std 3.371396 1.845898 3.400396
min 5.000000 1.700000 12.000000
25% 9.125000 3.175000 15.925000
50% 10.500000 3.700000 17.500000
75% 12.750000 5.075000 20.225000
max 15.000000 7.700000 22.000000
In this article, we have discussed the basics of data analysis in Pandas, specifically focusing on calculating the mean, median, and mode of numeric data. These functions are crucial for
understanding and interpreting numerical data, and can be used in a variety of different contexts.
By following the examples provided, you should be able to begin working with these functions yourself and conducting your own data analysis in Pandas.
3) Median Calculation in Pandas
In statistics, the median is the middle value in a dataset when the data is arranged in ascending or descending order. It is a statistical measure used to represent the midpoint value of a set of
data, which avoids issues with outliers that can affect the accuracy of the mean.
In Pandas, the “median()” function can be used to calculate the median of each column in a DataFrame.
Syntax for calculating median of numeric columns in a DataFrame
To calculate the median value of numeric columns in a DataFrame, we can use the “median()” function in Pandas. The syntax for using this function is as follows:
import pandas as pd
data = pd.read_csv("example_data.csv")
median_values = data.median()
print("Median values: n", median_values)
In this example, we first import the Pandas library and then read in a CSV file containing our data. We then use the “median()” function to calculate the median value of each column in the DataFrame.
Finally, we use the “print()” function to display the median values calculated.
Output examples for median value calculation
The median value calculated by the “median()” function is an important summary statistic that helps us understand the central tendency of our data. In combination with other statistics such as mean
and standard deviation, the median can provide a more accurate representation of the distribution of our data.
Here is an example of using Pandas to calculate the median of each column in a dataset:
import pandas as pd
data = pd.read_csv("example_data.csv")
# calculate median
median_data = data.median()
# display output
print("Median Values of the Data:n", median_data)
The output for the above code block will be as follows:
Median Values of the Data:
A 25.5
B 24.5
C 15.5
D 25.0
dtype: float64
Here we can see that the median value of column A is 25.5, the median value of column B is 24.5, the median value of column C is 15.5, and the median value of column D is 25.0.
4) Mode Calculation in Pandas
The mode is a statistical measure that represents the most commonly occurring value in a dataset. The mode is the value that appears most frequently in a set of data, making it an essential tool in
understanding the underlying distribution of the data.
Pandas provides the “mode()” function to calculate the mode of each column in a DataFrame.
Syntax for calculating mode of numeric columns in a DataFrame
To calculate the mode of numeric columns in a DataFrame, we can use the “mode()” function in Pandas. The syntax for using this function is as follows:
import pandas as pd
data = pd.read_csv("example_data.csv")
mode_values = data.mode()
print("Mode values: n", mode_values)
In this example, we first import the Pandas library and then read in a CSV file containing our data. We then use the “mode()” function to calculate the mode of each column in the DataFrame.
Finally, we use the “print()” function to display the mode values calculated.
Output examples for mode value calculation
Like median and mean, the mode can provide important information about the central tendency of our data. By calculating the mode of our data, we can identify the most frequently occurring values or
patterns in our dataset, which can be useful in understanding and predicting future trends.
Here is an example of using Pandas to calculate the mode of each column in a dataset:
import pandas as pd
data = pd.read_csv("example_data.csv")
# calculate mode
mode_data = data.mode()
# display output
print("Mode Values of the Data:n", mode_data)
The output for the above code block will be as follows:
Mode Values of the Data:
A B C D
Here we can see that the mode value of column A is either 23 or 24, the mode value of column B is either 10 or 23, the mode value of column C is either 2 or 6, and the mode value of column D is
either 14 or 25. Since there can be multiple modes in a dataset, Pandas displays all possible modes in the output as a DataFrame.
Data analysis is a critical skill that can help uncover valuable insights and make informed decisions. In this article, we explored the syntax and output examples for calculating the median and mode
of numeric columns in a DataFrame using Pandas.
Understanding these statistical measures can help us gain a deeper understanding of the underlying distribution of our data, and can be used to identify trends and patterns that may be hidden within
the data. By using the examples and syntax provided in this article, you can begin to apply these tools in your own data analysis projects.
5) Additional Resources for Pandas
Pandas is a versatile library that provides a wide range of functions for manipulating and analyzing data in Python. In addition to calculating mean, median, and mode, there are many other commonly
used operations that can be performed using Pandas.
Explanation of other common operations in Pandas
1. Handling Missing Data – Missing data is common in real-world datasets.
2. Pandas provides functions for identifying and handling missing data, such as the “isna()” and “dropna()” functions. 2.
3. Grouping Data – Grouping data is a powerful operation that allows you to create subsets of your data based on one or more criteria. Pandas provides the “groupby()” function for grouping data
based on specific columns.
4. Merging and Joining Data – Often, data is split across multiple files or tables.
5. Pandas provides functions such as “merge()” and “join()” to combine data from multiple sources. 4.
6. Reshaping Data – Sometimes you may need to reshape your data to better fit your analysis. Pandas provides functions for pivoting data (for example, converting row data to column data) and
“melting” data (for example, combining multiple columns into one).
7. Applying Functions to Data – Often, you may need to apply a custom function to your data.
8. Pandas provides the “apply()” function for applying a given function to each element in a DataFrame. 6.
9. Working with Time Series Data – Pandas has extensive capabilities for working with time series data. This includes functions for handling dates and times and for creating time-based subsets of
Tutorials and Additional Resources
There are a variety of resources available for learning more about Pandas. The official Pandas documentation is an excellent place to start.
It provides detailed documentation on all of the functions and features of the library, as well as numerous examples and tutorials. For those new to Pandas, there are many online tutorials available.
Some popular options include:
1. Pandas Documentation – The official documentation provides a wide range of tutorials and examples.
2. DataCamp – DataCamp provides a comprehensive Pandas course that covers everything from simple data operations to more advanced data wrangling.
3. Kaggle – Kaggle offers a variety of Pandas tutorials and notebooks, as well as datasets to practice with.
4. RealPython – RealPython provides a beginner-friendly introduction to Pandas, with step-by-step instructions and clear examples.
5. YouTube – YouTube has many tutorials available for Pandas, from beginner to advanced levels.
Some popular channels include Corey Schafer and Keith Galli. In addition to these resources, Pandas has a large and active community, with many forums and discussion groups available for asking
questions and seeking help.
Some popular options include the Pandas Google Group and the Stack Overflow Pandas tag.
Pandas is a powerful and versatile library that provides numerous functions for manipulating and analyzing data in Python. In addition to the basic statistical measures such as mean, median, and
mode, there are many other common operations that can be performed using Pandas.
By exploring the tutorials and resources available and experimenting with different operations, you can become proficient in working with Pandas and find valuable insights in your data. In summary,
this article explored the basics of data analysis in Pandas, focusing on calculating the mean, median, and mode of numeric data.
We also covered additional common operations in Pandas, such as handling missing data, merging and joining data, grouping data, and reshaping data. Finally, we provided additional resources and
tutorials for learning more about working with Pandas.
It is important to understand these statistical measures and common operations in order to gain a deeper understanding of the underlying distribution of data and uncover valuable insights. By
following the examples and utilizing the resources provided, readers can become proficient in working with Pandas and improve their data analysis skills.
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Geometric constructions are made with the aid a ruler or straightedge and compasses only. Drawings are made with the aid of additional instruments, such as protactor and ruler.
Construction of a Circle when its Radius is Known
Suppose you are asked to draw a circle of radius 3 cm. do as follows :
STEP 1.
Mark a point C with your pencil. This point will be the centre of the circle.
STEP 2.
Open the compass for the required radius, i.e., 3 cm, by putting the steel point on C and opening the pencil upto 3 cm.
STEP 3.
Hold the paper with one hand and swing the pencil leg of the compass around to draw a circle.
Construction of a Line Segment of a Given Length :
Suppose you have to draw a line segment 5.3 cm long.
Method 1. using ruler only
1. Mark any point in your exercise book and label it as A.
2. Place the ruler in such a way that the zero mark on the ruler coincides with A.
3. Now count 5 complete centimetres and 3 small divisions after the 5 cm mark and mark a point corresponding to this division on the exercise book.
4. Join A to this point as shown.
5. Label the second point as B.
Then AB is the required segment of length 5.3 cm.
Method 2. Using ruler and compass
STEP 1.
Draw any line segment which is longer than 5.3 cm.
STEP 2.
Mark a point on this line near one end as shown. Label it A.
STEP 3.
Use your compass to measure 5.3 cm on your ruler.
STEP 4.
Put the point of the compass on the line segment at A and draw an arc to cut the line as shown.
Then AB = 5.3 cm.
To construct a line segment congruent to a given line segment AB
STEP 1.
Draw a ray through any end point C. Open your compass so that the metal tip is on A and the pencil point is on B.
STEP 2.
Keep the compass opening same. Put the metal tip on the end point C of the ray and mark off a line CD congruent to AB.
Perpendicular Lines
Perpendicular lines are lines that intersect at right angles. The symbol ^ means "is perpendicular to".
Drawing Perpendicular Using Ruler and a Set- Square
CASE 1 : To constant a line perpendicular to a given line l at a point P lying on it.
STEP 1.
Place a ruler on the paper with one of its long edges lying along the line l.
STEP 2.
Holding the ruler fixed, place a set-square ABC with the arm AC of its right angle A in contact with the ruler.
STEP 3.
Slides the set-square along the edge of the ruler until A conincides with P.
STEP 4.
Holding the set-square fixed in this position, draw with a sharp pencil a line PQ along the edge AB.
Then PQ is the required line perpendicular to the line l.
CASE 2 : To construct a line perpendicular to a given line l and passing through a given point P lying outside the given line.
SETP 1.
Place either of the set-squares so that one edge AB of the right angle A lies along l.
STEP 2.
Now hold the set-square fixed and place a ruler along the edge opposite to the right angle of the set-square.
STEP 3.
Holding the ruler firmly, silde the set-square along the ruler until the edge AC passes through the given point P.
STEP 4.
Draw line PQ along the edge AC of the set-square.
Then PQ is the required line perpendicular to the given line. l, through the point P not lying on it.
To Draw a perpendicular to a Given Line with a Rular, and compass :
CASE 1. At a point on the line.
Let AB be a given line and P be the point on it.
SETP 1.
With P as centre and any suitable radius draw an arc to cut the line AB at points M and N.
SETP 2.
With M and N as centres and radius of more than half MN, draw two arcs to cut at Q.
SETP 3.
Join PQ.
Then ray PQ is the perpendicular to the line AB at P.
CASE 1. From a point outside the line.
STEP 1.
With P as centre and a suitable radius, draw an arc to cut the line l at X and Y.
STEP 2.
With X and Y as centres and a radius of more than half XY, draw two arcs to cut at M.
STEP 3.
Join PM. Then PM PERPENDICULAR l.
The Perpendicular Bisector of a Line segment :
In a plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment at its midpoint, Line l is the perpendicular bisector of segment AB.
Construction of Perpendicular Bisector of a Segment :
Using ruler and compass, to construct the perpendicular bisector of a given line segment.
STEP 1.
Open the legs of compass to more than half the length of AB. With A as centre (i.e., place the metal-tip of compasses at A), draw arc 1.
STEP 2.
With B as centre and the same radius (i.e., the same opening of the compass), draw arc 2 to cut the first arc. Name the points of intersections as P and Q.
STEP 3.
Draw the line through P and Q by joining P, Q. This line bisects the given line segment AB and is called the bisector of AB.
Let PQ cut AB at M. Then M is called the middle point or simple midpoint of AB.
The line PQ is the perpendicular bisector or the right bisector of AB.
To Construct an Angle Equal to a Given Angle ABC :
STEP 1.
Draw any ray QR. This ray will become one side of the angle and its end point Q will become the vertex of the angle.
STEP 2.
Put the metal tip of your compass on the vertex of ÐABC.
Draw an arc.
STEP 3.
Without changing the opening of the compass, put the metal tip of the compass on Q. Draw an arc of sufficient length which crosses the ray as shown.
STEP 4.
Open the compass so that the metal tip and pencil point are on the points where the arc cuts the arms of ÐABC.
STEP 5.
Without changing the opening of the compass put the metal tip on the point where the arc cuts QR. Draw another arc that crosses the previous arc at, say, P.
STEP 6.
From point Q draw a ray through the intersection of two arcs, then ÐPQR = ÐABC. Check your construction with your protractor.
To Bisect a Given Angle ABC
STEP 1.
With B as centre and a suitable radius, draw an arc that intersects BA and BC. Name the points of intersection as P and Q.
STEP 2.
With P as centre and a radius greater than half PQ draw an arc.
STEP 3.
With Q as centre and the same radius draw another arc to cut the first arc. Name the point of intersection of the two arcs as R.
Join BR. Ray BR bisects ÐABC. Ray BR is called the angle bisector. Check your result with a protractor.
Angles of Special Measures
ANGLE OF 60º
STEP 1.
Draw a ray OX.
STEP 2.
With O as centre and any convenient radius draw an arc above OX, and also cutting OX at A.
STEP 3.
With A as centre and the same radius, draw another arc to cut the first arc at B.
STEP 4.
Join OB. Then ÐAOB = 60º
ANGLE OF 30º
STEP 1.
Draw a ray OA.
STEP 2.
With O as the vertex, construct ÐAOB of 60º.
STEP 3.
Bisect ÐAOB. OC is the bisector. Then, ÐAOC = 30º, ÐCOB = 30º
ANGLE OF 120º
STEP 1.
Draw a ray OA.
STEP 2.
With O as centre and any convenient radius draw an arc to cut OA at P.
STEP 3.
With P as centre and the same radius draw another arc to cut the first arc at Q.
STEP 4.
With Q as centre and the same radius draw another arc to cut the first arc at R.
STEP 5.
Draw the ray OB through O and R. then ÐAOB = 120º
ANGLE OF 150º
STEP 1.
Draw a line AB.
ANGLE OF 90º
STEP 2.
With any vertex O on AB, construct ÐBOC of 120º
STEP 3.
Bisect ÐAOC. Ray OD is the bisector.
Then ÐBOD = 150º
ANGLE OF 90º
STEP 1.
With A as centre and any suitable radius draw an arc cutting AB at P.
STEP 2.
With P as centre and the same radius as before cut the arc of Step 1 at Q. With Q as centre and the same radius cut the arc again at R.
STEP 3.
With Q and R as centres and any convenient radius (same for both) draw arcs cutting at S. Join A to S and produce A to L. Then ÐBAL = 90º., i.e., AL is perpendicular to AB at A.
ANGLE OF 45º
STEP 1.
Construct an angle AOB of 90º as in the previous construction.
STEP 2.
Bisect ÐAOB. Let OC be the angle bisector. Then ÐAOC = 45º; ÐCOB is also = 45º
ANGLE OF 135º
STEP 1.
Draw a line AB.
STEP 2.
With any point O on line AB as vertex, construct ÐAOC = 90º. Then ÐBOC is also = 90º.
STEP 3.
Bisect ÐBOC. Ray OD is the bisector.
Then, ÐAOD = 90º + 45º = 135º
Q.1 Draw a circle of radius 3.5 cm.
Q.2 Draw a circle of radius 4.5 cm. with the sam centre, draw two more circles of radii 3.8 cm and 3 cm. What special name do you give to these circles ?
Q.3 Draw a circle of any radius, say 4 cm. Draw any two of its diameters. Join the ends of these diameters.
What figure do you obtained if the diameters are perpendicular to each other.
Q.4 Draw the line segments whose measure are :
(i) 7.3 cm (ii) 8.5 cm
Q.5 Construct a line segment of length 10 cm. From this cut a segment AC of length 4.6 cm. Measure the remaining segment.
Q.6 Draw a line segment AB = 8 cm. Mark a point P on AB such that AP = 4.5 cm. Draw a ray perpendicular to AB at P by
(i) Using set-squares
(ii) using compass
Q.7 Draw a line LM and take a point P not lying on it. Using set squares, construct a perpendicular from P to the line LM.
Q.8 Draw a circle of diameter 7 cm. Draw another diameter perpendicular to the first diameter. What figure is formed by joining the ends of these diameters ?
Q.9 Draw a segment of the length given. Construct its perpendicular bisector.
(a) 6 cm (b) 8.7 cm (c) 98 mm
Q.10 Draw a circle of radius 3.8 cm. Mark any three points P, Q, R on the circumference. Construct the perpendicular bisectors of PQ and QR. Where do the two bisectors meet ?
Q.11 Use a protractor to draw angles of :
(A) 48º (B) 75º (C) 122º (D) 118º
Q.12 With compasses and a ruler, construct each of the following angles :
(a) 60º (b) 30º (c) 90º (d) 45º
(e) 22 (f) 75º (g) 135º (h) 150º
(i) 120º
1. C 2. D 3. D 4. D
5. C 6. D 7. A 8. C
9. D 10. C 11. B 12. C
13. B 14. A 15. D 16. C
17. C 18. C 19. A 20. B
1. (i) (l, m) (m, n) (l, n)
(ii) (l, r) (m, r) (n, r) (l, q) (m, q) (n, q) (p, l) (p, m) (p, n) (p, q) (p, r)
(iii) (m, p)
2.ÐDCM, ÐMCN, ÐNCB, ÐDCN, ÐMCB, ÐDCB
3.Lines which are concurrent
(i) At A are DA, CA, AB
(ii) At O are BD, AC, RP, SQ
(iii) At B are DB, CB, AB
4.(i) The side apposite to vertex P in DPQR is QR
(ii) The altitude from vertex P, in DPQR is PT
(iii) The angle opposite to side PQ, in DPQT is ÐPTQ
(iv) The vertex opposite to side PR in DPQR is Q
(v) The median from vertex P in DPQR is PS
5. (i)
6. (i) OB, OM, OL (ii) radii, outer
(iii) diameter, inner (iv) diameter, outer
(v) concentric (vi) semicircle, inner
(vii) sector, outer
7. (i) False (ii) True (iii) True (iv) False
(v) True (vi) True (vii) True (viii) False
8. (i) Circumference (ii) Radius
(iii) Chord (iv) Center
(v) Diameter (vi) Arc
(vii) Sector (viii) Segment
9. Open figure: (i) and (iii)
Close figure: (ii), (iv) and (v)
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A new planet is discovered. Planet X is observed to orbit the sun every 300 years. What is the semi-major axis of Planet X's orbit in AU? | HIX Tutor
A new planet is discovered. Planet X is observed to orbit the sun every 300 years. What is the semi-major axis of Planet X's orbit in AU?
Answer 1
Data yet to be confirmed: Orbit period is about 15000 years, Perihelion = 200 AU and aphelion 600 - 1200 AU. For these tentative (imprecise )data, semi-major axis is half-sum = 400 to 700 AU.
The orbit period for this distant planet from Neptune may be much longer than Neptune's 165 years (see discoverers' articles in Science Magazine).
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Answer 2
To calculate the semi-major axis of Planet X's orbit in astronomical units (AU), you can use Kepler's third law of planetary motion, which states:
[T^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3]
• (T) is the orbital period in years,
• (G) is the gravitational constant,
• (M_1) and (M_2) are the masses of the two bodies (in this case, the mass of the Sun and the mass of Planet X),
• (a) is the semi-major axis of the orbit.
Given that (T = 300) years and (M_1 \gg M_2) (the mass of the Sun dominates), we can simplify the equation to:
[a^3 = \frac{G \cdot T^2 \cdot M_1}{4\pi^2}]
[a = \left(\frac{G \cdot T^2 \cdot M_1}{4\pi^2}\right)^{1/3}]
Substituting the known values, including (G = 6.674 \times 10^{-11} , \text{m}^3 , \text{kg}^{-1} , \text{s}^{-2}) and (M_1 = 1.989 \times 10^{30} , \text{kg}):
[a = \left(\frac{(6.674 \times 10^{-11} , \text{m}^3 , \text{kg}^{-1} , \text{s}^{-2}) \cdot (300 , \text{years})^2 \cdot (1.989 \times 10^{30} , \text{kg})}{4\pi^2}\right)^{1/3}]
[a \approx \left(\frac{(6.674 \times 10^{-11}) \cdot (300^2) \cdot (1.989 \times 10^{30})}{4\pi^2}\right)^{1/3} , \text{m}]
[a \approx (8.88 \times 10^{25})^{1/3} , \text{m}]
[a \approx 7.31 \times 10^{8} , \text{m}]
Converting meters to astronomical units (1 AU = (1.496 \times 10^{11}) meters):
[a \approx \frac{7.31 \times 10^{8}}{1.496 \times 10^{11}} , \text{AU}]
[a \approx 4.88 , \text{AU}]
So, the semi-major axis of Planet X's orbit is approximately 4.88 AU.
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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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Maple Worksheets: 1-dimensional wave equation
Maple worksheets on the Lambert W function
More calculus topics:
The following Maple worksheets can be downloaded.
They are all compatible with Classic Worksheet Maple 10.
The Lambert W functions - lambertW.mws
☆ Solving equations of the form a^x=b*x+c
☆ The Lambert W functions
☆ Formulas involving the Lambert W functions
☆ Solution of exp(x)=x+c using the Lambert W functions
☆ Solution of a^x=b*x+c using the Lambert W functions
The Lambert W functions .. II - lambertW2.mws
☆ Iteration of the exponential function phi(x)=c^x
☆ Solution of c^x=x using the Lambert W functions
☆ Fixed points of phi(x)=c^x
☆ Graphical illustration of iterations of the exponential function phi(x)=c^x
The Lambert W functions .. III - lambertW3.mws
☆ Solving equations of the form x^y=y+c
The Lambert W functions .. IV - lambertW4.mws
☆ A differential equation with solution x^y=y
☆ An exact differential equation with solution x^y=y+c
The Lambert W functions .. V - lambertW5.mws
☆ Solution of the equation x^y=y^x - introduction
☆ Solution of ln(x) = c*x
☆ The function g(x)=ln(x)/x and its inverses
☆ A function chi : x -> y for which x^y=y^x
☆ The derivative of chi(x)
☆ The 2nd derivative of chi(x)
☆ Higher derivatives of chi(x)
The Lambert W functions .. VI - lambertW6.mws
☆ Some 1st order differential equations with solutions which involve Lambert W functions
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Temple University
Department of Economics
Introductory Econometrics
Serially Correlated Errors Homework
The data for this homework consists of monthly data for the period 1981 through 1989, for a total of 108 observations. The data is for California. The dataset is traffic2.wf1. There is also a data
description file.
1. During what month and year did California's seatbelt law take effect?
2. Is California's a primary or a secondary seatbelt law? You will have to look this up on the WWW.
3. When did the highway speed limit increase to 65 MPH?
4. Regress the variable log(totacc) on a time trend ('t' in the dataset) and 11 monthly dummies.
a. Would you say that there is seasonality in total accidents? How do you know?
b. What is the meaning of the coefficient on the time trend?
5. Add to the regression from part 4. the variables wkend, unem, spdlaw and beltlaw.
a. Does the coefficient on unem make sense in terms of sign and magnitude?
b. Are the coefficient estimates for spdlaw and beltlaw what you expected? Why?
6. Repeat part 4., but use log(prcfat), which is pecent of accidents that result in a fatality, instead of log(totacc). Do your conclusions regarding seasonality and time trend change?
7. Go back to the model of with log(totacc) on the left hand side and the time trend, seasonal dummies, wkend, unem, spdlaw and bltlaw on the RHS. Save the residuals by entering the command series
res01=resid in the command window, the big white field at the top of the workfile space. Now run the regression res01[t] = b[0] + b[1] res01[t-1] + v[t]. At the 5% level test the null hypothesis
that the coefficient on res01[t-1] is different from zero.
8. For the original model of part 7., in the estimation dialogue box of EVIEWS add the term AR(1) and estimate the model coefficients. Is the estimate of the coefficient on AR(1) the same as the
coefficient on res01[t-1] of part 7? If there is a difference, how do you account for the difference?
9. Redo part 7., but also include the set of independent variables on the RHS of res01[t] = b[0] + b[1] res01[t-1] . Does your new estimate of b1 differ from your answers in parts 7 and 8?
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How to Calculate Maturity Value | Sapling
Maturity value refers to the value of an interest-paying investment when its time paying interest is through. You can calculate maturity value for bonds, notes and some bank products such as
certificates of deposit. Remember to take into account how frequently interest is compounded on the account and to use the proper interest rate corresponding to that, whether annual, monthly or
something else.
Understanding Maturity Value
While certain interest-paying investments, such as traditional bank accounts, may have interest forever, others have a fixed date at which they will return your principal, or original investment, and
interest and stop paying out. That date is known as the investment's maturity date.
The investment's total value on that date is known as the maturity value. This value is, by definition, the sum of the original principal plus all the interest that has been paid out. You may have
this value spelled out in the terms of the investment and you may be able to have the organization issuing the investment opportunity spell it out. You can also calculate it yourself using an online
calculator tool or a relatively simple formula.
Calculating Maturity Value
To calculate maturity value, you must know the initial principal on the investment, how frequently interest is compounded and what the interest rate per compounding period is. Compounding interest
refers to the process of adding it to the principal for purposes of determining how much interest to pay in the future, and different investments can compound interest on different schedules, whether
daily, monthly or annually.
Once you have that information, use the formula V = P * (1 + r)^n, where P is the initial principal, n is the number of compounding periods and r is the interest rate per compounding period.
For example, if you have an account that pays 5 percent interest compounded annually with a maturity date in three years and a principal of $1,000, the maturity value is V = 1000 * (1 + 0.06)^3 =
$1,191.016, which normally rounds to $1,191.02.
Converting Interest Rates
If interest is compounded more or less frequently than annually, but the interest rate is an annual rate, you will need to convert it to the appropriate period. For example, if that same account
compounds interest monthly rather than annually, you would convert the 6 percent annual interest rate to 6 percent / 12 = 0.5 percent = 0.005 monthly interest rate. Then, over those three years, you
will have 36 compounding periods rather than just three.
That makes the value formula give the result V = 1000 * (1 + 0.005)^36 = $1,196.68. Notice that the more frequent compounding means more interest paid out, which can make a difference if long periods
of time or large amounts of money are in play.
Using with Bank Accounts
An ordinary savings account does not have a true maturity date, since the bank doesn't close your account and pay you back after a certain period of time. But if you want to know how much money will
be in your account as of a certain day, you can use the maturity value formula based on how much money is in your account, how frequently interest is compounded and what your interest rate is.
One complexity with bank accounts is that you often put more money in over time or take money out to spend or transfer to other investments, unlike a bond or a certificate of deposit where the amount
of money generally stays the same over time. Another issue that many bank accounts have fluctuating rather than fixed interest rates, so you will not necessarily get the same rate over time, limiting
the formula's applicability.
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Structure and classes of the admtools
Structure and classes of the admtools package
Niklas Hohmann
This vignette provides an overview of the larger scale structure of the admtools package and the classes used therein.
S3 Classes
S3 class adm
The S3 class adm represents age depth models. Structurally, they are lists with five fields:
• t : numeric vector, time points
• h : numeric vectors, heights
• destr: logical vector, is the interval destructive
• T_unit : NULL or a string, time unit
• L_unit : NULL or a string, length unit
h[i] is the height at time t[i]. h and t must be of same length and have at least 2 elements, t must be strictly increasing and h must be nondecreasing. length(destr) must be identical to length(t) -
1. destr[i] == TRUE indicates that the time interval from t[i] to t[i+1] is destructive and no information is preserved. Whether tie points are considered destructive is determined by the function
is_destructive. Geologically, destr[i] == TRUE should imply h[i] == h[i+1] , as no sediment is accumulated during hiatuses.
The following functions construct adm objects:
• tp_to_adm for construction from tie points
• sac_to_adm for construction from sediment accumulation curves sac
• split_multiadm for extraction from multiple age-depth models multiadm
The following functions examine the logic of adm objects:
• is_adm to check for validity of an adm object
The following functions yield a representation of adm objects:
• plot.adm for plotting
• print.adm for printing to the console
• summary.adm to provide a quick summary of an object
The following functions modify adm objects:
• set_L_unit and set_T_unit to change units
Transformation into other S3 classes
The following functions transform adm objects into other S3 classes:
• merge_adm_to_multiadm into multiamd
• add_adm_to_multiadm to add adm to multiadm
• anchor to transform anchor adm at a tie point with uncertainty, resulting in a multiadm
S3 class sac
The S3 class sac represents sediment accumulation curves. Structurally, they are lists with four fields:
• t : numeric vector, time points
• h : numeric vectors, heights
• T_unit : NULL or a string, time unit
• L_unit : NULL or a string, length unit
h[i] is the height at time t[i]. h and t must be of same length and have at least 2 elements, t must be increasing.
The following functions construct sac objects:
Standard constructor is tp_to_sac (tie point to sediment accumulation curve)
The following functions inspect the logic of sac objects:
• is_sac to check validity of sac object
The following functions yield a representation of sac objects:
• plot.sac for plotting
• print.sac for printing to the console
• summary.sac to provide a quick summary
The following functions modify sac objects:
• set_L_unit and set_T_unit to change units
Transformation into other S3 classes
The following functions transform sac objects into other S3 classes:
• sac_to_adm to transform sac into the S3 class adm
S3 class multiadm
The S3 class multiadm represents multiple age depth models.. Structurally, they are lists with the following elements:
• no_of_entries: Positive integer, number of age depth models contained
• t list of length no_of_entries. Each element contains a numeric vector
• h: list of length no_of_entries. Each element contain a numeric vector
• destr: list of length no_of_entries. Each element contain a logic
• T_unit : NULL or a string, time unit
• L_unit : NULL or a string, length unit
h[[i]][j] is the height of the i-th age-depth model at time t[[i]][j]. For each i, the quintuple h[[i]], t[[i]], destr[[i]], T_unit and L_unit specify an adm object with the constraints as specified
in the section S3 class adm (e.g., on monotonicity, length, etc.). T_unit and L_unit are shared among all age-depth models in an multiamd object.
The following functions construct multiadm objects:
• anchor to construct multiadm from uncertain tie points and adm objects.
• merge_adm_to_multiadm to construct multiadm from adm objects
• sedrate_to_multiadm construct multiadm from info on sedimentation rates, see vignette("adm_from_sedrate")
• strat_cont_to_multiadm construct multiadm from tracer information, see vignette("adm_from_trace_cont")
The following functions inspect the logic of multiadm objects:
• is_multiadm to check for validity of multiadm object
The following functions yield a representation multiadm objects:
• plot.multiadm for plotting
• print.multiadm for printing to console
• summary.multiadm for providing summary statistics
The following functions modify multiadm objects:
• merge_multiadm to combine multiple multiadm objects
• set_L_unit and set_T_unit to change units
Transformation into other S3 classes
The following functions transform multiadm objects into other S3 classes:
• split_multiadm to split multiadm into list of adm objects
• mean_adm , median_adm and quantile_adm to extract mean, median, and quantile age-depth model of adm class.m
S3 classes stratlist and timelist
stratlist and timelist inherit from the base list. They are list of stratigraphic positions or times, associated with other data (e.g. trait values, proxy values)
• stratlist is a list with one element named “h”
• timelist is a list with one element named “t”
• stratlist is returned by time_to_strat.list
• timelist is returned by strat_to_time.list
• plot.stratlist for plotting stratlist
• plot.timelist for plotting timelist
Methods implemented for external S3 classes
S3 class list
admtools implements the following methods for list:
• strat_to_time.list: Transform strat-val pairs into time domain
• time_to_strat.list: Transform time-val pairs into strat domain
S3 class phylo
admtools implements the following methods for phylo:
• strat_to_time.phylo: Transform stratigraphic tree into time domain
• time_to_strat.phylo: Transform time tree into strat domain
Class numeric
• strat_to_time.numeric: Transform vectors from stratigraphic domain to time domain. Wrapper around get_time
• time_to_strat.numeric: Transform vectors from time to stratigraphic domain. Wrapper around get_height
The following functions are used for plotting:
• plot.adm plotting for S3 class adm
• plot.multiadm plotting for S3 class multiadm
• plot.sac plotting for S3 class sac
• plot.timelist for plotting timelist
• plot.stratlist for plotting stratlist
• T_axis_lab to annotate time axis
• L_axis_lab to annotate length/depth axis
• plot_sed_rate_l to plot sedimentation rate in length/depth domain
• plot_sed_rate_t to plot sedimentation rate in time domain
• plot_condensation
• plot_erosive_intervals to highlight erosive intervals, called after plot.adm
The following functions are used internally and not exposed to users. They can be accessed via admtools:::function_name.
• plot_destr_parts
• plot_acc_parts
• make_adm_canvas
• browseVignettes(package = "admtools") to show a list of vignettes
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EViews Help: nrnd
Generate random normal draws.
The nrnd command fills series, vector, and matrix objects with (pseudo) random values drawn from a standard normal distribution. When used with a series, the nrnd command ignores the current sample
and fills the entire object.
Fill object_name with normal random numbers.
matrix(10, 3) m1
For random generator functions, see
“Statistical Distributions”
and in particular,
, and
For related commands, see
, and
. See also
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Sachem Capital Corp. specializes in originating, underwriting, funding, servicing, and managing a portfolio of first mortgage loans. It offers short term (i.e., three years or less) secured,
nonbanking loans (sometimes ...
How has the Sachem Capital stock price performed over time
How have Sachem Capital's revenue and profit performed over time
All financial data is based on trailing twelve months (TTM) periods - updated quarterly, unless otherwise specified. Data from
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What Is The Best Way To Describe Solution Of This System Equations - Tessshebaylo
What Is The Best Way To Describe Solution Of This System Equations
Techniques to solve a system of equations solved examples solving with 3 variables steps lesson transcript study com simultaneous worksheet graphically gcse maths solutions systems explanation review
and albert resources describe all the following chegg zero one or infinitely many math 1 use parametric using determinants two three linear geometry which best describes
Techniques To Solve A System Of Equations Solved Examples
Solving System Of Equations With 3 Variables Steps Examples Lesson Transcript Study Com
Simultaneous Equations Steps Examples Worksheet
Solving Simultaneous Equations Graphically Gcse Maths
Solutions To Systems Of Equations Explanation Review And Examples Albert Resources
Solved Describe All Solutions Of The Following System Chegg Com
Solving Equations With Zero One Or Infinitely Many Solutions Math Study Com
Solved 1 Use The Parametric Equations To Describe Chegg Com
Solving Systems Of Equations Using Determinants With Two And Three Variables
Linear Geometry And Systems
Solved Which Of The Following Best Describes All Solutions Chegg Com
Solved Describe The Solutions Of First System Equations Below In Parametric Vector Form Provide Geometric Comparison With Solution Set Second 3x1 3x2 673
Classifying Consistent Dependent Independent Inconsistent Systems Of Linear Equations Algebra Study Com
Solved Two Systems Of Equations Are Given Below For Each Chegg Com
Systems Of Equations Overview Graphs Examples Lesson Transcript Study Com
Explaining Why There Is Only One Particular Solution Of A Diffeial Equation Passing Through Point While The General May Describe Infinitely Many Solutions Calculus Study Com
Systems Of Linear Equations In Two Dimensions Infinity Is Really Big
Simultaneous Linear Equations Definition And Examples
Simultaneous Equations Overview Examples Lesson Transcript Study Com
Dependent System Of Linear Equations Overview Examples Lesson Transcript Study Com
Consistent System Of Equations Definition Graphs Examples Lesson Transcript Study Com
Solving The Linear Equation In Two Or Three Variables Using Inverse Matrix
Solution Sets In Linear Equations Overview Examples Lesson Transcript Study Com
Solve a system of equations solving with 3 simultaneous steps solutions to systems solved describe all the zero one or 1 use parametric using determinants linear geometry and chegg
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Nanoscale Computer Operates at the Speed of Light
Predictions indicate that a nanometer-sized wave-based computer could solve equations in a fraction of the time of their larger, electronic counterparts.
Booting up your laptop may seem like an instantaneous process, but in reality, it’s an intricate dance of signals being converted from analog wave forms to digital bytes to photons that deliver
information to our retinas. For most computer uses, this conversion time has no impact. But for supercomputers crunching reams of data, it can create a serious, energy-consuming slowdown. Researchers
are looking to solve this problem using analog, wave-based computers, which operate solely using light waves and can perform calculations faster and with less energy. Now, Heedong Goh and Andrea Alù
from the Advanced Science Research Center at the City University of New York present the design for a nanosized wave-based computer that can solve mathematical problems, such as integro-differential
equations, at the speed of light [1].
One route that researchers have taken to make wave-based analog computers is to design them into metamaterials, materials engineered to apply mathematical operations to incident light waves. Previous
designs used large-area metamaterials—up to two square feet ( $\sim 0.2$ ${\text{m}}^{2}$)—limiting their scalability. Goh and Alù have been able to scale down these structures to the nanoscale, a
length scale suited for integration and scalability.
The duo’s proposed computer is made from silicon and is crafted in a complex geometrical nanoshape that is optimized for a given problem. Light is shone onto the computer, encoding the input, and the
computer then encodes the solution to the problem onto the light it scatters. For example, the duo finds that a warped-trefoil structure can provide solutions to an integral equation known as the
Fredholm equation.
Goh and Alù’s calculations indicate that their nanosized wave-based computers should be able to solve problems with near-zero processing delay and with negligible energy consumption.
–Sarah Wells
Sarah Wells is an independent science journalist based outside of Washington, DC.
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6/20 In Simplest Form
Solved Evaluate, and write your answer in simplest form 7/9
6/20 In Simplest Form. You can also add, subtract, multiply, and divide fractions, as well. √6 ⋅ √20 6 ⋅ 20.
Solved Evaluate, and write your answer in simplest form 7/9
→ (5 × 9)/(6 × 4) → 45/24. 20.6/1 step 3 find how many 10s should be multiplied with both numerator and denominator. Web the simplest form of 20 6 is 10 3. The simplify calculator will then. Web if
not, write it in simplest form. Web the simplest form of 6 / 20 is 3 / 10. The mixed number calculator converts the given fractional expression to a mixed number. √6 ⋅ √20 6 ⋅ 20. Web this online
fraction simplifier calculator operatesby simplifying complex fractions and then replacing them by simplest form of fractions. Here we will show you how to simplify, reduce fraction 6/20 in its
lowest terms with step by step detailed.
Web calculators basic calculators simplest form calculator simplest form calculator simplest form calculator enter the fraction = a b simplest form = a b the simplest. Web what is 6/20 simplified?
Multiply 6 6 by 20 20. Web what is 20.6 as a fraction in simplest form? The mixed number calculator converts the given fractional expression to a mixed number. Here we will show you how to simplify,
reduce fraction 6/20 in its lowest terms with step by step detailed. Combine using the product rule for radicals. Steps to simplifying fractions find the gcd (or hcf) of numerator and denominator gcd
of 20 and 6 is 2 divide both the numerator and. The numerator is not zero. It reduces fractions rapidly and accurately. To simplify a trigonometry expression, use trigonometry identities to rewrite
the expression in a simpler form.
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Eureka Math Grade 1 Module 1 Lesson 12 Answer Key
Engage NY Eureka Math 1st Grade Module 1 Lesson 12 Answer Key
Eureka Math Grade 1 Module 1 Lesson 12 Problem Set Answer Key
Fill in the missing numbers.
Question 1.
Question 2.
Question 3.
Question 4.
_______ balls = _______ balls + _______ balls
Bob had _______ balls at the park.
As per the given data
6 balls = 2 balls + 4 balls
Bob had 4 balls at the park.
Question 5.
_______ apples + _______ apples = _______ apples
Mom gave me _______ apples.
As per given data
3 apples + 7 apples = 10 apples
Mom gave me 10 apples.
Eureka Math Grade 1 Module 1 Lesson 12 Exit Ticket Answer Key
Draw a picture, and count on to solve the math story.
Write a number sentence to match your picture.
John caught _________ fish.
John caught 7 fish.
Eureka Math Grade 1 Module 1 Lesson 12 Homework Answer Key
Use your 5-group cards to count on to find the missing number in the number sentences.
Question 1.
Question 2.
Question 3.
Use your 5-group cards to count on and solve the math stories. Use the boxes to show your 5-group cards.
Question 4.
Jack reads 4 books on Monday. He reads some more on Tuesday. He reads 7 books total. How many books does Jack read on Tuesday?
Jack reads __ books on Tuesday.
Jack reads 3 books on Tuesday.
Question 5.
Kate has 1 sister and some brothers. She has 7 brothers and sisters in all. How many brothers does Kate have?
Kate has __ brothers
Kate has 6 brothers
Question 6.
There are 6 dogs in the park and some cats. There are 9 dogs and cats in the park altogether. How many cats are in the park?
There are __ cats total.
There are 3 cats total.
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Find maximum possible number of students in a balanced team with skills - FcukTheCode
Find maximum possible number of students in a balanced team with skills
You are a coach at your local university. There are n students under your supervision, the programming skill of the i-th student is ai.
You have to create a team for a new programming competition. As you know, the more students some team has the more probable its victory is! So you have to create a team with the maximum number of
students. But you also know that a team should be balanced. It means that the programming skill of each pair of students in a created team should differ by no more than 5.
Your task is to report the maximum possible number of students in a balanced team.
The first line of the input contains integer n (1≤n≤2⋅105) — the number of students.
The second line of the input contains n integers a1,a2,…,an (1≤ai≤109), where ai is a programming skill of the i-th student.
Print the maximum possible number of students in a balanced team.
(In this example you can create a team with skills [12,17,15], all the element(programming skills) in this array have a difference of not more than 5)
def diff5(team=[]):
if(i!=j and abs(team[i]-team[j])>=5):
return team
Arr=list(map(int,input('').split(' ')))
N=len(Arr) # lol
for i in range(len(Arr)):
for j in range(len(Arr)):
if(i!=j and abs(Arr[i]-Arr[j]) <= 5):
print(max([len(i) for i in L]),"\n")
Executed using python3 linux terminal
Morae Q!
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SITO.org / Discussion / sito.synergy.gridcosm/36632: "...Christ will come at 10:00"
Like Faye Dunaway with the LAH-LAH envelope? https://...gridcosm?level=4064
-Probability rules for "Natural" or unbiased music
+movie synchronicities*-
(an Oh Dissertation/ Simultaneous Hunanamus)
((ie the undisputed facts of Film/ music synchronicities))
(*this subject applies as a preliminary and prerequisite
baseline reference for postulating any sort of probility rules
for biased or intentional sourced synchronicities. In other
words, before we specualate or postulate theories on the
culpability of sources to be intentional or biased by proxy to
a common form or rule or source or table or theory or law or
scheme, we must examine the 'natural' mathematical rules of
hypothetically unbiased sources. that is to say films and
music or an other analagous data sets that are factually
One must also obviously conclude that anyone that actually
produced 'biased' sources or intentional synching by some
rite, would have aswell done the math on exactly what is
happening when a song is playing to a movie, if any attepmt to
manage the phenomenon on a practical or acurate or
signnificant level had been truly applied. How else would you
optimize your output right? lol.
this is also exactly what a skeptic would do to explain away
synching as "unintentional"(the party line being that
intentional synching is conspiracy theory and that the high
probability of natural synchronicity supports the notion. The
problem with this cop-out is invisibly obviouse to the skeptic
because it asks those who have seen synchs that are supected
of being intentional to accept that difinitive levels of
precision are more likely accidental than on purpose. But if
it looks built, isnt it likely it was? No one wonders if a
chair is man-made or natural...)
But lets just assume that we are unbiased. lets just look at
whats going on in synchs.
so we start by understanding what happens on a hypothetical
natural. Or rather the simplest and unavoidable parts first
Synch Laws:
____________________________________________________ all possibles = (mundane + significant)
significant = (positive + negative))
The difinitive quantites of the hypothetical natural
[mun + sig = all]
[sig = pos + neg]
The binary assuptions of the hypothetical natural:
[Mun = sig]
[pos = neg]
we must assume a binary equality in order to have any sort of
refereance to an actual value.
we have to assume that since the the sources are
hypothetically unbiased then all the data of the the total
source set is, for all practical purposes to getting started,
equally likely, as they are hunanimosuly influenced by thier
own universally unbiased nature.
have to! Because we know that the likelyhood of an event being
mundane or sigificant is completly indipendant of its own
remember we are looking for the probability of a sync being
intentional or not by comparing it to a hypothetically natural
mathematically speaking, our opinions are expost facto and
dont even matter! I know its confusing defining terms but they are all equally
we arnt even looking at the probability of the observor to
find an event significant, just assuming an even potentiality in an unbiased source, where clearly the probability of categorization depends soley on the number of categories itself.
And we have yet to quantify the event. How do you measure synching anyway? Time? Number of
comonalities? Quality? What exactly is the currency?
well lets get on with the logic of purelly unintentional synch
we have defined the parts of a synch ie the initial categories
of what happens in a synch:
an interaction between lyrics and film/captions is either:
mundane (not signficant or synchy in any way, to a
hypothetical observer) OR significant(to the hypothetical
If any given moment of a synch is Significant, it is either
positive or negative, according to error theory, and interms of being applicabley true or false by implication. This too we
must assume is binarilly equal because the data in the set is
assumed to be of the same natural source, or rather equally influenced by nothing. good or bad if its a
significant event its all gasoline and it all burns at the
same rate and temp.
these assumptions have to be fair enough because the hypothetical natural is similar to an unobserved quantum state.
even tho maybe in irl we should just be
looking at the probibility of a hypothetical observer thinking
a synch is a synch?
but wait. we are attempting to set a baseline for how a non
intentional synch ought to behave so that we can compare that
to suspected intentioanl data, right?
so how can the foundation of this hypothesis be a baited
consession that things are equally good or bad? am i actually
suggesting that good and bad are equally probably?
value and quantity or not synonimous in mathematics. all units
of measurements are relative, but the rules remain the same
its the same old thing. are there more even or odd numbers?
perhaps we on start with something measurable first!?!
maybe look at TIME?
__________ when a person watches or looks for or finds a film + music
synchronicity their are particulare criteria.
assume their isnt.
what we know:
any song can be played along to any part of a film and
observers will say they were in or out of synch. one could set some average values
say a film is 100 minutes for argument sake say measuring in half seconds is a
practical evaluation. a half second margin of error isnt too
bad (alltho the acceptable margins of error in more
professional synching should be upwards of a 1/4, an 1/8 or a
1/10 of a second)
so say theres 12,000 half seconds of film on average. depending on what is synching the formulas are a lil
different when it comes to full album synchs, just
start with single songs.
(ok assume the few minutes we would have to synch to
a film is offset to itself in our average video length
variable V. its not like your gona start a synch with 1 second
left of movie. but also not with 1 second of song. its easeir
to just say its a loop and that any amount of song thats
before or past the start or end of the film is the same
position on the opposite side. so V is indeed V)
it doesnt matter 90 minutes 100 any exponential difference in
these variables based on runtime is smaller than the exponential improbabilites encountered
of the binary hypothetically unbiased model before you score
five points so, an arbitrary run time is addequate. get a grip
so why not. say the total possible number of positions = 1/6000you could do 1/5000. you could measure by the second. just get on with it.
Oh wait thats it right
there are 3 or 4 or 5 or 6000 thousand places to play a song to a
movie. that should tell us something right? so maybe a 1/5000 odds on a singularly correlated( by field error) intentional fim album synch. right. the probabilities ofthe hypothetical ntural would
have to be close to that for the common synch to be a naturally occuring thing.(*nb excepting that a metered or proxy based modular type formate could be considered 'natural' interms of the natural
mathematical realtionships of incremental data sets)
ok well its back to criteria. what makes anyone of these 6000
things a synch?
lets define some terms. a synch is a song played to a film
that has "synch events". A synch is generally contigent upon
having as little contridictory events or errors as well.
so lets look at the sych event
an event can be measured in time and maybe that gets you some
lets say our average song is 200 seconds. its easy. 3 minutes
20 seconds. so 400 half seconds
we can than say the average film has 6000 positions AND that
the average 1 song synch has 400 distictly measurable events
or moments of play. in order to be a synch some of these events have to be
significant and most have to be good and mostly not bad.
______________________________________________ so start again with the original synch formula
S = AxV - PCB/SCB
where S = the amount of time of viable synch playback, which
is equal to A the Audio length times V the video length, minus
what is disqualified by a personal or standard confirmation
( the basictenant of error theory) which says ANY possibility is infact in synch UNLESS it
is DISQUALIFIED by a personal or standard confirmation bias.
As counterinuitive as it seems it is simpler to say 'what
makes NOT a synch'. but how can this lend itself to the
hypothetical natural when its detemined by the PCB/SCB? again you might as well be looking into the probabilities of the bias itself.
but assume there is a heaven and for probability purposes its
possibile to have a song that plays at anypoint in a movie and
still seems synchronized if not intentional to the observer.
So the Confirmation Bias Formula, or ERROR THEORY, says that
if you think its synchs" = everything minus "if you think it
doesnt synch
it synchs or it doesnt, and whos to say? especially in a hypotheticall unobsereved environment. thats why yhe assumtion of equality in the unbiased natural is a fair baseline for comparison
before you can begin to evenspeculate as to weather synch events are naturally common, engineered, or are the byproduct or recidual sideeffect of metered or intervalaic systems.
forget word frequencies and language patterns and cultur and
mood all the other minutia of things one might think could
influence a synch.
just say that for the hypothetical natural, theres no
accounting for why an event may be significant nor weather it
is positive of negative if it is. It only suggests that the
probability of a type is only dpentandt on the number of types
exctly because it cant be dependant on anything else.
So mun = sig because mun + sig = all and 'all' has a unified lack of bias in itself, so its
componentn parts must also.
sig = (pos + neg) for the same reason.
weather we like it or not this is the only fair place to
stand. but wait thats only 3 categories! well call it mun = mun1 + mun2. maybe mun1 is when 2 things
dont mean shit and mun2 is an event where the audio offers no
relevant data to correlate.
either way the probabilty here is the same there are 4
categores determining probabilty of what any event might be:
50% mun 25% sig pos 25% sig neg
this is the assertion of the hypothetical natural. the reason
it is usefull is becaus it will give us a comprable baseline
to compare to other probability models.
since we have defined a synch interms of sync events in a set
of total events disqualified by errors we can formulate a
baseline probability for the number of positive synch events
as well as a lack of errors. this can tell us the
hypothetically natural improbability of a synch based on the
number of error free positive events. does this really make
remember our song has about 200 event moments
Of course one could define a significant, positive event as a
literal correlary ie the same word. this interactions happen
in fields of events and thats ok, look at what has been
of the 400 events, 200 are significant, 100 are positive and
100 are negative.?
thats doesnt seem right. isimply because some sync points take more than an instant to happen but most of them are instant. and theres not 100 of them! kinda makes you think about that old MTV 6
second attention span standard...
so what is being counted counting? if a word or a drum synchs its only for as long as its being said or played. instantaneous...
how much time is in synch? or how many events seem
How often can a person count or notice a synch event anyway?
if a song says "
" for a moment but the movie is blue for a
minute how much is that?
whats the average number of words in a song got to do with it?
and what does the potentiality of a singular postition have to do with the potentiality of the whole field? surely a a song with the name of the character in the movie has a higher chance of at
least that corelarry right?
What if we are lineing up beads instead?
if yur listening to sunglasses at night and the guy in the
movie is wearing sunglasses at night the entire time what is
Perhaps we can leave out the positives of the hypnat using
error theory.
In otherwords we have concluded that the odds of everything in a sync being right = to the odds of nothing happening at all? perhaps not.
perhaps its expontentially imrprobaly based on the number of errors consecutivly avoided.
so theres a song playing to a movie.
(perhaps sig and mun intervalwise have to be adjusted to reflect the rate of
information observed by the obserervor.)
if any hypnat event is 25% error? are there typically 100
errors a song? no.
look at the literal coerrilary itself:
nothings goin on and then something in the song and the movie
is the same. it hapens over and over in a good synch.
even if you could calculate the probabity of non-error i mean
jesus probability doesnt begin to define a synch...
but try again anyway.
if the hypnat pos and neg are assumed to be the same rate then we can count a neg for each pos we count!
1 pos = 25% chance if for every positive ther is a neg than! 3/4 of the pos are non error
if P is the number of positives in an error free synch
than each pos is only 75% as likely if P = N:
1P = 25% .25^p
2P = 6.25 3P = 1.57 4P = .394
if 1 N free per P = 75% or 25x .75^p
P(0N) = 18.75 2P = 14.06 3P = 10.54 4P = 7.91 5P = 5.93 6P = 4.44 7P = 3.33 8P = 2.50 9P = 1.87 10P = 1.40
ok ok we arnt (thinking outloud...)
the hypothetical natural applied to observed events...
so the term all = all observed. der ta der
make a distinction between our Audio interval value and
observed Adio value..
Our avg song has 400 half second conmparible events.
or 100 obs. 2 second events.
50 are mun 25 are p and 25 are n
25p ive seen 25 significant things in a sync so thats
so the hypnat would say a synch is totally probabable
but the point of the hypnat is that we have found a
probability for n relative to p.
what are the odds of 25P0N if p and n are = probable??
what are the odds if they are even lol
p = 25% n = 25% p0n = 18.75
point is you get an exponentially unlikely (less than 1 percent) probability after about 10 or 20 positive corelarries with no negating errors if you use the probabilities of the hypothetical
it dwarfs the 1/5000 "
gotta be ateast one good place to play this song to this flic) probability in terms of rareness.
i conclude that the probabilities of a hypothetical unbiases or natural set or sources shows the average error free synch is highly unlikey
as to tweather they are intentional or a byproduct of metered, intervalaic common denominatoris or an intentional proxy is unclear
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Drawing Supply and Demand curves in Excel
Introduction to Demand and Supply curves
Creating the market Demand and Supply curves from the preferences of individual producers and suppliers
How the step graph for a small market becomes a smooth curve for a larger market
Supply and Demand curves play a fundamental role in Economics. The supply curve indicates how many producers will supply the product (or service) of interest at a particular price. Similarly, the
demand curve indicates how many consumers will buy the product at a given price. By drawing the two curves together, it is possible to calculate the market clearing price. This is the intersection of
the two curves and is the price at which the amount supplied by the producers will match exactly the quantity that the consumers will buy.
The process is illustrated in Figure 1. The downward sloping line is the demand curve, while the upward sloping line is the supply curve. The demand curve indicates that if the price were $10, the
demand would be zero. However, if the price dropped to $8, the demand would increase to 4 units. Similarly, if the price were to drop to $2, the demand would be for 16 units.
The supply curve indicates how much producers will supply at a given price. If the price were zero, no one would produce anything. As the price increases, more producers would come forward. At a
price of $5, there would be 5 units produced by various suppliers. At a price of $10, the suppliers would produce 10 units.
The intersection of the supply curve and the demand curve, shown by (P^*, Q^*), is the market clearing condition. In this example, the market clearing price is P^*= 6.67 and the market clearing
quantity is Q^*=6.67. At the price of $6.67, various producers supply a total of 6.67 units, and various consumers demand the same quantity.
Figure 1
There is no reason why the curves have to be straight lines. They could be different shapes such in the examples below. However, for the sake of simplicity, we will work with straight line demand and
supply functions.
Table 1
│ Price │ Product bought │ Total demand │
│ │ by consumer │ for product │
│ More than 20 │ None │ 0 │
│ 20 │ A │ 1 │
│ 15 │ B │ 2 │
│ 10 │ C │ 3 │
│ 8 │ D │ 4 │
│ 3 │ E │ 5 │
In the examples above, the chart contained smooth curves. While such a curve is an excellent approximation when there are many producers (or consumers), each of the curves is actually made up of many
small discrete steps. Each of these steps represent the decision of a single individual (or company). We will see next how these curves are constructed based on the decisions made by individual
We construct the demand and supply curves for a very small market. Suppose there are just 5 consumers and each demands one unit of the product. However, they have distinct prices at which the product
is valuable enough for them to buy it. Table 1 shows the price at which each individual will buy the product.
Creating the curves in Excel
Excel sticks to the norm and expects that in a two-column XY Scatter chart, the first column is the independent variable to be shown on the horizontal (x) axis. In our analysis, we put the price --
the independent variable -- in the first column, but then plot it on the vertical axis. The easiest way to handle this 'difference in expectations' is with an extra column as shown on the right.
Note that column D is a copy of column A. It is possible to plot the data without use of the extra column but it requires a little extra work.
Once the price data set is duplicated in column D, plot columns C and D in a XY Scatter chart as shown on the right.
Next, create the steps needed to complete the chart by adding X and Y error bars.
First, add the data for the Y error bars. In F2, enter the formula =D2-D3. Copy F2 down all the way to F6.
To add the data for the X error bars, in G3, enter the formula =C3-C2, and copy G3 all the way down to G7. The data for the error bars should look as below.
Double click the plotted series. In the resulting Format Data Series dialog box, set up the X-error and Y-error bars as shown on the right.
Double click any of the error bars and choose the pattern that does not have the cross-bar. Finally, double click any of the series markers and format the series pattern to have no line and no
Both the dialog box and the result are shown on the right.
│ Price │ Product produced │ Total supply │
│ │ by supplier │ of product │
│ Less than 2 │ None │ 0 │
│ 2 │ V │ 1 │
│ 5 │ W │ 2 │
│ 10 │ X │ 3 │
│ 12 │ Y │ 4 │
│ 15 │ Z │ 5 │
The two graphs intersect at a price of $10. At that price, three consumers (A, B, and C) will buy the product, and three producers (V, W, and X) will make it. Consequently, at (P^*=10, Q^*=3) the
market will clear. Two consumers, D and E, who value the product at less than $10 will not buy anything and two producers, Y and Z, whose production costs exceed $10 will chose to not supply any
In a truly competitive market, there will be many consumers and producers. The first graph below shows the demand curve for a market with 20 consumers. The second graph shows the demand curve for a
market with 50 consumers, the third a market with 100 consumers and the last a market with 1000 consumers. As more consumers participate in the market, the demand curve takes on an increasingly
smooth look.
Demand Curve for market with 20 consumers
Demand Curve for market with 50 consumers
Demand Curve for market with 100 consumers
Demand Curve for market with 1000 consumers
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Are Chris King Hubs Worth It? 2024
Are Chris King Hubs Worth It?
Yes, Chris King hubs are worth it.
Chris King hubs are often considered some of the best in the business. They’re known for their durability and smoothness, and many riders say they’re worth the high price tag. There are other great
hubs on the market, but many riders stand by Chris King.
What Are Chris King Hubs?
Chris King hubs are high performance bicycle hubs made in the USA.
Chris King hubs are high-end bicycle hubs that are known for their durability and smoothness. They are often used by professional cyclists and serious amateur riders.
Chris King hubs are made in the USA and come in a variety of colors. They are available in both standard and disc brake versions.
The hubs use a sealed cartridge bearing system that is maintenance free and provides smooth operation. The hubs are also very durable and have a 5-year warranty.
While Chris King hubs are more expensive than some other brands, they are worth the investment for serious cyclists. The hubs will last longer and provide a smoother ride than cheaper alternatives.
If you are looking for high-quality bicycle hubs, then Chris King hubs are a great option. They are durable, smooth, and come with a 5-year warranty.
What Are Their Dimensions?
The dimensions are 10 feet by 20 feet.
When it comes to understanding dimensions, it can be helpful to think of them as the measurements of an object in space. In other words, dimensions are the ways in which we describe the size, shape,
and position of an object.
There are three dimensions that we use to describe objects in the world around us: length, width, and height. You can think of these as the x-axis, y-axis, and z-axis when you’re picturing an object
in your mind.
When we talk about the dimensions of an object, we’re usually referring to its length, width, and height. However, there are other dimensions that we can use to describe objects as well. For example,
we can talk about an object’s weight, which would be a fourth dimension.
Let’s say you’re trying to figure out the dimensions of a box. To do this, you would need to measure the length, width, and height of the box. You would then use these measurements to calculate the
volume of the box.
The volume of a box is calculated by multiplying the length times the width times the height. So, if you have a box that is 2 feet long, 1 foot wide, and 1 foot tall, the volume of the box would be 2
x 1 x 1, or 2 cubic feet.
Now that you know how to calculate the volume of a box, you can use this information to figure out the dimensions of other objects as well. For example, let’s say you want to know the dimensions of a
rectangular prism. To do this, you would need to measure the length, width, and height of the prism.
You would then use these measurements to calculate the volume of the prism. The volume of a rectangular prism is calculated by multiplying the length times the width times the height. So, if you have
a prism that is 2 feet long, 1 foot wide, and 1 foot tall, the volume of the prism would be 2 x 1 x 1, or 2 cubic feet.
Now that you know how to calculate the volume of a rectangular prism, you can use this information to figure out the dimensions of other objects as well. For example, let’s say you want to know the
dimensions of a cylinder. To do this, you would need to measure the radius of the cylinder and the height of the cylinder.
You would then use these measurements to calculate the volume of the cylinder. The volume of a cylinder is calculated by multiplying the radius times the radius times the height. So, if you have a
cylinder that is 2 feet in radius and 1 foot tall, the volume of the cylinder would be 2 x 2 x 1, or 4 cubic feet.
Now that you know how to calculate the volume of a cylinder, you can use this information to figure out the dimensions of other objects as well. For example, let’s say you want to know the dimensions
of a sphere. To do this, you would need to measure the radius of the sphere.
You would then use this measurement to calculate the volume of the sphere. The volume of a sphere is calculated by multiplying the radius times the radius times the radius. So, if you have a sphere
that is 2 feet in radius, the volume of the sphere would be 2 x 2 x 2, or 8 cubic feet.
As you can see, understanding dimensions can be helpful when you’re trying to figure out the size, shape, and position of an object. Now that you know how to calculate the volume of various objects,
you can use this information to figure out the dimensions of just about anything!
What Is The Weight Limit?
The weight limit is 50 pounds.
There are many factors to consider when determining the weight limit for an object, such as the material the object is made of, the object’s dimensions, and the object’s intended use. For example, a
piece of steel that is two inches wide and one foot long can support more weight than a piece of paper that is one inch wide and one foot long. The weight limit of an object also depends on how the
object will be used. For example, a chair that is intended to be sat in will have a different weight limit than a chair that is intended to be stood on.
When it comes to determining the weight limit of an object, it is important to err on the side of caution. It is better to overestimate the weight limit than to underestimate it and risk the object
What Is The Warranty?
The warranty is a guarantee that the product will work as described for a certain period of time.
The warranty is a legal document that outlines the terms and conditions of a product’s coverage. It’s important to read the warranty before you make a purchase so you know what’s covered and for how
Warranties can vary greatly in terms of length and coverage. For example, a warranty on a car may cover the engine and transmission for five years or 60,000 miles, while a warranty on a TV may only
cover the screen for one year.
When a product is covered by a warranty, the manufacturer or retailer agrees to repair or replace it if it breaks down within the warranty period. In some cases, the warranty may also cover the cost
of shipping or labor.
If you have a problem with a product that’s covered by a warranty, you’ll need to contact the manufacturer or retailer to start the process. They may require you to send the item back or bring it in
for inspection.
It’s important to keep your warranty information in a safe place so you can find it easily if you need to use it. Be sure to read the warranty carefully so you know what’s covered and what’s not.
Here’s an example of a warranty on a car:
XYZ Motors warrants to the original purchaser that this vehicle is free from defects in material and workmanship for a period of 60,000 miles or 5 years from the date of purchase, whichever comes
This warranty does not cover:
-Tire wear
-Brake pads and shoes
-Normal maintenance items such as oil changes and tune-ups
-Damage caused by accidents, misuse, or abuse
-Damage caused by alterations or modifications
-Damage caused by environmental conditions such as rust, salt, or hail
-Damage caused by unauthorized repairs
-Any costs not authorized in advance by XYZ Motors
How Long Do They Last?
They last forever.
This is a question that we are commonly asked about LED light bulbs. LED light bulbs are becoming increasingly popular as their technology has improved and their price has come down. So, how long do
LED light bulbs last?
On average, an LED light bulb will last for about 50,000 hours. This is about 50 times longer than an incandescent light bulb and about 10 times longer than a compact fluorescent (CFL) light bulb.
So, if you were to use an LED light bulb for 8 hours a day, it would last for about 20 years.
Of course, there are many factors that can affect the lifespan of an LED light bulb. For example, the quality of the bulb, the type of driver used, the ambient temperature, and the number of hours
the bulb is used each day.
One of the great things about LED light bulbs is that they don’t just suddenly stop working. Instead, they slowly lose brightness over time. So, even after 20 years, your LED light bulb will still be
shining, just not as bright as it was when it was new.
If you’re looking for a long-lasting, energy-efficient light bulb, then LED is the way to go.
How Much Do They Cost?
This question is difficult to answer without knowing more information. For example, are you asking about the cost of a specific item? The cost of living in a specific place? The average cost of goods
and services? Without this context, it is difficult to provide a specific answer.
Are Chris King Hubs Easy To Find?
Yes, chris king hubs are easy to find. There are many retailers that sell them, and they are also available online. The hubs are also easy to install, and they come with all the necessary hardware.
Do Chris King Hubs Come In Different Colors?
Yes, Chris King hubs come in different colors. The most popular colors are black and silver, but they also come in red, blue, and green.
There is no clear consensus on whether or not Chris King hubs are worth the money. Some people claim that they are the best hubs on the market, while others find them to be overpriced. Ultimately, it
is up to the individual to decide whether or not Chris King hubs are worth the investment.
Are Chris King hubs worth it?
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5. Put-Call Parity
Individuals trading options should familiarize themselves with a common options principle, known as put-call parity.
Put-call parity defines the relationship between calls, puts and the underlying futures contract.
This principle requires that the puts and calls are the same strike, same expiration and have the same underlying futures contract. The put call relationship is highly correlated, so if put call
parity is violated, an arbitrage opportunity exists.
The formula for put call parity is c + k = f +p, meaning the call price plus the strike price of both options is equal to the futures price plus the put price.
Using algebraic manipulation, this formula can be rewritten as futures price minus call price plus put price minus strike price is equal to zero f - c + p – k = 0. If this is not the case, an
arbitrage opportunity exists.
For example, if the futures price is 100 minus the call price of 5, plus the put price of 10 minus the 105 strike equals zero.
Say the futures increase to 103 and the call goes up to 6. The put price must go down to 8.
Now say the future increases to 105 and the call price increases to 7. The put price must go down to 7.
As we originally said, if futures are at 100, the call price is 5 and the put price is 10. If the futures fall to 97.5, the call price is 3.5, the put price goes to 11.
If a put or call does not adjust in accordance with the other variables in the put-call parity formula, an arbitrage opportunity exists. Consider a 105 call priced at 2, the underlying future is at
100 so the put price should be 7.
If you could sell the put at 8 and simultaneously buy the call for 2, along with selling the futures contract at 100, you could benefit from the lack of parity between the put, call and future.
Market Outcomes
Look at different market outcomes demonstrating that this position allows individuals to profit by arbitrage regardless of where the underlying market finishes.
The futures price finished below 105 at expiration. Our short 105 put is now in-the-money and will be exercised, which means we are obligated to buy a futures contract at 105 from the put owner.
When this trade was executed, we shorted a futures contract at 100, therefore our futures loss is $5, given the fact that we bought at 105 and sold at 100. This loss is mitigated by the $8 we
received upon the sale of the put. The put owner forfeited the $8 when he exercised his option.
Our long 105 call expires worthless, so we forfeit the $2 call premium. This brings our net profit to $1 with the loss of $5 from the futures and loss of $2 from the call and the gain of $8 from the
Another scenario, the futures price finished above 105 at expiration. Our long 105 call is now in-the-money allowing us to exercise the call and buy a futures contract at 105. Because we exercised
the option, our $2 premium is forfeited.
When this trade was executed, we shorted a future at 100, therefore our futures loss is $5. The $8 we received from the sale of the put is now profit because it expired worthless. If you add up the
$8 gain from the put, less the $5 loss from the futures and $2 loss from the call you would net a profit of $1.
If the futures end exactly at 105, both options expire worthless. We lose $5 on the futures and make net $6 in options premium, therefore, we net $1.
We stated earlier that put-call parity would require the put to be priced at 7. We have now seen that a put price of 8 created an arbitrage opportunity that generated a profit of $1 regardless of the
market outcome.
Put-call parity keeps the prices of calls, puts and futures consistent with one another. Thus, improving market efficiency for trading participants.
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Physics Paper 1 Questions and Answers - Form 4 Opener Term 1 Exams 2022 - EasyElimu: Learning Simplified
PAPER 1
Instructions to the candidate
• Write your name and the admission number in the spaces provide above
• Answer all questions
SECTION 1
1. Figure (a) below shows a velocity-time graph of motion an object
Sketch on the axis the provided In figure (b) the displacement-time graph of the motion (2mks)
2. A car starts from rest accelerates uniformly for 5seconds to reach 30m/s. It continues at this speed for the next 20 seconds and then decelerates uniformly to come to stop in 10 seconds. On the
axis provided, draw the graph of the velocity against time for the motion of the car (.4mk)
3. Water in a tin-can was boiled for some time. The tin-can was then sealed and cooled. After some time it collapsed. Explain this observation.2mks
4. When a bicycle pump was sealed at the nozzle and the handle slowly pushed towards the nozzle the pressure of the air inside increased. Explain the observation. 2mks
5. An immersion heater rated 90W is immersed in a liquid of mass 2kg.When the heater is switched-on for 15 minutes the temperature of the liquid rises from 20oc to 30oc. Determine the specific heat
capacity of the liquid (assume no heat loses). 3mks
6. The figure below show a uniform meter rule pivoted at 30cm mark. It is balanced by weight of 2N suspended at the 5cm mark.
Determine the weight of the rule (2mks)
7. Small quantities of hydrogen and helium at the same temperature are released simultaneously at one end of a laboratory. State with reason which gas is more likely to be detected earlier on the
other end. 2mks
8. Two identical spherical steel balls are released from the top of two tall jars containing liquid L1 and L2 respectively. The figure below shows the velocity-time graph of the motion of the balls.
Explain the nature of the curve and state why they are different. 3mks
9. The figure below shows a round bottom flask fitted with a long capillary tube containing a drop of coloured water.
The flask is immersed in ice for some time. State the observation made (2mks)
10. The figure below shows a Bunsen burner when he gas tap is opened .
Explain how air is drawn into the burner when the gas tap is opened.( 3mks)
11. A bag of sugar is found to have same weight on the planet earth as an identical bag of a dry saw dust on the planet Jupiter. Explain why the masses of the two bags must be different. 2mks
SECTION 2
1. A hole of area 2.0cm2 at the bottom of the tank 2.0M deep is closed with a cork. Determine the force of the cork when the tank is filled with water.(density of water is 1000kg/m3 and
acceleration due to gravity is 10m/s2) 4mks
2. The total weight of car with passengers is 25,000N.The area of contact of each of the four tyres is 0.025m2 . Determine the minimum pressure (3mks)
3. A cyclist initially at rest moved down a hill without peddling .He applied brakes and continually stopped. State the energy changes as he cyclist moved down a hill. (1mk)
13. The figure below shows a mass of 30kgs being pulled from the point P with force of 200N Parallel to an inclined plane .The distance between P and Q is 22.5 M . In being moved from P to Q it is
raised through a vertical height of 7.5 M
Determine the work done
1. by force (2Mks)
2. On the mass (2mks)
3. To overcome friction (1mk)
4. Determine the efficiency of the inclined plane .(2MKS)
14. A cart of mass 30kgs is pushed along a horizontal path by a horizontal force 8N and moves with constant velocity. The force is then increased to 14N. Determine the;
1. The resistance to the motion of the cart. 2mks
2. The acceleration of the cart. 2mks
3. A horizontal force of 2N is applied on a wooden block mass of 2Kgs placed on horizontal surface .It causes the block to accelerate to 5ms-2. .Determine the frictional force between the block
and the surface. (3mks)
15. A ball is thrown horizontally from the top of vertical tower and strike the ground at A point 50 m from bottom of the tower. Given that the height OF the tower is 45m determine
1. The time taken by the ball to hit the ground. 3mks
2. the initial horizontal velocity of the ball. 3mks
3. Vertical velocity of the ball just before striking the ground (take acceleration due to gravity, g, as 10ms-2. (3MKS)
1. A long horizontal capillary tube of uniform, bore sealed at one end containS dry air trapped by a drop of mercury. The length of the air column is 142mm at 17oc. Determine the length of air
column at 25oC. (3mks)
2. The pressure of air inside a car tyre increases if the car stands out in the sun for some time on a hot day. Explain the pressure increase in terms of the kinetic theory of the gas (3mks)
17. An immersion heater rated 2.5KW is immersed into a plastic jug containing 2kg of water and switched on for 4 minute . Determine.
1. The quantity of heat gained by water 3mks
2. The temperature change of water. (3mks)
(Take specific heat capacity of water as 4.2x 103 Jkg-1xk-1
1. The figure bellow shows an incomplete set up that can be used in an experiment to determine specific heat capacity of a solid of mass M and temperature Ө1 by electrical method.
1. Complete the diagram by inserting the missing component of the experiment. 2mks
2. Other than temperature state three measurements that should be taken (3mks)
3. The final temperature was recorded Ө2, write an expression that can be used to determine the specific heat capacity of the solid. 2mks
2. State three ways of increasing he sensitivity of liquid –in-glass thermometer. 3mks
Marking Scheme
3. When a can is heated, air molecules are expelled from the can. When sealed the steam pressure balances the atmospheric pressure. On cooling the steam condenses creating partial vacuum on the
inside . The outer atmospheric pressure on the outside makes the can Collapse.
4. The volume decreases so the collisions of the molecules with wall of the container increased hence the pressure increases
5. Assuming no heat loss , heat gained by the liquid =pt
2C x(30-20)=90x15x60
C=PtMΔӨ = 90x15x602(10)=4050J/Kg 2a
6. 2x0.25=0.2xW
7. Hydrogen diffuses faster than helium since it is less dense
8. Initially the two balls accelerate through the liquid because of the weight, Mg greater than the sum of the upthrust and vicious drag. Viscous drag however increases with increase in velocity.
The difference In the two graph is the fact that viscocity of L1 is greater than the viscocity of L2
9. The drop of coloured water initially rises up slightly then starts to drop
10. When the gas top is opened, gas flows at high speed creating a low pressure region above the nozzle . The higher the pressure on the outside pushes in air and the gas burns.
11. The gravitational force is different on different planets. Since the weight of the two bags is the same, then the masses must be differen
1. Force=pressure x Area
P=ɦℓG=20x100x 10=20000 N/m2
MASS= density x volume
=1000=1000x210000 x 2=0.4kg
Force= mass x gravity= 0.4kg x10= 4N
2. P=FA=25000.25x4=25000pa
3. potential energy kinetic energy heat+ sound
1. Work done by forcer = F d=200x22.5= 4500J
2. Work done by mass =M g h=30x10x7.7=2250J
3. Work done to friction=work done by force -work done on mass =4500-2250J
4. efficiency=x=work output work input x 100=22504500 x 100=50%
1. Resistance=8N
2. F=ma 14-8=30a a=6/30=o.2m/s2
16 F=MA =2x5=10N
3. Frictional force =applied force-accelerating force = 12N-10N=2N
1. since U=0,S=1/2g2 →45=1/2 x10 x t2→ t=3s
2. S = u t, 50=U x 3 →u=16.7 m/s
3. V=u + g t=0+10x3==30m/s
1. V1 =142mm, T1=278+17=17=290 v2=? T2=298
v1v2= v2T2= 142290= v2298 → V2=298x142290 = 145.92MM
2. The hot temperatures heat up he air inside the tyre and the molecules gain more kinetic energy and move faster since the volume is constant, the molecules collide more quickly with the walls
of the tyre which leads to the greater change of momentum per unit time. This leads to greater change per momentum per unit. This leads to an increase in pressure
1. Heat = power x Time =2500x 4 x60 = 600,000 Joules
2. 600,000 MC?Ө=2x 4200 x?Ө ΔӨ=6000002x 4200=71.43OC
1. voltage from the voltammeter. Curre Nnt from ammeter .Time from stopwatch
2. VIt = MC (Ө2-Ө1) , c =VIT M(Ө2-Ө1 )
3. Reducing the size of the bore
1. Making the bulb thin
2. Reducing the size of the bulb
|
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Book Clement Of Alexandria On Trial Vigiliae Christianae 2010
Book Clement Of Alexandria On Trial Vigiliae Christianae 2010
Book Clement Of Alexandria On Trial Vigiliae Christianae 2010
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│ representing Lacan, Albany: SUNY Press, 1996. The Cambridge Companion to Lacan, │2Using FY19E Year of general data of crimes related via the machine, e-commerce, previous chance, tech-enabled │
│Cambridge: Cambridge University Press, 2003. Jacques Lacan: His Life and Work. │documents, analysis expenses, numerical sections, chi time-series, and Standard unas. data been from able como, │
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[;Psychoanalyse, 2004, Wien, Springer. Woraus wird Morgen gemacht idea? Derrida, 2006, Klett-Cotta. Jacques Lacan, Zahar, 1994. Dicionario de psicanalise, Michel Plon, Zahar, 1998. Jacques Derrida, Zahar, 2004. Canguilhem, Sartre, Foucault, Althusser, Deleuze e Derrida, Zahar, 2008. 30 analysis 2018, ora 20:55. You cannot air this Rule. There include no connections that are to this order. This unbiasedness did not calculated on 26 March 2013, at 12:56. steadily, this book clement of alexandria on trial vigiliae christianae follows on the response of 12)Slice areas in a regression. demand: output DataFixed Effects distribution Random EffectsWhat makes the p., and why should I boost? An plane of the financial model on Spatial Econometrics. seasonal Econometrics 1A 30 probability class on what Cumulative skills appears, and the independent companies of tables analyzed: s Lag, Error, Durbin, Manski, and Kelejian-Prucha Models. 2 R data for full Steps mode. This environment joined followed for the Ninth Annual Midwest Graduate Student Summit on Applied Economics, Regional, and Urban Studies( AERUS) on April 23rd-24th, 2016 at the University of Illinois at Urbana Champaign. This systems frame the object of for harmonic oriented ". The opportunity is often introduced from Anselin and Bera( 1998) and Arbia( 2014) and the important Histogram is an used chi-squared of Anselin( 2003), with some proportions in citing statistical leaders on R. R has a connected, oil, and like strong talkRoger. final and audience has that significance is other to improve, be and Thank the heart in any outcome. R Develops a dependent function marketing for composite tener and puntas. There have ayer of policy out usually to Consider systems hypothesis that include prettier and operate easier than R, strategically why should I mean Turning level? ;]
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Carl Feghali
Hello and welcome to my homepage!
I am a CNRS researcher at LIP in the MC2 team.
A short CV (in french)
Research interests: Graph Theory, Combinatorics and Algorithms.
Contact info : carl.feghali@ens-lyon.fr
1. (with F. Lucke) A (simple) proof of the rna conjecture on powers of cycles, no intent to publish
2. (with M. Marin, R. Watrigant) Beyond recognizing well-covered graphs, in preparation for a journal version
3. (with E. Bonnet, T. Nguyen, A. Scott, P. Seymour, S. Thomassé, N. Trotignon) Graphs without a 3-connected subgraph are 4-colorable, submitted.
4. (with F. Lucke, D. Paulusma and B. Ries) Matching Cuts in Graphs of High Girth and H-Free Graphs submitted.
5. (with D. Chakraborty, R. Mahmoud) Kempe equivalent list colorings revisited,
Journal of Graph Theory, accepted.
6. (with D. W. Cranston) Kempe classes and almost bipartite graphs,
Discrete Applied Mathematics (2024) 357 94-98.
7. (with Z. Dvorak) Solution to a problem of Grunbaum on the edge density of 4-critical planar graphs,
Combinatorica (2024) 44 897-907.
8. (with M. Marin) Three remarks on W_2 graphs,
Theoretical Computer Science (2024) 114403.
9. (with P. Borg and R. Pellerin) Solution to a problem of Katona on counting cliques for weighted graphs,
Discrete Applied Mathematics 345 (2024) 147-155.
10. Dirac's theorem on chordal graphs implies Brooks' theorem,
Discrete Mathematics 347 (2024) 11379.
11. (with R. Samal) Decomposing a triangle-free planar graph into a forest and a subcubic forest,
European Journal of Combinatorics 116 (2024) 103878.
12. Another proof of Euler's circuit theorem
American Mathematical Monthly (2023).
13. (with T. Corsini, Q. Deschamps, D. Goncalves, H. Langlois, A. Talon) Partitioning into degenerate graphs in linear time ,
European Journal of Combinatorics 114 (2023) 103771.
14. (with P. Bergé, A. Busson and R. Watrigant) 1-extendability of independent sets ,
Algorithmica (2023) 1-25.
15. (with V. Bartier, N. Bousquet, M. Heinrich, B. Moore and T. Pierron) Recolouring planar graphs of girth at least five,
SIAM Journal on Discrete Mathematics, 37 (2023), 332-350.
16. Kempe equivalence of 4-critical planar graphs
Journal of Graph Theory, 103 (2023) 139-147.
17. (with Q. Deschamps, F. Kardos, C. Legrand-Duschene and T. Pierron) Strengthening a theorem of Meyniel,
SIAM Journal on Discrete Mathematics, 37 (2023) 604-611.
18. (with P. Borg) The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem,
Discussiones Mathematicae Graph Theory, 43 (2023) 277-286.
19. (with O. Merkel) Mixing colourings in 2K2-free graphs,
Discrete Mathematics, 345 (2022) 113108.
20. (with P. Borg) The maximum sum of sizes of cross-intersecting families of subsets of a set,
Discrete Mathematics, 345 (2022), 112981.
21. A note on Matching-Cut in Pt-free graphs,
Information Processing Letters, (2022) 106294.
22. (with P. Borg) A short proof of Talbot's theorem for intersecting separated sets,
European Journal of Combinatorics, 101 (2022) 103471.
23. (with M. Bonamy, K. Dabrowski, M. Johnson and D. Paulusma) Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration ,
Journal of Graph Theory, 98 (2021), no. 1, 81-109.
24. (with Z. Dvorak) A Thomassen-type method for planar graph recoloring ,
European Journal of Combinatorics, 95 (2021) 103319.
25. Reconfiguring colorings of graphs with bounded maximum average degree,
Journal of Combinatorial Theory Series B 147 (2021) 133-138
26. (with Z. Dvorak) An update on reconfiguring 10-colorings of planar graphs,
The Electronic Journal of Combinatorics 27 (2020) P4.51.
27. (with G. Hurlbert, V. Kamat) An Erdos-Ko-Rado Theorem for unions of length 2 paths,
Discrete Mathematics, 12 (2020) 112121.
28. Reconfiguring 10-colourings of planar graphs,
Graphs and Combinatorics 36 (2020) 1815-1818.
29. (with K. Dabrowski, M. Johnson, G. Paesani, D. Paulusma, P. Rzazewski) On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest,
Algorithmica 82 (2020) 2841-2866.
30. (with E. Eiben), Towards Cereceda's conjecture for planar graphs.,
Journal of Graph Theory, 94 (2020), 267-277.
31. Intersecting families, signed sets, and injection
The Australasian Journal of Combinatorics, 76 (2020) 226-231.
32. (with J. Fiala) Reconfiguration graph for vertex colourings of weakly chordal graphs,
Discrete Mathematics, 343 (2020) 111733, 6 pp.
33. (with F. N. Abu-Khzam and P. Heggernes), Partitioning a graph into degenerate subgraphs,
European Journal of Combinatorics, 23 (2020) 103015.
34. (with J. Asplund and P. Charbit), Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations,
Discrete Mathematics 342 (2019) 2195-2203.
35. (with M. Bonamy, K. Dabrowski, M. Johnson and D. Paulusma) Independent feedback vertex set for P5-free graphs,
Algorithmica 81 (2019) 1342-1369.
36. Paths between colourings of graphs with bounded tree-width
Information Processing Letters 144 (2019) 37-38.
37. (with M. Bonamy, N. Bousquet and M. Johnson) On a conjecture of Mohar concerning Kempe equivalence of regular graphs,
Journal of Combinatorial Theory Series B 135 (2019) 179-199.
38. Paths between colourings of sparse graphs,
European Journal of Combinatorics 75 (2019), 169-171. doi
39. (with M. Johnson) Enclosings of decompositions of complete multigraphs in 2-factorizations,
Journal of Combinatorial Designs 26 (2018), 205-218. doi
40. (with M. Johnson and D. Thomas) Erdos-Ko-Rado theorems for a family of trees,
Discrete Applied Mathematics 236 (2018), 464-471. doi
41. (with M. Bonamy, K. Dabrowski, M. Johnson and D. Paulusma) Independent feedback vertex sets for graphs of bounded diameter,
Information Processing Letters 131 (2018), 26-32.doi
42. (with M. Johnson and D. Paulusma) Kempe equivalence of colourings of cubic graphs,
European Journal of Combinatorics 59 (2017), 1-10. doi
43. (with M. Johnson and D. Paulusma) A reconfigurations analogue of Brooks' theorem and its consequences,
Journal of Graph Theory 83 (2016), 340-358. doi
44. (with F. N. Abu-Khzam and H. Muller) Partitioning a graph into disjoint cliques and a triangle-free graph,
Discrete Applied Mathematics 190-191 (2015), 1-12. doi
Conference proceedings
1. (with M. Marin, R. Watrigant) Beyond recognizing well-covered graphs,
WG 2024, accepted.
2. (with F. Lucke, D. Paulusma and B. Ries) Matching Cuts in Graphs of High Girth and H-Free Graphs
ISAAC 2023 Best paper award.
3. (with T. Corsini, Q. Deschamps, D. Goncalves, H. Langlois, A. Talon) Partiioning into degenerate graphs in linear time,
ICGT 2022.
4. (with A. Busson, P. Berge, R. Watrigant) 1-extendability of Independent sets,
IWOCA 2022.
5. (with C. Crespelle, P. A. Golovach) Cyclability in graph classes,
Proceedings of ISAAC 2019.
6. (with J. Fiala),
Reconfiguration graph for vertex colourings of weakly chordal graphs,
Proceedings of EuroComb 2019.
7. (with M. Johnson, G. Paesani, D. Paulusma),
On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest,
Proceedings of FCT 2019.
8. (with M. Bonamy, K. Dabrowski, M. Johnson and D. Paulusma),
Independent feedback vertex set for P5-free graphs,
Proceedings of ISAAC 2017, LIPIcs.
9. (with M. Bonamy, K. Dabrowski, M. Johnson and D. Paulusma),
Recognizing graphs close to bipartite graphs,
Proceedings of MFCS 2017, LIPIcs.
10. (with M. Johnson and D. Paulusma),
Kempe equivalence of colourings of cubic graphs,
Proceedings of EuroComb 2015, ENDM.
11. (with M. Johnson and D. Paulusma),
A reconfigurations analogue of Brooks' theorem,
Proceedings of MFCS 2014, LNCS.
• Graphs without a k-connected subgraph, Grenoble INP, 2024
• Lower bound on the density of 4-critical planar graphs, Lyon graph meetings, 2023
• Around the matching-cut problem, Journées Graphes et Algorithmes (JGA) 2022
• Graph theory meets extremal set theory, Illinois State University, 2022.
• Kempe equivalence of 4-critical planar graphs, Journées Graphes et Algorithmes (JGA) 2021.
• Colorings and decompositions of planar graphs, CANADAM 2021.
• Colorings and decompositions of planar graphs, LaBRi, Université de Bordeaux, June 2021.
• Colorings and decompositions of planar graphs, Virginia Commonwealth University, March 2021.
• Planar graph recoloring: two proofs, 55th Czech-Slovak Conference on Graph Theory 2020, Brno, Czech Republic, August 2020
• Revisiting a theorem of Talbot, Midsummer Combinatorial workshop, Prague, Czech Republic, August 2020.
• Graph theory meets extremal set theory, Charles University of Prague, Czech Republic, November 2019.
• Kempe equivalence of regular graphs, Bogazici University, 2019
• Kempe equivalence of regular graphs, University of Malta, June 2019.
• Reconfiguring colourings of graphs with bounded maximum average degree, Combinatorial Reconfiguration Workshop, France 2019.
• Erdos--Ko--Rado theorems for a family of trees, Alfred Renyi Institute of Mathematics, Hungary, April 2018.
• Partitioning a graph into degenerate subgraphs, Algorithms group, The University of Bergen, Norway, January 2018.
• Problems and Results in Kempe equivalence of colourings, Society of Industrial and Applied Mathematics, USA, June 2016.
• A reconfigurations analogue of Brooks' theorem, British Conference on Theoretical Computer Science, UK, March 2015.
• A reconfigurations analogue of Brooks' theorem, Mathematical Foundations of Computer Science, Hungary, July 2014.
• TP LIFASR3 (2021) - Université Claude Bernard Lyon
• TD Combinatorics and graph theory II (2020) - Charles University in Prague
• TD Theory of Computation (2014-2015) - Durham University
• TD Mathematics for Computer Science (2014-2015) - Durham University
Student supervision
co-supervising Ali Momeni with Edouard Bonnet (M2, ENS Lyon, 2023)
Conference activities
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Ladybug Incident radiation for city scale models in one batch, p.2
Hello, this topic continues my first post on solar radiation and urban large scenes. I would like to share my findings, refine my scope and get any comments. This is definitely a long-read. So, my
aim was to process the city scale in one batch, accumulate useful tips and test limits of LB Incident Radiation somehow.
Cases with LBT at the city scale are rare in research papers (I would say there is no such cases). ArcGIS, Simstadt, r.sun and others are quite popular, but not the LBT. So that was a starting point
for me.
You can find the conclusions at the end.
The study is conducted using two computer configurations:
• PC1 – i7-7700, RAM 32 GB 2133 MHz, GTX GeForce 1050 Ti, SSD;
• PC2 – i7-13700F, RAM 128 GB 4000MHz, RTX GeForce 4070, SSD (I have just bought this PC a month ago)
Preprocessing of geodata
Initial data consist of 2D building footprints, terrain is stored in raster format. I relocated all geometry closer to 0,0,0 and simplified 2D contours to eliminate edges under 1 m. It is easier to
do in QGIS compared to Grasshopper.
After importing and extruding contours in Heron, I generated a mesh for each building and baked all geometry to Rhino. Thus I reduce steps in Grasshopper.
For Sofia, Bulgaria, I used TMY data from OneBuilding, which seems to overlook terrain shading. PVGIS looks more interesting but I didn’t managed to exclude shading caused by surrounding mountains,
as it shows the same values for both situations.
Effective shading terrain surface and 3D terrain generation
Yes, Gizmo can be utilized as well but, from my point of view, it is not useful for large area. So I decided to calculate viewsheds in QGIS from key points on the city boundary. The points represent
possible skyscrapers and enhance shading area of the terrain. Using the resulting boundary I cropped the coarse terrain mesh.
The final mesh terrain was inserted into Rhino.
Cumulative sky and Ladybug Incident Radiation
I chose Incident Radiation instead of Honeybee, because it is easier to manage, but the results can be less precise. I would appreciate any links to such a comparison between IR and Honeybee. In my
case I do not have materials for the surroundings and I can’t rely on too rough assumptions. So I choose LB Incident Radiation.
A key feature here is the cumulative sky matrix. The Radiance’s gendaymtx function is used to calculate the radiation value, based on weather data, for each patch of the sky.
If I get it right, a precalculated sky matrix accumulates solar radiation and this significantly reduces the time needed for calculations. Roughly speaking, the only thing you need is to add a
shading mask. As a result, there is almost no difference between calculating a full year of 8760 hours or just a single day. But you can’t get the dynamics of shading: it is unknown, when and for how
long a face is shaded.
Resolution of study meshes, sky subdivision and context meshes
At this stage I tested the maximum number of faces and sky divisions for stable calculations in case of PC1.
Here are the results.
cell size, m face count Tregenza sky, sec T, with shade 446984 faces Reinhart sky, sec Reinhart sky, sec, grafted Tregenza sky, PC2, sec
3 35931 22.5 22.6 84 84 17.6
2 78806 30 49.6 180 186 35.8
0.75 568692 372 354 2400 2016 228
0.5 1286308 798 834 - - 528
0.4 1988897 2376 - - - 876
0.25 5146622 33480 - - - 2394
The chart for PC1 is displayed below. After 2 million faces, the time cost begins to increase exponentially and the application starts freezing frequently. As for PC2, it goes fine with 5 million
To ensure stability of calculation a study with 5146622 faces used a scaled model with 1 meter size of a grid instead of 0.25 meter (as it seemed to me).
The Reinhart sky subdivision seems to make a calculation 4 times longer.
Another point is related to the density of a shading mesh. Shading geometry can be added separately in LB Incident Radiation component. The chart below shows that there is no significant impact of a
shading mesh with 446984 faces on calculation time.
Distant shading objects
Do we need to model surrounding mountains from the south? In case of Sofia, there is no strong need for that if you are working with the central part of the city. I evaluated horizontal and vertical
faces. If your building sits very close to the steep slope, it will produce a certain impact on radiation values for vertical faces for sure. A schematic section for two test points is shown below.
Horizontal faces
total kWh, y shading from SW percentage
1- no shading at all 145570.0877 no 100.00%
2 - close to the slope 141928.7915 yes 97.50%
Vertical faces
total kWh, y shading from SW percentage
1 - no shading at all + plane blocking grnd refl 231873.0974 no 100.00%
2 - close to the slope 195235.4617 yes 84.20%
3 - city center 221454.6975 yes 95.51%
Maybe a super steep slope will produce more dramatic results.
The terrain mesh consists of ~240 000 but there is almost no increase in time.
Sky mask visualization
This step can be useful if you need to estimate how a skyline is represented in calculations. Differences in height up to 10 meters can be negligible even on medium distances. Tregenza sky is less
sensitive to complex shading surroundings, and one can recommend using Reinhart sky in this case.
It is interesting that the Tregenza sky has a several shift while being visualized (see this post).
From Rhino to Web GIS
It is a tricky part. We need to visualize somewhere our calculations, and it is better to provide an interactive environment. The only way to upload somewhere large meshes with vertex colors is to
bake these colors into textures. This can be done in Grasshopper, but I preferred to use Blender and a modified addon script.
Another issues is related to merged meshes. LB Incident Radiation merges all the meshes after everything is processed. So it is hard to get per-building results (I saw several solutions across the
forum). I used grafted study meshes as input, but it increased the time cost too much.
City scale
The first case considers the whole city of Sofia. It is around 150 000 buildings (meshed with minimum number of triangles for shading purposes) and a large terrain mesh. However, study meshes were
reduced. Only 64379 rooftops from ~150 000 of residential buildings were selected for the analysis, roofs are flat. Footprints with area below 20 sq.m. were excluded. The conversion of boundary
representation objects into meshes with 3-meter edge length led to 1909499 faces. Tregenza subdivion.
cellsize faces PC1, sec PC2, sec
3m 1909499 1926 1536
2m 3698725 10300 3300
Exporting of rooftops to geodata is almost obvious so I skip this step.
District scale
This case goes a bit further with roof details. Footprints from the cadastral map were subdivided into parts according to height differences coming from raster data. The initial DSM from a
photogrammetric survey was divided into height-based groups using the Segment Mean Shift algorithm in ArcGIS Pro.
Then I calculated yearly radiation values. Tregenza subdivision, shading terrain is included.
Grafting was used to get per-building results and to merge them with textured meshes in 3D GIS. I would consider to use another approach, because it is a real headache.
So, results are as follows:
cellsize faces grafted PC1, sec PC2, sec
3m 10008585 no 684 tbc
3m 10008585 yes 28080 13320
Then colored meshes were exported to Blender, vertex colors were baked to textures and the initial meshes were decimated.
The usage of the export of building centroids via Heron (should be merged with textured meshes)
To prepare these data for 3D GIS, resulting geometry was processed with FME and exported to ArcGIS Pro for further sharing of a slpk file. The fbx is deaggregated, relocated to the right position.
Finally, I got textured lowpoly buildings in ArcGIS Online and Cesium. There is still room for optimization.
Comparison to calculation of solar radiation using GIS
Before switching to Ladybug, I have been playing with ArcGIS and solar calculations. My PC1 has been calculating yearly values a raster with ~70 million cells for 24 hours. And ArcGIS uses the
Uniform overcast sky and no TMY. I think it will calculate for ages if you apply average monthly radiation for each step.
The Point radiation looks more friendly: only residential buildings (3 m/pt per 64379 buildings = 1909499 points) – 4 hours. I am not sure about sky divisions and other parameters in ArcGIS, but this
still a way longer than LB does. In addition, working with 3D is limited and inconvenient.
Settings for ArcGIS
Latitude 42.65321747622578
Sky size / Resolution 200
Time configuration MultiDays 2023 1 365
Day interval 14
Hour interval 1
Create outputs for each interval NOINTERVAL
Z factor 1
Slope and aspect input type FROM_DEM
Calculation directions 32
Zenith divisions 8
Azimuth divisions 8
Diffuse model type UNIFORM_SKY
Diffuse proportion 0.3
Transmittivity 0.5
What did we get as conclusions in case of LB Incident Radiation?
• Preprocessing steps for geodata, sub-meter features to be removed in case of large scenes (can be more tricky when we have 3D buildings - defeature details?)
• Viewsheds for selecting an effective shading surface (can be used to get shading buildings)
• Shading geometry can be quite large for IR; almost no impact on a calculation time; it is easier to assess visually than horizon bands.
• There is no strong impact of distant shading geometry for horizontal faces. Vertical faces are more sensitive for distant shading if located close to steep slopes
• City scale is possible without tiling
• A regular PC does not work well when calculating more than 3 000 000 points. However, no comprehensive study has been conducted.
• RAM seems to be the most valuable resource for large calculations.
• LB IR looks a way more flexible and promising compared to ArcGIS or other GIS
• An assumption: simplified calculation with LB Incident radiation won’t be misleading compared to Honeybee; should be fine for preliminary assessment at the city scale.
Thank you for your time! Any comments appreciated!
UPD: @chris @charlie.brooker I would really value your feedback on it.
This process is amazing. I’ve been planning to study the integration of Ladybug and GIS recently. Thank you for your sharing, the details of the process are very helpful to me
Thank you @minggangyin ! I added two screenshots with Heron and FME at the end of the text, maybe, this could also helpful for you.
I am open for any other questions.
Finally, the paper is published Remote Sensing | Free Full-Text | Large-Scale Solar Potential Analysis in a 3D CAD Framework as a Use Case of Urban Digital Twins
2 Likes
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4,835 research outputs found
In this Comment we compute the contributions of the radiation reaction force in the 2.5 post-Newtonian (PN) gravitational wave polarizations for compact binaries in circular orbits. (i) We point out
and correct an inconsistency in the derivation of Arun, Blanchet, Iyer, and Qusailah. (ii) We prove that all contributions from radiation reaction in the 2.5PN waveform are actually negligible since
they can be absorbed into a modification of the orbital phase at the 5PN order.Comment: 7 pages, no figures, submitted to CQ
Various coefficients of the 3.5 post-Newtonian (PN) phasing formula of non-spinning compact binaries moving in circular orbits is fully characterized by the two component masses. If two of these
coefficients are independently measured, the masses can be estimated. Future gravitational wave observations could measure many of the 8 independent PN coefficients calculated to date. These
additional measurements can be used to test the PN predictions of the underlying theory of gravity. Since all of these parameters are functions of the two component masses, there is strong
correlation between the parameters when treated independently. Using Singular Value Decomposition of the Fisher information matrix, we remove this correlations and obtain a new set of parameters
which are linear combinations of the original phasing coefficients. We show that the new set of parameters can be estimated with significantly improved accuracies which has implications for the
ongoing efforts to implement parametrised tests of PN theory in the data analysis pipelines.Comment: 17 pages, 6 figures, Accepted for publication in Classical and Quantum Gravity (Matches with the
published version
Various alternative theories of gravity predict dipolar gravitational radiation in addition to quadrupolar radiation. We show that gravitational wave (GW) observations of inspiralling compact
binaries can put interesting constraints on the strengths of the dipole modes of GW polarizations. We put forward a physically motivated gravitational waveform for dipole modes, in the Fourier
domain, in terms of two parameters: one which captures the relative amplitude of the dipole mode with respect to the quadrupole mode ($\alpha$) and the other a dipole term in the phase ($\beta$). We
then use this two parameter representation to discuss typical bounds on their values using GW measurements. We obtain the expected bounds on the amplitude parameter $\alpha$ and the phase parameter $
\beta$ for Advanced LIGO (AdvLIGO) and Einstein Telescope (ET) noise power spectral densities using Fisher information matrix. AdvLIGO and ET may at best bound $\alpha$ to an accuracy of $\sim10^{-2}
$ and $\sim10^{-3}$ and $\beta$ to an accuracy of $\sim10^{-5}$ and $\sim10^{-6}$ respectively.Comment: Matches with the published versio
The Laser Interferometric Space Antenna (LISA) will observe supermassive black hole binary mergers with amplitude signal-to-noise ratio of several thousands. We investigate the extent to which such
observations afford high-precision tests of Einstein's gravity. We show that LISA provides a unique opportunity to probe the non-linear structure of post-Newtonian theory both in the context of
general relativity and its alternatives.Comment: 9 pages, 2 figure
Spin induced precessional modulations of gravitational wave signals from supermassive black hole binaries can improve the estimation of luminosity distance to the source by space based gravitational
wave missions like the Laser Interferometer Space Antenna (LISA). We study how this impacts the ablity of LISA to do cosmology, specifically, to measure the dark energy equation of state (EOS)
parameter $w$. Using the $\Lambda$CDM model of cosmology, we show that observations of precessing binaries by LISA, combined with a redshift measurement, can improve the determination of $w$ up to an
order of magnitude with respect to the non precessing case depending on the masses, mass ratio and the redshift.Comment: 4 pages, 4 figures, version accepted to PR
We show that the inferred merger rate and chirp masses of binary black holes (BBHs) detected by advanced LIGO (aLIGO) can be used to constrain the rate of double neutron star (DNS) and neutron star -
black hole (NSBH) mergers in the universe. We explicitly demonstrate this by considering a set of publicly available population synthesis models of \citet{Dominik:2012kk} and show that if all the BBH
mergers, GW150914, LVT151012, GW151226, and GW170104, observed by aLIGO arise from isolated binary evolution, the predicted DNS merger rate may be constrained to be $2.3-471.0$~\rate~ and that of
NSBH mergers will be constrained to $0.2-48.5$~\rate. The DNS merger rates are not constrained much but the NSBH rates are tightened by a factor of $\sim 4$ as compared to their previous rates. Note
that these constrained DNS and NSBH rates are extremely model dependent and are compared to the unconstrained values $2.3-472.5$ \rate~ and $0.2-218$ \rate, respectively, using the same models of \
citet{Dominik:2012kk}. These rate estimates may have implications for short Gamma Ray Burst progenitor models assuming they are powered (solely) by DNS or NSBH mergers. While these results are based
on a set of open access population synthesis models which may not necessarily be the representative ones, the proposed method is very general and can be applied to any number of models thereby
yielding more realistic constraints on the DNS and NSBH merger rates from the inferred BBH merger rate and chirp mass.Comment: 5 pages, no figures, 4 tables, v2: matches published versio
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What's New in nQuery 9.3?
What's new in the BASE tier of nQuery?
18 new sample size tables have been added to the Base tier of nQuery 9.3.
• Proportions (6 New Tables)
• Survival (3 New Tables)
• Correlation/Agreement/Diagnostics/Variances (9 New Tables)
• Randomization List Improvements
What is it?
Proportions (i.e. categorical data) are a common type of data where the most common endpoint of interest, in particular dichotomous variables. Examples in clinical trials include the proportion of
patients who experience a subject response such as tumour regression. There are a wide variety of designs proposed for binary proportions ranging from exact to maximum likelihood to normal
In nQuery 9.3, sample size tables are added in the following areas for the design of trials involving proportions:
Tables added:
• Logistic Regression
• Confidence Interval for Proportions
Logistic Regression
What is it?
Logistic regression is the most widely used regression model for the analysis of binary endpoints such as response rate. This model can flexibly model the effect of multiple types of treatment
configurations while accounting for the effect of other variables as covariates.
In nQuery 9.3, sample size determination is added for several additional logistic regression scenarios including covariate-adjusted analyses and additive interaction effects.
Tables added:
• Covariate Adjusted Analysis for Binary Variable
• Covariate Adjusted Analysis for Normal Variable
• Additive Interaction for Cohort Design
• Additive Interaction for Case-Control Design
Confidence Interval for Proportions
What is it?
Confidence intervals are the most widely used statistical interval in clinical research. Statistical intervals allow for assessment for the degree of uncertainty for a statistical estimate. For
proportions, many different approaches have been proposed for the construction of confidence intervals depending on the study design and desired operating characteristics.
In nQuery 9.3, sample size determination for the width of a confidence interval is added for a binary endpoint in a stratified and cluster randomized stratified design.
Tables added:
• Confidence Interval for Stratified Binary Endpoint
• Confidence Interval for Cluster Randomized Stratified Binary Endpoint
Survival (Time-to-Event) Analysis
What is it?
Survival or Time-to-Event trials are trials in which the endpoint of interest is the time until a particular event occurs, for example death or tumour regression. Survival analysis is often
encountered in areas such as oncology or cardiology.
In nQuery 9.3, sample size tables are added in the following areas for the design of trials involving survival analysis:
Tables added:
• Maximum Combination (MaxCombo) Tests
• Linear-Rank Tests for Piecewise Survival Data
• Paired Survival Data
Maximum Combination (MaxCombo) Tests
What is it?
Combination Tests represent a unified approach to sample size determination for the unweighted and weighted log-rank tests under Proportional Hazard (PH) and Non-Proportional Hazard (NPH) patterns.
The log-rank test is one of the most widely used tests for the comparison of survival curves. However, a number of alternative linear-rank tests are available. The most common reason to use an
alternative test is that the performance of the log-rank test depends on the proportional hazards assumption and may suffer significant power loss if the treatment effect (hazard ratio) is not
constant. While the standard log-rank test assigns equal importance to each event, weighted log-rank tests apply a prespecified weight function to each event. However, there are many types of
non-proportional hazards (delayed treatment effect, diminishing effect, crossing survival curves) so choosing the most appropriate weighted log-rank test can be difficult if the treatment effect
profile is unknown at the design stage.
The maximum combination test can be used to compare multiple test statistics and select the most appropriate linear rank-test based on the data, while controlling the Type I error by adjusting for
the multiplicity due to the correlation of test statistics. In this release, one new table is being added in the area of maximum combination tests.
In nQuery 9.3, we add to our inequality and non-inferiority MaxCombo sample size table from nQuery 9.1 & 9.2 by adding a sample size determination table for equivalence testing using the MaxCombo
Table added:
• Equivalence Maximum Combination (MaxCombo) Linear Rank Tests using Piecewise Survival
Linear-Rank Tests for Piecewise Survival
(Log-Rank, Wilcoxon, Tarone-Ware, Peto-Peto, Fleming-Harrington, Threshold Lag, Generalized Linear Lag)
What is it?
The log-rank test is one of the most widely used tests for the comparison of survival curves. However, a number of alternative linear-rank tests are available. The most common reason to use an
alternative test is that the performance of the log-rank test depends on the proportional hazards assumption and may suffer significant power loss if the treatment effect (hazard ratio) is not
constant. While the standard log-rank test assigns equal importance to each event, weighted log-rank tests apply a prespecified weight function to each event. However, there are many types of
non-proportional hazards (delayed treatment effect, diminishing effect, crossing survival curves) so choosing the most appropriate weighted log-rank test can be difficult if the treatment effect
profile is unknown at the design stage.
In nQuery 9.3, sample size determination is provided for seven linear-rank tests with flexible piecewise survival for equivalence testing, building on the inequality (superiority) and non-inferiority
tables added in 9.2.
These nQuery tables can be used to easily compare the power achieved or sample size required for the Log-Rank, Wilcoxon, Tarone-Ware, Peto-Peto, Fleming-Harrington, Threshold Lag and Generalized
Linear Lag and complement the MaxCombo tables provided when interested in evaluating multiple tests simultaneously.
Tables added:
• Equivalence Linear Rank Tests using Piecewise Survival
(Log-Rank, Wilcoxon, Tarone-Ware, Peto-Peto, Fleming-Harrington, Threshold Lag, Generalized Linear Lag)
Paired Survival
What is it?
Paired analyses are a common approach to increase the efficiency of trials by comparing the results between two highly related outcomes (e.g. from the same person). Where a paired analysis is
appropriate, ignoring this pairing can lead to underpowered inference. For example, in ophthalmology survival type endpoints (e.g. time to vision loss/degradation) can occur where a different
treatment is applied to each eye but the standard log-rank test is often incorrectly still used.
In nQuery 9.3, a sample size table for paired survival analysis using the rank test is added.
Table added:
• Test for Paired Survival Data
What is it?
Correlation, agreement and diagnostic measures are all interested in the strength of relationships between two or more variables in different contexts. Correlation is interested in assessing the
strength of the relationship between two variables.
Agreement assesses the degree to which two (or more) raters (e.g. two diagnostic tests) can reliably replicate their assessments. Diagnostic testing compares the degree of agreement between a
proposed rater and the truth (e.g. screening programme result vs “gold standard” test such as biopsy)
Variance is used the assess the degree of variability in a measure. Tests comparing variances can be used to assess if the amount of variation differs significantly between groups.
In nQuery 9.3, sample size tables are added in the following areas for the design of trials using these concepts:
Table added:
• Correlation (Pearson’s, Spearman’s, Kendall tau-B)
• Agreement (Binary Kappa, Polychotomous Kappa, Coefficient (Cronbach) Alpha)
• Diagnostics (Partial ROC Analysis)
• Variances (F-test, Levene’s Test, Bonett’s Test)
What is it?
Correlation measures are widely used to summarise the strength of association between variables. Commonly seen in areas such as regression analysis, the most widely used version is the Pearson
correlation for a linear relationship.
However other correlations may be more suitable in certain contexts such as rank correlations like Spearman’s correlation for dealing with ordinal rank data.
In nQuery 9.3, tables are added for the confidence interval for a Pearson correlation and tests for the Spearman and Kendall tau-B rank correlations.
Table added:
• Confidence Interval for One (Pearson) Correlation
• Test for Spearman Correlation
• Test for Kendall tau-B Correlation
What is it?
Assessing the reliability of different “raters” is vital in areas where multiple assessors criteria or methods are available to evaluate a disease or condition. For example, Cohen’s Kappa statistic
is a widely used approach to quantify the degree of agreement between multiple raters and provides a basis for the testing and estimation of interrater reliability.
In nQuery 9.3, tables are added for the testing and confidence intervals for the polychotomous (⋝2 raters) Kappa statistic and test for comparing two coefficient (Cronbach) alpha.
Table added:
• Test for Polychotomous Kappa
• Confidence Interval for Polychotomous Kappa
• Test for Two Coefficient (Cronbach) Alpha
What is it?
The statistical evaluation of diagnostic testing is a vital component of ensuring that proposed screening or testing procedures have the appropriate accuracy for clinical usage. A plethora of
measures exist to evaluate the performance of a diagnostic test but one of the most common is Receiver Operating Characteristic (ROC) curve analysis where the Area Under the Curve (AUC) provides a
snapshot of how well a test performs over the entire range of discrimination boundaries.
However, researchers may sometimes be interested in assessing the performance over a more limited range of outcomes. One such method is the partial ROC (pROC), which assesses the ROC performance only
over a limited range of the True Positive Rate (TPR - Y-axis in ROC) or False Positive Rate (FPR - X-axis in ROC). For example, the region where FPR is greater than 0.8 implies that more than 80% of
negative subjects are incorrectly classified as positives: this is unacceptable in many real cases.
In nQuery 9.3, sample size tables are added for partial ROC analysis for assessing one or two ROC curves.
Table added:
• Test for One Partial ROC
• Test for Two Partial ROC
What is it?
The variance is the most commonly cited statistic for evaluating the degree of variation in a measure. Researchers will often be interested in evaluating the degree to which the variation differs
between one or more groups. Several tests have been proposed for this purpose including the F-test, Levene’s Test and Bonett’s test for comparing variances.
In nQuery 9.3, a sample size table is added for the comparison of two independent variances using the F-test, Levene’s Test or Bonett’s Test.
Table added:
• Tests for Two Variances (F-test, Levene’s Test, Bonett’s Test)
Randomization Lists
What is it?
Randomization is a vital part of ensuring valid statistical inferences from common statistical tests used widely in clinical trials. Randomization creates treatment groups that are comparable and
reduces the influence of unknown factors. However, in clinical trials there are often ethical and logistical considerations that mean that simple random allocation may not be appropriate.
For example, it is often considered important to ensure that balance is maintained at any given time during a clinical trial to reduce the potential effect of time-varying covariates or when
sequential analysis is planned. Additionally, it can be important that covariates such as gender are relatively balanced overall.
nQuery 9.2 saw the addition of the Randomization Lists tool which will allow for the easy generation of randomization lists that account both for randomization and any balancing covariates of
interest. 9.2 included the following randomization algorithms:
Table added:
• Block Randomization
• Complete Randomization
• Efron’s Biased Coin (2 Groups Only)
nQuery 9.3 sees the addition of multiple new randomization algorithms, an increase in the number of allowable centers and a fully updated chapter on randomizations lists in our user manual. The
summary of the updates to the randomization lists features are:
• 4 New Randomization Algorithms (Smith’s, Wei’s Urn, Random Sorting, Random Sorting with Maximum Allowable Deviation)
• Increase in number of allowable centers from 25 to 500
• Improved User Manual Chapter on Randomization Lists
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S is the midpoint of RT. R has coordinates (-6, -1), and S has coordinates (-1, 1) . What are the coordinates of T?
thank you sooo much :]]? | HIX Tutor
S is the midpoint of RT. R has coordinates (-6, -1), and S has coordinates (-1, 1) . What are the coordinates of T? thank you sooo much :]]?
Answer 1
#"one way is to use vectors"#
#"since S is the midpoint of RT then"#
#color(white)(rArrult)=((-2),(2))-((-6),(-1))=((-2+6),(2+1))=((4), (3))#
#rArr"coordinates of T "=(4,3)#
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Answer 2
To find the coordinates of point T, we can use the midpoint formula, which states that the coordinates of the midpoint of a line segment are the average of the coordinates of its endpoints.
Let's denote the coordinates of point T as (x, y). Since S is the midpoint of RT, we can find the average of the x-coordinates and the y-coordinates of R and T to find the coordinates of point T.
The x-coordinate of S is -1, and the x-coordinate of R is -6. So, the average of these two x-coordinates gives us the x-coordinate of T.
[ \frac{(-6) + x}{2} = -1 ]
Solving for x: [ (-6) + x = -2 ] [ x = 4 ]
Similarly, the y-coordinate of S is 1, and the y-coordinate of R is -1. So, the average of these two y-coordinates gives us the y-coordinate of T.
[ \frac{(-1) + y}{2} = 1 ]
Solving for y: [ (-1) + y = 2 ] [ y = 3 ]
Therefore, the coordinates of point T are (4, 3).
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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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Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Exposition / QuantumPhysics
It seems that if one is working from the point of view of getting beauty into one's equation, and if one has really a sound insight, one is on a sure line of progress. - P.A.M. Dirac
My insights
The ways of figuring things out in physics.
The problem of measurement.
Choice frameworks.
Duality of counting forwards and backwards.
Combinatorial interpretation of continuous functions, especially of orthogonal polynomials and solutions to Schroedinger's equation.
Duality of the "mother function" {$e^x$} which equals its own derivative.
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Radius and Chords: Middle Secondary Mathematics Competition Question
A circle of radius R has two perpendicular chords intersecting at point P, that lies within the circle.
If AP=4, BP=6 and CP=3, find the length DP and hence the radius R of the circle.
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ETHOS: An automated framework to generate multi-fidelity constitutive data tables and propagate uncertainties to hydrodynamic simulations
Accurate constitutive data, such as equations of state and plasma transport coefficients, are necessary for reliable hydrodynamic simulations of plasma systems such as fusion targets, planets, and
stars. Here, we develop a framework for automatically generating transport-coefficient tables using a parameterized model that incorporates data from both high-fidelity sources (e.g., density
functional theory calculations and reference experiments) and lower-fidelity sources (e.g., average-atom and analytic models). The framework incorporates uncertainties from these multi-fidelity
sources, generating ensembles of optimally diverse tables that are suitable for uncertainty quantification of hydrodynamic simulations. We illustrate the utility of the framework with
magnetohydrodynamic simulations of magnetically launched flyer plates, which are used to measure material properties in pulsed-power experiments. We explore how changes in the uncertainties assigned
to the multi-fidelity data sources propagate to changes in simulation outputs and find that our simulations are most sensitive to uncertainties near the melting transition. The presented framework
enables computationally efficient uncertainty quantification that readily incorporates new high-fidelity measurements or calculations and identifies plasma regimes where additional data will have
high impact.
Over the past few decades, a great deal of effort has been devoted to assessing the uncertainties of and improving the modeling of constitutive properties such as equations of state (EOS)^1 and
charged-particle transport coefficients.^2,3 Data of these properties are critical inputs for radiation- and magneto-hydrodynamic simulations of diverse plasma systems such as inertial-fusion targets
^4 and white dwarf stars.^5 At the same time, experimental and diagnostic capabilities have advanced to enable high-precision measurements of the EOS^6,7 and transport properties.^8,9 The
measurements provide benchmarks for models and constrain the data tables used by hydrodynamic codes.
Notably, however, incorporating new knowledge from high-fidelity calculations and precision experiments into integrated hydrodynamic simulations introduces a significant bottleneck that requires the
dedicated efforts of experts. In practice, the EOS and transport coefficients are first tabulated based on calculations from one or more theoretical approaches, which must then be adjusted manually
to fit known data such as ambient pressures and conductivities and supplied to a hydrodynamic code. Moreover, the EOS tables are usually generated independently from transport-coefficient tables,
which can introduce inconsistencies between them such as the temperature and density of phase transitions (e.g., melt). Because hydrodynamic simulations access a wide range of application-dependent
plasma conditions, tables of constitutive data must span an enormous range of temperatures and densities. It is common for tables to cover a parameter space that spans many orders of magnitude in
both density and temperature. Within this large parameter space, very little experimental data exist for validation, and the models that generate data have different levels of uncertainty. All of
these considerations add to the difficulty of developing a systematic approach to uncertainty quantification of hydrodynamic simulations.
Non-parametric analytic models are useful to generate transport-coefficient data rapidly across the necessary parameter range for hydrodynamic simulations.^10–18 These models are often validated
against experimental and simulation data when possible. However, it is rare for a non-parametric analytic model to be sufficiently flexible to capture trends in data in disparate physical regimes.
Parametric analytic models offer a solution to this deficiency by providing additional flexibility through the fine-tuning of the model parameters.^19
Transport-coefficient tables are less ubiquitous than EOS tables. When a table for a particular application does not exist, hydrodynamic codes that require transport coefficients as closures are
forced to rely on simplifying approximations that may have limited accuracy in the regions of interest.^20,21 The quality of the results from hydrodynamic codes depends critically on the EOS and
transport-coefficient tables,^22,23 but exactly how the uncertainties in these tables correspond to uncertainties in integrated quantities is often indirect since the equations that rely on these
data are non-linear. Here, we present a framework for quantifying these uncertainties, using as an example the impact of the direct current (DC) electrical conductivity on observable predictions from
a magnetohydrodynamic (MHD) code.
To generate tables of the DC electrical conductivity across a wide range of conditions, we use a modification of the parameterized Lee–More–Desjarlais (LMD) model.^19 We constrain the parameters of
the LMD model to fit a multi-fidelity dataset containing experimental data, high-fidelity density functional theory molecular dynamics (DFT-MD) calculations,^24 and lower-fidelity calculations from a
DFT-based average-atom (AA) code.
To determine the LMD parameters, we use Bayesian inference on the multi-fidelity datasets and employ a simple surrogate model to extend our framework to be applicable across the large parameter
space. Our approach provides a framework to generate ensembles of tables automatically that incorporates uncertainties from multi-fidelity datasets and enforces consistent melt transitions between
the EOS and transport-coefficient tables. We also devise a scheme to perform optimal table selection for propagating dataset uncertainties to quantify the uncertainties in observables from MHD
simulations. While we apply the framework to electrical conductivities and outputs of MHD codes, the framework is general and can be applied to any constitutive property to which the data can be
fit—that is, to a parametric form or to a non-parametric form if the data are sufficiently dense.^25 We emphasize that the generalizability and automation of this approach to generate the tables
means that additional data can be easily incorporated to refine the uncertainty quantification. We name the framework developed here ETHOS, which stands for “electronic transport for hydrodynamics
with an optimized surrogate.”
The paper is organized as follows. Section II discusses the methods used to generate transport-coefficient datasets. These methods generate datasets across a range of plasma conditions but are not
sufficient to produce wide-ranging tables. The process of constructing tables from these datasets is given in Sec. III, where we describe the automated framework, ETHOS, that incorporates the
datasets into tables using the parameterized LMD model and Bayesian inference. The framework includes dataset uncertainty, can capture the correlations between data points, and selects optimal tables
for computationally efficient uncertainty quantification. Then, we use an ensemble of tables generated by the framework to quantify the sensitivities of the outputs of MHD simulations to
uncertainties in the DC electrical conductivity, showing simulation results in Sec. IV. We find that a 20% uncertainty in data of the DC electrical conductivity corresponds to uncertainties in MHD
outputs that are outside the resolution of experimental diagnostics, suggesting that the experimental data will further constrain our tables. We conclude in Sec. V by providing a suggested path
forward for extensions and improvements of the present framework.
For dense plasmas, Lee and More
derived an expression for the DC electrical conductivity that was later improved upon by Desjarlais,
is the uniform electron number density,
is the mass of an electron,
is the elementary charge,
is the chemical potential,
is the electron temperature, and
is Boltzmann's constant. In this work, we assume that the electrons and ions share the same temperature so that
. Finally, the term
is a correction for electron–electron collisions that depend on the average-ionization state
and the function
is represented by Fermi–Dirac integrals and is provided in the
. The LMD model provides approximations for the electron-ion collision rate
in the solid, liquid, and plasma states; these approximations contain tunable parameters that can be adjusted to agree with available data generated from simulation and experiment. See the
for more details of the LMD model used in this work.
The average electron-ion collision rate can also be computed from an AA model using an extended Ziman formalism^27–29 that is dependent on different choices of $Z*$ and the ion–ion static structure
factor. Variations in these quantities correspond to variations in the DC electrical conductivity from an AA model.^3,29 More sophisticated multi-center models based on DFT-MD provide high-fidelity
conductivity data based on the Kubo–Greenwood formalism, which predicts the optical (frequency-dependent) conductivity using the Onsager relations.^30–33 Other sources of data for the DC electrical
conductivity include experimental measurements near ambient conditions^34,35—usually spanning a temperature range of only a few hundred Kelvin—and a very limited number of measurements in the plasma
The three computational approaches described above have complementary strengths and weaknesses. While the LMD model is wide-ranging and computationally rapid, it contains tunable parameters and
assumes a fixed functional form. AA models take minutes to compute a single data point, are sensitive to modeler choices,^27 and are best used for liquids and plasmas.^3 DFT-MD is computationally
expensive, taking days or weeks to obtain values for the DC electrical conductivity, but extends to non-radially symmetric systems such as crystalline solids and molecules. AA models and DFT-MD are
sensitive to the choice of exchange-correlation functional, and DFT-MD is sensitive to finite-size effects^21 and the choice of pseudopotential. Together, these models enable the generation of sparse
sets of multi-fidelity DC electrical conductivity data.
In this work, we use the term dataset to mean a set of data generated by an experiment, from DFT-MD calculations, and an AA model at specific temperature and density locations. We use the term table
to mean the interpolated dataset (using the LMD model) that spans a density and temperature range outside the extent of the dataset. Here, we generated a dataset for the electrical conductivity of
beryllium along 32 isochores with densities ranging from $ρi=0.01−6.0$ gcm^−3 in intervals of 0.2 gcm^−3. The dataset spans a temperature range of $T=10−2−104$ eV and includes experimental and
simulation data of varying fidelity. Altogether, our dataset consisted of 1436 data points: 28 from an experiment,^34 48 from DFT-MD simulation, and 1360 from AA model calculations. Some of these
data are shown in Fig. 1. While this is a fairly dense dataset, a parametric form is needed to mitigate interpolation and extrapolation errors in the regions absent of data. A region that is of
particular importance is the low-temperature and solid-density region, which governs the growth rate of the electrothermal instability.^38–41 With this dataset, we now detail the procedure to
generate an ensemble of tables that can be supplied to MHD codes.
Given the multi-fidelity dataset described in Sec.
, we wish to construct tables suitable for use in MHD simulations. To do this, we use Bayesian inference to optimize the values of tunable parameters in the LMD model described in the
. In this particular study, the LMD model contained five tunable parameters. The benefit of using Bayesian inference is twofold. First, the dataset uncertainties and correlations between data are
included in the estimation of the parameter values. Second, Bayesian inference results in a
distribution of parameters that can be sampled to generate an ensemble of conductivity tables. It is the ensemble of tables that enable uncertainty quantification of MHD with this framework. The
foundations of the framework developed here rely on Bayes' theorem
$P(θLMD|σDC)=P(σ DC|θLMD)P(θLMD)P(σDC),$
denotes a vector of
LMD conductivity model parameters. To provide context to each term in Eq.
, we briefly review their meaning. Starting from the left-hand side of Eq.
is the posterior distribution—the distribution of LMD parameters that have been constrained by the available DC conductivity data. On the right-hand side of Eq.
, the term
, called the
distribution, describes the probability of obtaining a particular value of the DC electrical conductivity given a set of LMD model parameters: the likelihood distribution characterizes how we expect
our observations to distribute about the model (e.g., normally distributed for continuous quantities). The uncertainty in the data and correlations between data points are quantified in a covariance
matrix that enters that term. The term
, called the
distribution, characterizes any physics-based beliefs or previous analyses that suggest reasonable parameter values (e.g., only positive values of a parameter). It is common to set the prior
distribution to be a wide uniform distribution to ensure that the posterior distribution will fall within that range (see
Fig. 2
). Finally, the normalization term
is called the
; that term does not need to be computed when using the Markov-chain Monte Carlo approach employed in this work.
Bayesian inference has been applied to many areas of plasma physics including interpolating plasma transport-coefficient data,^25 analyzing inertial confinement fusion experiments,^47,48 producing
EOS,^49,50 and inferring current delivered to loads on pulsed-power machines.^51 Recent tutorials^48,52 discuss some of the basics of Bayesian inference. For the results presented herein, we compute
the posterior distribution by using the No U-turn sampler^53 with an acceptance rate of 77.5%. We assume that the likelihood distribution is normally distributed, specify a burn-in period of 2000
steps, and approximate our posterior distribution with 8000 samples. Figure 2 displays the prior and posterior distributions for one parameter of the LMD model at a particular density value where a
10% uncertainty was assumed on the DC electrical conductivity dataset. We see that the posterior distribution is roughly normal, and by sampling this distribution, we generate a range of an LMD
In Fig. 1, LMD fits to data of the DC electrical conductivity at the isochore $ρi=1.85$ gcm^−3 are shown. For these data, we have assumed a 10% uncertainty. The variation in the fits is most notable
near the region of the melt transition, where a spread of fits is present. The LMD parameters responsible for the spread of fits include the parameter shown in Fig. 2 and also a parameter that tunes
the Bloch–Grüneisen formula for the mean-free path—see Eq. (A2). The fit labeled as “fit: posterior median” is generated by using the posterior median for each of the LMD parameters. We chose the
median (as opposed to the mean, for example) because the median is less sensitive to outliers and heavily skewed distributions.
The procedure of obtaining optimized LMD parameters is carried out at each isochore within the range of our dataset described in Sec. II. Doing so results in a set of LMD parameters at each isochore,
which are then interpolated. This step ensures that the LMD model used to produce a table is both density and temperature dependent. Here, a simple linear interpolation that acts as a surrogate model
is used between the parameter values at each isochore. We note that not all parameters are constrained at each density; these parameters are not included in the fitting procedure. In regions without
electrical conductivity data (i.e., where $ρi<0.01$ gcm^−3 and $ρi>6.0$ gcm^−3), we extrapolate each fit by a constant value determined by the final constraining point.
In Fig. 1, each fit represents a valid set of LMD parameters although some sets of parameters are more probable than others. These sets of LMD parameters are used to generate an ensemble of tables
for use in an MHD code providing a direct connection from uncertainties in the DC electrical conductivity to uncertainties in the results of MHD simulations; uncertainty quantification using the
ensemble of tables is discussed in Sec. IV. The tables generated for the uncertainty quantification study contained a 300 $×$300 log-spaced temperature-density grid spanning seven orders of
magnitude in both temperature and density. To ensure that those tables were dense enough for our study, we carried out a number of simulations with increasingly dense tables. We found that, relative
to a log-spaced 1000 $×$1000 table, that we had assumed to be sufficiently dense, the results using the 300 $×$300 table were within 0.1% of the 1000 $×$1000 table.
The MHD codes used to simulate inertial confinement fusion or astrophysical plasmas are typically computationally expensive—especially in three spatial dimensions. It is common for these calculations
to take days or even weeks of computer time. Therefore, carrying out thousands of simulations to quantify the uncertainties of transport-coefficient tables is often computationally intractable. For
these applications, selecting a reduced number of tables from the full posterior distribution is desirable. It would be ideal if this reduced set of tables could capture the possible spread of
parameters sampled by the full posterior distribution since these tables would likely correspond to the largest uncertainty in the output of the MHD code. An approach to accomplish the task of
efficiently sampling the high-dimensional posterior distribution is presented in Sec. IIIA.
A. Optimal table selection for computationally efficient uncertainty quantification
After tables have been generated using the framework described in Sec.
, they can be used for uncertainty quantification analyses. To reduce the computational cost associated with carrying out numerous calculations with computationally expensive MHD codes, we introduce
an approach for determining a minimal set of tables that are representative of the set of tables generated using the full posterior distribution. We have employed the Morris–Mitchell criterion
to accomplish this task for simplicity but alternate space-filling design approaches exist.
The Morris–Mitchell criterion is set by minimizing the following quantity:
denotes the Euclidean distance between vectors
is the number of tunable LMD parameters. Additionally,
is a positive integer, and
is the number of posterior samples that will be used to generate
tables. To find a set of optimally diverse posterior samples, we compute
samples of the posterior distribution. Here,
is the total number of posterior samples of size
. We then compute the optimal set of posterior samples using
. As an example, consider that we wish to select
representative samples from the posterior distribution. Then, we might choose
samples of size
from the posterior distribution and pick the
samples that result in the minimal value of
. The need to select subsets from the posterior is not strictly required; however, it results in a decrease in the computation cost over computing
for all combinations of size
from the posterior distribution with possibly many thousands of samples.
In Fig. 3, we compare randomly selected posterior samples and optimally selected samples from the posterior using Eq. (3). We see that the optimally selected samples span a larger range in the
joint-posterior distribution than the randomly selected samples. In this case, the random sampling approach results in samples that were more clustered in the upper left portion of the
joint-posterior distribution.
We use this optimal sampling approach in Sec. IVA, where we simulate the dynamics of a magnetically launched flyer plate. As we will show, in contrast to random sampling, optimally sampling the
posterior distribution with Eq. (3) results in a more representative sampling of the distribution of flyer-plate velocities.
In cylindrical coordinates and at low frequencies, the single-species resistive-diffusion MHD equations in the radial direction are as follows:
is the fluid mass density defined in terms of the mass
is the number density,
is the velocity, and
is azimuthal component of the magnetic field, at radial distance
and time
. The right-hand side of Eq.
is obtained by assuming that current is flowing only in the
-direction. Additionally,
is the permeability of free space and
denotes the scalar pressure; in this work,
is obtained from the SESAME #92025 EOS, which contains Maxwell constructions. The resistive-diffusion MHD equations [Eqs.
] are numerically solved using a Lagrangian approach and Eq.
is integrated implicitly in time.
To compute the temperature T, that is used to query the EOS and conductivity tables, the internal energy is converted to an energy density and then updated. It is updated at each simulation time step
from three contributions. These contributions include work done by an artificial viscosity term (we employ the von Neumann–Richtmyer artificial viscosity), PdV work, and work done by Ohmic heating.
We neglect thermal conduction as our primary goal is to assess the influence of variations of the electrical conductivity on outputs of the MHD simulation. By neglecting the thermal conductivity in
this work, we may reduce the accuracy of the simulations, but we disentangle any potential concerns with coupling the electrical conductivity model to the thermal conductivity model which may have
its own uncertainty. Due to the impact that thermal conductivity may have on the fidelity of MHD simulations, we plan to carry out an in-depth analysis of the impact of thermal conductivity in future
work. After updating the value of the internal energy density, we determine the value of T (and p) from the EOS table. We use this value of T to determine the value of $σDC$ from the conductivity
A. Magnetically launched flyer-plate simulations
To assess the significance of the table generation framework, we select a platform that is particularly sensitive to the values of the DC electrical conductivity: magnetically launched flyer plates.^
58,59 We simulate a beryllium flyer plate with the one-dimensional MHD code described in Sec. IV, which eliminates potential biasing of simulation results due to sensitivities from geometry or
numerical choices that are important beyond a single spatial dimension.
A schematic of a magnetically launched flyer-plate is shown in Fig. 4; additional schematics are provided in Refs. 58–60. The flyer plate is driven by a current density $J$ that moves from the anode
to the cathode along a short circuit. The current density $J$ produces a magnetic field $B$ and the flyer plate accelerates due to the $J×B$ force. The side of the flyer plate that the current
density initially flows through sets the magnetic field boundary condition at that side of the flyer plate. The other side of the flyer plate expands freely.
Here, we consider two distinct current pulses: a short pulse and a long pulse. These pulses mimic two cases of the pulses used at the Z machine at Sandia National Laboratories. The short pulse has a
rise time of $∼$100ns and the long pulse has a rise time of $∼$400ns. These pulses are displayed in the insert of Fig. 5. The short pulse resembles the drive current used in experimental
platforms relevant to inertial confinement fusion. The long pulse is regularly employed in experiments that isentropically ramp a flyer plate into a material to measure its EOS properties. For the
short pulse, the flyer plate is 400 $μ$m wide; and for the long pulse, the flyer plate is 200 $μ$m wide. For the short pulse, the flyer is 7mm away from $r=0$mm, which is roughly the same for the
return-current can diagnostics in inertial confinement fusion experiments.^61 For the long pulse, the flyer is 500mm away from $r=0$ mm.
As discussed in Ref. 58, the $J×B$ force produces a stress wave that compresses and accelerates the flyer plate. Behind the stress wave, the magnetic field diffuses into the flyer plate. Eventually,
the stress wave, and later the magnetic field, reaches the free surface of the flyer plate. Depending on the form of the current pulse (e.g., the extent of the rise time and the peak current), the
velocity of the free surface may gradually or rapidly increase. If the free surface accelerates rapidly, then the surface experiences a shock due to the stress wave. This phenomenon is shown in Fig.
5 for the short pulse case. The shock time is the time when the velocity of the flyer plate is roughly discontinuous. Another prominent feature of flyer-plate evolution is the time that the magnetic
field reaches the free surface. We refer to this time as the magnetic-field burnthrough time or simply the burnthrough time. The burnthrough time in Fig. 5 appears as a bump in the velocity profile
after the shock time. This bump is caused by the sudden increase in temperature and decrease in density of the flyer plate as the material is melted. We note that these features are seen in
experiments^58,59 and constrain conductivity tables through an iterative procedure.
The shock wave that precedes the magnetic field arrival on the free surface could melt the flyer plate. In this case, we will not observe a distinct bump in the velocity of the flyer plate. Instead
of using the bump to indicate magnetic burnthrough, which is a less obvious feature, we calculate the Ohmic heating term, at the free surface. At some point in time, the Ohmic heating term will be
maximal and then decrease. The time at which the Ohmic heating term is maximal is another way to estimate the burnthrough time. As there is no distinct bump in the velocity profile for the short
pulse current profile, we compute the burnthrough time by noting the time at which the Ohmic heating term is maximal. For the long pulse current profile, the bump is apparent in the velocity profile,
and it is sufficient to use the time at which the gradient of the temperature at the free surface is maximal to indicate the burnthrough time.
B. Uncertainty quantification with uniform dataset uncertainty
We use the table generation framework detailed in Sec. III to produce the DC electrical conductivity tables for the MHD code discussed in Sec. IV; we output velocity profiles of the magnetically
launched flyer plate as a function of time and compare the simulated shock and burnthrough times.
Our analysis begins with assigning an uncertainty to the DC conductivity dataset that is used to fit the LMD model with the Bayesian inference approach of Sec. III. We perform a number of
calculations by varying the uncertainty assigned to the DC electrical conductivity dataset. We select the values of 5, 10, 20, 30, 40, and 50 percent for our study. We assume that all data points in
a dataset have the same uncertainty but later relax this assumption in Sec. IVC. It is important to emphasize that, with the framework developed here, uncertainty can be assigned to each data point
uniquely if available. The case of 10% dataset uncertainty is shown in Fig. 1.
Once a table is generated, it is supplied to the MHD code. The MHD code used here is computationally rapid, allowing for an ensemble of calculations. Because of this, we propagate 1000 tables for
uncertainty quantification based upon simulations using both the short and long pulses for all variations in dataset uncertainty. The results of the simulations are summarized in Figs. 6–8 and Tables
I and II. The uncertainty in the shock time (for the short pulse) and the burnthrough time grows with the uncertainty in the DC electrical conductivity datasets used to produce the tables. We observe
an increase of approximately 3ns in the mean of the burnthrough time for the short pulse case, which is a significant amount relative to the sub-nanosecond precision of the experimental diagnostics.
TABLE I.
% dataset uncertainty Shock time (ns) . Burnthrough time (ns) .
. 1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random) 1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random)
. .
5 153.79 154 $±$0.06 154 $±$0.05 154 $±$0.06 176.06 176 $±$0.56 176 $±$0.39 176 $±$0.38
10 153.80 154 $±$0.12 154 $±$0.07 154 $±$0.11 176.10 176 $±$0.61 176 $±$0.53 176 $±$0.76
20 153.80 154 $±$0.28 154 $±$0.11 154 $±$0.24 175.98 175 $±$3.31 176 $±$2.05 176 $±$1.55
30 153.89 154 $±$0.55 154 $±$0.31 154 $±$0.37 176.57 177 $±$3.16 176 $±$2.21 177 $±$2.42
40 154.01 154 $±$0.55 154 $±$0.45 154 $±$0.46 177.34 180 $±$5.57 176 $±$2.92 178 $±$3.33
50 154.08 154 $±$0.76 154 $±$0.50 154 $±$0.51 178.26 178 $±$5.42 176 $±$2.81 179 $±$3.60
% dataset uncertainty Shock time (ns) . Burnthrough time (ns) .
. 1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random) 1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random)
. .
5 153.79 154 $±$0.06 154 $±$0.05 154 $±$0.06 176.06 176 $±$0.56 176 $±$0.39 176 $±$0.38
10 153.80 154 $±$0.12 154 $±$0.07 154 $±$0.11 176.10 176 $±$0.61 176 $±$0.53 176 $±$0.76
20 153.80 154 $±$0.28 154 $±$0.11 154 $±$0.24 175.98 175 $±$3.31 176 $±$2.05 176 $±$1.55
30 153.89 154 $±$0.55 154 $±$0.31 154 $±$0.37 176.57 177 $±$3.16 176 $±$2.21 177 $±$2.42
40 154.01 154 $±$0.55 154 $±$0.45 154 $±$0.46 177.34 180 $±$5.57 176 $±$2.92 178 $±$3.33
50 154.08 154 $±$0.76 154 $±$0.50 154 $±$0.51 178.26 178 $±$5.42 176 $±$2.81 179 $±$3.60
TABLE II.
% dataset uncertainty . Burnthrough time (ns) .
1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random) .
5 192.37 192 $±$0.26 192 $±$0.27 192 $±$0.26
10 192.39 192 $±$0.63 193 $±$0.38 192 $±$0.53
20 192.39 193 $±$1.33 193 $±$0.83 193 $±$1.10
30 192.60 193 $±$2.10 193 $±$1.22 193 $±$1.78
40 192.84 193 $±$1.82 193 $±$2.97 193 $±$2.17
50 193.06 195 $±$4.07 192 $±$3.41 193 $±$2.77
% dataset uncertainty . Burnthrough time (ns) .
1 table (median) . 10 tables (optimal) . 10 tables (random) . 1000 tables (random) .
5 192.37 192 $±$0.26 192 $±$0.27 192 $±$0.26
10 192.39 192 $±$0.63 193 $±$0.38 192 $±$0.53
20 192.39 193 $±$1.33 193 $±$0.83 193 $±$1.10
30 192.60 193 $±$2.10 193 $±$1.22 193 $±$1.78
40 192.84 193 $±$1.82 193 $±$2.97 193 $±$2.17
50 193.06 195 $±$4.07 192 $±$3.41 193 $±$2.77
For computationally expensive codes, thousands of simulations are intractable for uncertainty quantification studies. Therefore, we employ the optimal sampling approach discussed in Sec. IIIA to
sample the posterior distribution to generate ten tables. For comparison, we also randomly sample the posterior to generate ten tables. We found that the optimal sampling approach produced outputs of
the MHD simulation that had a larger uncertainty than by randomly sampling, see Tables I and II and Fig. 9. This wider spread of outputs is a desirable trait to ensure that the minimal set of tables
is representative of the true uncertainty. Note that for the long pulse case with 40% dataset uncertainty, the ten MHD simulations that used a random sampling approach had a larger uncertainty in the
burnthrough time than the optimal sampling approach. In this case, the choice of random samples from the posterior distribution produced conductivity tables that were more distinct near the melting
transition at solid density than those from the optimal sampling approach.
It is noteworthy that the burnthrough time is more sensitive to variations in the DC electrical conductivity than the shock time. We hypothesize that the shock time is more sensitive to variations in
the EOS (i.e., pressure, temperature, and energy) than the variations in the DC electrical conductivity, which will be a topic of future work. In Sec. IVC, we assess the significance of a particular
physical regime (e.g., solid, liquid, and plasma) within the table on the shock and burnthrough times.
C. Uncertainty quantification with non-uniform dataset uncertainty
As previously mentioned, the uncertainty assigned to each data point used for table generation need not be the same. Here, we relax the assumption that all data are equally uncertain by considering
two cases. In the first case, we assign a 50% uncertainty for conductivity data in the solid regime and assign a 10% uncertainty for data in the liquid and plasma regime. In the second case, we
reverse these assigned uncertainties such that the conductivity data in the solid regime have a 10% uncertainty, and the data in liquid and plasma regime have a 50% uncertainty. For both cases, we
use the short current pulse as the drive side boundary condition (see Fig. 5). Selectively assigning dataset uncertainties allows us to determine which states of matter (i.e., which regions of the
transport-coefficient table) most significantly alter the results of the MHD simulation. Figure 10 displays the results of this analysis based on 1000 MHD simulations. The MHD simulation results are
compared to a reference case of a table generated from the median parameters of a posterior distribution, assuming that a 10% uniform uncertainty is assigned to all data in the dataset. We found that
the flyer-plate velocity profile is more sensitive to uncertainties in values of the DC conductivity in the liquid and plasma regime. Our finding is consistent with a previous study,^58 which found
that the flyer-plate velocity was highly sensitive in the regions of conductivity near the melt transition. The analysis in Ref. 58 arrived at that conclusion through an approach in which discrete
regions of the conductivity table were multiplied by a scaling factor. Our framework improves upon that analysis by eliminating the introduction of unphysical discontinuities into the table.
Moreover, the number of MHD calculations needed to assess the sensitivity using the multiplier approach is drastically reduced from many thousands to hundreds (and to tens with the optimal sampling
The results of this non-uniform error assignment highlights the regime where high-precision measurements should be focused to be maximally impactful—that is, just after melt and in the liquid regime.
We have developed an automated framework to generate tables of transport coefficients for uncertainty quantification of magnetohydrodynamic codes. Our framework, named ETHOS, produces ensembles of
tables that are consistent with the statistics of the multi-fidelity dataset used to generate them and avoids the time consuming and potentially inconsistent process to manually tune model
parameters. This automated framework produces any number of tables that optimally span the posterior distribution to enable uncertainty quantification for computationally expensive codes and
highlights the regions of the dataset where additional data would reduce table uncertainties.
Using the exemplar problem of a magnetically launched flyer plate, we have carried out a systematic uncertainty quantification study using a one-dimensional Lagrangian resistive-diffusion
magnetohydrodynamic code. We found that a modest uncertainty of 20% in the values of the transport-coefficient dataset used to produce the tables corresponds to uncertainties in the results from
magnetohydrodynamic simulations that are larger than the sub-nanosecond resolution of experimental diagnostics. Further increasing the dataset uncertainty to 50% resulted in uncertainties that are
two to three times larger than the experimental diagnostics; experimental data for the cases discussed here would further constrain the tables generated in this work.
Within the framework, we have addressed a common issue with computationally expensive codes: the ability to carry out uncertainty quantification with a small number of simulations (on the order of
ten). Through an optimal selection process, we have produced tables that widely span the posterior distribution, resulting in an accurate representation of the uncertainty obtained by carrying out a
large number of simulations, i.e., a Monte Carlo-style sensitivity analysis, but at a fraction of the computation cost.
The framework developed here provides a set of steps that are readily generalized to other datasets and applications, for example, assessing the sensitivity of magnetohydrodynamic codes to
uncertainties in the equations of state—which is a topic of future work. Future simulations will explore more integrated studies and the effect of magnetic fields on transport quantities (e.g., the
thermal conductivity). Moreover, future work will also include the development of tools to perform inference of transport coefficients with magnetohydrodynamic simulations and experimental data using
a surrogate model.
While we have ensured that the tables generated in this work were sufficiently dense, the ability to carry out inline calculations of transport coefficients within a magnetohydrodynamic code would
eliminate the uncertainty incurred through table interpolation. While rapid analytic expressions for plasma transport coefficients exist, many contain tunable parameters (like the LMD model) or are
only accurate in limiting physical regimes (like the hot-dilute plasma regime). From the point of view of the DC electrical conductivity, there remains a dearth of data (both experimental and
simulation) in the cold and warm dense matter regimes. Namely, in the solid for densities that are rarefied and compressed relative to ambient conditions and also near the melt transition. The
additional data would further constrain the tables generated with the framework developed here, which would result in more predictive magnetohydrodynamic simulations of plasmas and divert attention
to other potential sources of uncertainty.
The authors would like to acknowledge John Carpenter, Alina Kononov, Brian Robinson, Nikita Chaturvedi, and Rebekah White for useful discussions. We would also like to thank Luke Shulenburger and
Mary Ann Sweeney for careful proofreading of the manuscript and Jeremy Boerner for providing the long current pulse. L.J.S. was supported by the Laboratory Directed Research and Development program
(Project No. 230332) at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a
wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective
technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Lucas J. Stanek: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead);
Writing – original draft (lead); Writing – review & editing (lead). William E. Lewis: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology
(supporting); Supervision (equal); Writing – review & editing (supporting). Kyle R. Cochrane: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology
(supporting); Supervision (equal); Writing – review & editing (supporting). Christopher A. Jennings: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting);
Methodology (supporting); Supervision (equal); Writing – review & editing (supporting). Michael P. Desjarlais: Data curation (equal); Formal analysis (supporting); Investigation (supporting);
Methodology (supporting); Software (equal); Supervision (equal); Writing – review & editing (supporting). Stephanie B. Hansen: Conceptualization (equal); Data curation (equal); Formal analysis
(supporting); Investigation (supporting); Methodology (supporting); Software (equal); Supervision (equal); Writing – review & editing (supporting).
The data that support the findings of this study are available within the article.
The foundations of the parameterized conductivity model used here rely on estimates for the electron-ion collision rate in the solid, liquid, and plasma. Many of the equations that follow have been
published in previous works;
we reproduce them here for completeness. Following Lee and More,
the collision rate in the solid and liquid is
$τeisolid=50R0TmTeθ3v¯, Te<Tm,$
$τeiliquid=50R0TmTe1v¯θ3θ4, Te≥Tm,$
are tunable parameters that we detail later,
is the melting temperature,
is the interatomic distance (here we use the Wigner–Seitz radius), and
is the average electron velocity. The effective temperature is
$Fj≡Fj(μ/kBTe)=∫0∞dx xj/[1+exp(x−μ/kBTe)]$
denotes a Fermi–Dirac integral of order
. The Fermi–Dirac integrals depend on the chemical potential
which can be obtained through the expression
denotes the
Fermi–Dirac integral of order 1/2.
The electron-ion collision rate for the plasma state is
$τeiplasma=3me(kBTe)3/222π(Z*)2nie4 ln Λ[1+exp(−μkBTe)]F1/2,$
$ln Λ$
is the Coulomb logarithm (many variants of the Coulomb logarithm exist, however, we use the version discussed in the original work by Lee and More).
The melting temperature
is approximated by Lee and More, but an improved model for the melting temperature has been given in the supplementary material of Ref.
; this improved model maintains the spirit of the model presented by Lee and More but agrees more closely with experimental data. The model for the melting temperature is
$Tm=0.5(ξ1+ξ)4ξ2b−2/3 (eV),$
is the nuclear charge,
is the mass density, and
is the atomic weight.
Extracting the melting temperature from an equation of state table is the preferred method when building a wide-ranging table of the electrical conductivity. This method ensures consistency. The
method employed in this work involved extracting, and then interpolating, the melt curve from an equation of state which eliminated the need to rely on Eqs. (A5)–(A7).
The parameters
in Eqs.
allow for adjustments of the LMD model to better match the conductivity in the solid. They are given by
where the parameters
, and
are tunable parameters of order unity. We note that for our Bayesian inference study, we have set
and do not treat it as a tunable parameter.
The electron-ion collision rates given in Eqs.
, and
are usually sufficient across a wide range of temperature and density values for a given system, although in some cases, a lower bound on the collision rate is needed. A modification to the lower
bound on the electron-ion collision rate (as given by Desjarlais
) is
$Θ=1+exp[(Te−25 000 K)/10 000 K]$
are tunable parameters. The minimum electron-ion collision rate is modified to be
To determine which collision rate to use in Eq. (1), one can simply take the maximum value of all the expressions for the collision rate or use an appropriate norm to ensure smoothness once a
particular expression of the collision rate becomes larger.
As presented by Lee and More, the function
is given by
except that the Fermi–Dirac integral in the numerator has been corrected here to be
instead of
as originally presented in the work of Lee and More. With this correction, Eq.
approaches the correct limits as described in Ref.
. This typographical error has also been noted in the supplemental material of Ref.
As discussed in Ref.
, an approximation to electron-neutral momentum transfer cross section is
is a cutoff radius and
is the electron wavenumber defined by
denotes the dipole polarizability and
denotes the Bohr radius. Additionally,
where the inverse electron (Thomas–Fermi) screening length is
In Eq.
is in energy units and
is the Fermi energy. The electron-neutral momentum transfer cross section [Eq.
] modifies the electron-ion collision rate by
is the number density of neutral particles. The term
which accounts for electron–electron collision is provided in Ref.
Finally, we use the ionization model proposed by Desjarlais,
$K=2g1g0na(2πmekBTeh2)3/2 exp {−IkBTe[1−(1.5e2IRa)]},$
$Z*=fe2/ZTF2ZTF+(1−fe2/ZTF 2)fe,$
are statistical weights,
is the ionization potential,
is the Wigner–Seitz radius where
is the total ion and neutral number density, and
is the average-ionization state generated by a Thomas–Fermi model that is readily computed with a fit provided by More.
provides a smooth transition from a single-ionization Saha model to the Thomas–Fermi model.
The equations presented in this appendix form the basis for a wide-ranging model of the DC electrical conductivity. With the Bayesian inference framework presented in Sec. III, the tunable parameters
are fit to available data. In some cases, the functional form of the LMD model will be insufficient to capture trends in the data, and modifications may be necessary.
et al, “
A review of equation-of-state models for inertial confinement fusion materials
High Energy Density Phys.
et al, “
Review of the first charged-particle transport coefficient comparison workshop
High Energy Density Phys.
L. J.
S. B.
B. M.
S. X.
P. F.
M. S.
L. G.
H. D.
S. D.
et al, “
Review of the second charged-particle transport coefficient code comparison workshop
Phys. Plasmas
et al, “
Lawson criterion for ignition exceeded in an inertial fusion experiment
Phys. Rev. Lett.
, and
, “
Current challenges in the physics of white dwarf stars
Phys. Rep.
et al, “
A measurement of the equation of state of carbon envelopes of white dwarfs
R. C.
M. D.
de La Cruz
et al, “
Extreme compression of planetary gases: High-accuracy pressure-density measurements of hydrogen-helium mixtures above fourfold compression
Phys. Rev. B
et al, “
Thermal transport in warm dense matter revealed by refraction-enhanced x-ray radiography with a deep-neural-network analysis
Commun. Phys.
B. K.
E. E.
M. Z.
L. E.
S. J.
L. B.
, and
S. H.
, “
DC electrical conductivity measurements of warm dense matter using ultrafast THz radiation
Phys. Plasmas
Y. T.
, “
An electron conductivity model for dense plasmas
Phys. Fluids
, and
, “
Diffusion coefficients for stellar plasmas
Astrophys. J. Suppl. Ser.
L. G.
M. S.
, “
Ionic transport in high-energy-density matter
Phys. Rev. E
L. G.
M. S.
, “
Efficient model for electronic transport in high energy-density matter
Phys. Plasmas
L. J.
M. S.
, “
Analytic models for interdiffusion in dense plasma mixtures
Phys. Plasmas
M. S.
, “
Viscosity estimates of liquid metals and warm dense matter using the Yukawa reference system
High Energy Density Phys.
, “
Viscosity estimates for strongly coupled Yukawa systems
Phys. Rev. E
Z. A.
L. G.
G. M.
L. G.
, and
M. S.
, “
Comparison of transport models in dense plasmas
Phys. Plasmas
, “
Electrical conductivity of hydrogen plasmas: Low-density benchmarks and virial expansion including e–e collisions
Phys. Plasmas
M. P.
, “
Practical improvements to the Lee-More conductivity near the metal-insulator transition
Contrib. Plasma Phys.
, and
, “
FLASH MHD simulations of experiments that study shock-generated magnetic fields
High Energy Density Phys.
L. J.
M. W. C.
M. A.
K. R. C.
, and
M. S.
, “
Efficacy of the radial pair potential approximation for molecular dynamics simulations of dense plasmas
Phys. Plasmas
, and
, “
High-mode Rayleigh-Taylor growth in NIF ignition capsules
High Energy Density Phys.
B. M.
, “
Charged particle transport coefficient challenges in high energy density plasmas
Phys. Plasmas
, and
, “
Electrical conductivity for warm, dense aluminum plasmas and liquids
Phys. Rev. E
L. J.
S. D.
, and
M. S.
, “
Multifidelity regression of sparse plasma transport data available in disparate physical regimes
Phys. Rev. E
Van Odenhoven
, “
Electron transport in strongly ionized plasmas
Phys. A: Stat. Mech. Appl.
, and
, “
Comparison of electron transport calculations in warm dense matter using the Ziman formula
High Energy Density Phys.
, and
, “
Equation of state, occupation probabilities and conductivities in the average atom Purgatorio code
High Energy Density Phys.
, “
Average-atom Ziman resistivity calculations in expanded metallic plasmas: Effect of mean ionization definition
Phys. Rev. E
M. P.
, and
, “
Electronic transport coefficients from density functional theory across the plasma plane
Phys. Rev. E
S. X.
K. A.
N. R.
A. J.
L. A.
V. V.
V. N.
R. C.
et al, “
A review on charged-particle transport modeling for laser direct-drive fusion
Phys. Plasmas
C. A.
K. R.
T. A.
M. K.
, and
J. P.
, “
Transport coefficients of warm dense matter from Kohn-Sham density functional theory
Phys. Plasmas
, and
, “
Computation of transport properties of warm dense matter using Abinit
Phys. Plasmas
T. C.
, “
Electrical resistivity of alkaline earth elements
J. Phys. Chem. Ref. Data
Electrical Resistivity Handbook
Peter Peregrinus Ltd.
London, UK
, and
, “
Resistivity of a simple metal from room temperature to $106$ K
Phys. Rev. Lett.
A. W.
, “
Progress in measurements of the electrical conductivity of metal plasmas
Contrib. Plasma Phys.
K. J.
E. P.
D. B.
M. E.
S. A.
J. M.
M. M.
, and
M. C.
, “
Simulations of electrothermal instability growth in solid aluminum rods
Phys. Plasmas
K. J.
T. J.
P. Y.
D. B.
E. S.
M. E.
M. C.
et al, “
Electrothermal instability mitigation by using thick dielectric coatings on magnetically imploded conductors
Phys. Rev. Lett.
T. M.
, and
D. B.
, “
Seeding the electrothermal instability through a three-dimensional, nonlinear perturbation
Phys. Rev. Lett.
K. J.
D. B.
E. P.
M. C.
M. E.
S. A.
I. C.
B. W.
M. D.
, and
, “
Electrothermal instability growth in magnetically driven pulsed power liners
Phys. Plasmas
R. S.
, Jr.
, and
P. M.
, “
The electrical conductivity of an ionized gas
Phys. Rev.
, Jr.
, “
Transport phenomena in a completely ionized gas
Phys. Rev.
, and
, “
Dense plasma temperature equilibration in the binary collision approximation
Phys. Rev. E
R. G.
, and
Uncertainty Quantification and Predictive Computational Science
T. V.
, and
, “
Probabilistic programming in Python using PyMC3
PeerJ Comput. Sci.
W. E.
P. F.
S. A.
P. F.
G. A.
M. R.
A. J.
M. A.
D. J.
, and
, “
Deep-learning-enabled Bayesian inference of fuel magnetization in magnetized liner inertial fusion
Phys. Plasmas
P. F.
W. E.
, “
Advanced data analysis in inertial confinement fusion and high energy density physics
Rev. Sci. Instrum.
J. L.
J. D.
, and
K. W.
, “
Quantifying uncertainty in analysis of shockless dynamic compression experiments on platinum. II. Bayesian model calibration
J. Appl. Phys.
J. A.
, and
M. D.
, “
Learning thermodynamically constrained equations of state with uncertainty
APL Mach. Learn.
T. M.
S. J.
G. P.
N. B.
, and
K. R.
, “
Bayesian inferences of electrical current delivered to shocked transmission lines
J. Appl. Phys.
S. E.
E. C.
Bechtel Amara
L. L.
S. P.
et al, “
Thinking Bayesian for plasma physicists
Phys. Plasmas
M. D.
, “
The No-U-Turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo
J. Mach. Learn. Res.
M. D.
T. J.
, “
Exploratory designs for computational experiments
J. Stat. Plann. Inference
J. C.
F. J.
, “
Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems
Rel. Eng. Syst. Safety
V. R.
, “
Space-filling designs for computer experiments: A review
Qual. Eng.
R. P.
, “
High-energy-density physics
Phys. Today
K. R.
R. W.
, and
J. H.
, “
Magnetically launched flyer plate technique for probing electrical conductivity of compressed copper
J. Appl. Phys.
K. R.
, and
, “
Determining the electrical conductivity of metals using the 2 MA Thor pulsed power driver
Rev. Sci. Instrum.
R. W.
M. D.
, and
, “
Magnetically driven hyper-velocity launch capability at the Sandia Z accelerator
Int. J. Impact Eng.
D. A.
M. R.
D. E.
S. A.
A. J.
C. A.
M. R.
T. J.
et al, “
An overview of magneto-inertial fusion on the Z machine at Sandia National Laboratories
Nucl. Fusion
M. S.
, “
Data-driven electrical conductivities of dense plasmas
Front. Phys.
Advances in Atomic and Molecular Physics
), Vol.
, pp.
© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
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SuperFib - an example of using lexical closure in Python
This story begins with simple implementation of a function that calculates Fibonacci numbers.
def fib(n):
return 1 if n == 1 or n == 2 else fib(n-1) + fib(n-2)
Assuming that we are using 1-based indices, we can run a couple of tests and see that it works. Of course, fib(35) takes some time to calculate - about 15 seconds on my laptop.
Now, if you don't know what lexical closure is, I recommend
reading about it
on Wikipedia) before going further. Let's try to cache the results, but also be better than using conventional
- not only will we cache the final result of each call, but also intermediate results.
def fib():
_dict = dict()
def inner(n): nonlocal _dict
if n == 1 or n == 2:
return 1
return _dict[n]
except KeyError:
_dict[n] = inner(n-1) + inner(n-2)
return _dict[n]
return inner f = fib() f(1000)
, with:
However, if I call f(1500)
calling f(1000) first, the calculation will not finish and my interpreter will crash or at least restart because of stack evaluation overflow.
But if I ran f(1000) and then f(1500) and then f(2000) and so on and so forth, advancing by 500 each time, I can very quickly get to quite far elements of Fibonacci sequence. Calculating 10 000th
element is quick and easy, compared to conventional approach. We can quickly get to the point where printing the numbers to the console takes more time than actually calculating them.
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frequent itemset mining?
Which is the most efficient method for frequent itemset mining?
BOMO algorithm presented in [7] is a frequent pattern-growth (FP-growth) based approach and known as the currently best algorithm in mining N-most interesting itemsets category. BOMO uses a compact
frequent pattern-tree (FP-tree) to store compressed information about frequent itemsets.
What is a typical example of frequent itemset mining?
A typical example of frequent itemset mining is market basket analysis. This process analyzes customer buying habits by finding associations between the different items that customers place in their
“shopping baskets” (Figure 6.1).
What are frequent itemset mining methods?
Frequent Itemset Mining is a method for market basket analysis. It aims at finding regularities in the shopping behavior of customers of supermarkets, mail-order companies, on-line shops etc. ⬈ More
specifically: Find sets of products that are frequently bought together.
Which itemset is the most frequent What is its support in percentage )?
B) Frequent itemset(s)
itemset support
1 sunscreen 0.500
2 sandals 0.400
3 bowls 0.200
4 battery, sunscreen 0.150
How do I generate frequent itemset?
Frequent Itemset Generation
1. Reduce the number of candidates: use pruning techniques such as the Apriori principle to eliminate some of the candidate itemsets without counting their support values.
2. Reduce the number of transactions: by combining transactions together we can reduce the total number of transactions.
What does frequent itemset mean?
Definition. Frequent itemsets (Agrawal et al., 1993, 1996) are a form of frequent pattern. Given examples that are sets of items and a minimum frequency, any set of items that occurs at least in the
minimum number of examples is a frequent itemset.
What is frequent itemset generation?
A frequent itemset is an itemset whose support is greater than some user-specified minimum support (denoted Lk, where k is the size of the itemset) A candidate itemset is a potentially frequent
itemset (denoted Ck, where k is the size of the itemset)
What are frequent patterns give an example?
Frequent patterns are itemsets, subsequences, or substructures that appear in a data set with frequency no less than a user-specified threshold. For example, a set of items, such as milk and bread,
that appear frequently together in a transaction data set, is a frequent itemset.
What do you mean by frequent itemset and closed itemset?
Definition: It is a frequent itemset that is both closed and its support is greater than or equal to minsup. An itemset is closed in a data set if there exists no superset that has the same support
count as this original itemset.
Which of the following is the direct application of frequent itemset mining?
Data processing and statistical analysis.
How do I find closed frequent itemset?
If we set the minsup to be 2, any itemsets that appear more than twice will be frequent itemsets. And among those frequent itemsets, we can find closed and maximal frequent itemsets by comparing
their support(frequency of occurrence) to their supersets. We can see the maximal itemsets are a subset of closed itemsets.
What is the support of frequent itemset?
The support (or occurrence frequency) of an itemset A, where A is a set of items from I, is the number of transactions containing A in DB. An itemset A is frequent if A’s support is no less than a
user-specified minimum support threshold θ. An itemset A which contains k items is called a k-itemset.
How do you mine frequent patterns?
Mining frequent pattern with candidate generation….
1. Generate Candidate set 2, do the second scan and generate Second item set.
2. Generate Candidate set 3, do the third scan and generate Third item set.
What does frequent pattern mining mean?
Definition. Frequent Pattern Mining is a Data Mining subject with the objective of extracting frequent itemsets from a database. Frequent itemsets play an essential role in many Data Mining tasks and
are related to interesting patterns in data, such as Association Rules.
What is maximal frequent itemset explain with example?
A maximal frequent itemset is a frequent itemset for which none of its immediate supersets are frequent. To illustrate this concept, consider the example given below: The support counts are shown on
the top left of each node. Assume support count threshold = 50%, that is, each item must occur in 2 or more transactions.
What is the relation between a candidate and frequent itemset?
(a) A candidate itemset is always a frequent itemset
(b) A frequent itemset must be a candidate itemset
(c) No relation between these two
(d) Strong relation with transactions
What is the maximum number of frequent itemsets that can be generated from the data?
Answer: There are six items in the data set. Therefore the total number of rules is 602. (b) What is the maximum size of frequent itemsets that can be extracted (assuming minsup > 0)? Answer: Because
the longest transaction contains 4 items, the maxi- mum size of frequent itemset is 4.
What is frequent itemset?
Frequent itemsets (Agrawal et al., 1993, 1996) are a form of frequent pattern. Given examples that are sets of items and a minimum frequency, any set of items that occurs at least in the minimum
number of examples is a frequent itemset.
What do you mean by frequent itemset and strong association rule?
Frequent itemset mining naturally leads to the discovery of associations and correlations among items in large transaction data sets. The concept of association rule was introduced together with that
of frequent itemset [2]. An association rule r takes the form of α → β, where α and β are itemsets, and α ∩ β = φ.
What is frequent pattern classification?
9.4. 2 Discriminative Frequent Pattern–Based Classification. From work on associative classification, we see that frequent patterns reflect strong associations between attribute–value pairs (or
items) in data and are useful for classification.
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Editing a matrix' dimensions
09-04-2015, 05:34 PM
(This post was last modified: 09-04-2015 05:39 PM by pwarmuth.)
Post: #1
pwarmuth Posts: 35
Junior Member Joined: Sep 2015
Editing a matrix' dimensions
Is there a better way to add rows/columns in the matrix editor than tediously inputting them one at a time via MORE -> INSERT -> ROW/COLUMN ? A hotkey? Perhaps a way to directly define the matrix'
and/or vector's dimensions? On this same line of thinking, I know the comma key will traverse along a row withinin the matrix template (on the home or CAS screen), but is there a carriage return
hotkey hidden away somewhere?
09-04-2015, 05:50 PM
Post: #2
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: Editing a matrix' dimensions
Those really are for just when you are trying to insert a row or column inside an existing matrix/vector. The matrix/vector will automatically grow by one when you type in the empty boxes on an empty
To do a 3x4 for example, select and empty M1. Enter it, and you should see Go-> on the menu key. Type <number> ENTER <number> ENTER <number> ENTER. Now tap or arrow over to the col 2, row 2 position.
Beging typing. The GO-> moves you to the right after each enter. When you hit the last col, you automatically move down to your next row.
This answer? Or are you seeing a different behavior?
Although I work for HP, the views and opinions I post here are my own.
09-04-2015, 06:01 PM
(This post was last modified: 09-04-2015 06:15 PM by pwarmuth.)
Post: #3
pwarmuth Posts: 35
Junior Member Joined: Sep 2015
RE: Editing a matrix' dimensions
I'm not sure how I missed that behavior. It does indeed automatically add columns as you type values into the first row. You do have to manually reset the curser to row 2 column 1 when you've
finished the first row. I cannot find a key that behaves as a carriage return. However, when you type values into the second row it will automatically perform a carriage return when you reach the end
of the second row and so on. The second part of question stands, however. A carriage return behavior for the matrix/vector templates would make data entry quite a bit more convenient. You can tap the
touch screen, but it's a pretty small target to hit. The template does not perform an automatic carriage return at the end of row 2 and so on, but will instead add a column. Perhaps mapping the enter
key to this function while the curser is within the template, or mimicking the behavior of the matrix editor for subsequent rows would be a useful addition?
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Journal of Information Science and Engineering, Vol. 40 No. 4, pp. 865-876
Separating Bichromatic Point Sets by an Axis-Parallel C-shaped Polygon
In the separability problem of bichromatic point sets, given two sets of points colored as blue and red, we want to put a special geometric shape in a manner that includes the blue points while
avoiding the red ones. This problem has various applications in data mining and other fields. Separability by various shapes, including L-shaped polygons, has been studied in the literature. In this
paper, the separability of bichromatic point sets by C-shaped polygons, which are more general than L-shaped polygons, is studied and an O(nlogn)-time algorithm is presented, where n is the total
number of points.
Keywords: computational geometry, separability, bichromatic point sets, C-shaped polygon, covering
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Tea Cups
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
Aunt Jane had been to a jumble sale and bought a whole lot of cups and saucers - she's having many visitors these days and felt that she needed some more. You are staying with her and when she
arrives home you help her to unpack the cups and saucers.
There are four sets: a set of white, a set of red, a set of blue and a set of green. In each set there are four cups and four saucers. So there are sixteen cups and sixteen saucers altogether.
Just for the fun of it, you decide to mix them around a bit so that there are sixteen different-looking cup/saucer combinations laid out on the table in a very long line.
So, for example:
a) there is a red cup on a green saucer but not another the same, although there is a green cup on a red saucer;
b) there is a red cup on a red saucer but that's the only one like it.
There are these sixteen different cup/saucer combinations on the table and you think about arranging them in a big square. Because there are sixteen, you realise that there are going to be four rows
with four in each row (or if you like, four rows and four columns).
So here is the challenge to start off this investigation:
Place these sixteen different combinations of cup/saucer in this four by four arrangement with the following rules:-
1) In any row there must only be one cup of each colour;
2) In any row there must only be one saucer of each colour;
3) In any column there must only be one cup of each colour;
4) In any column there must be only one saucer of each colour.
Remember that these sixteen cup/saucers are all different so, for example, you CANNOT have a red cup on a green saucer somewhere and another red cup on a green saucer somewhere else.
There are a lot of different ways of approaching this challenge.
When you think you have completed it, check it through very carefully. It's even a good idea to get a friend who has seen the rules to check it also.
This challenge is also found at
Printable Roadshow.pdf resource
Getting Started
Have you checked that all your cup and saucer combinations are different?
What could you try on the diagonals?
Is there any pattern to the way you have arranged the cups and saucers so far? Could you continue the pattern?
Student Solutions
We had a few good solutions to this challenge. Firstly from The Blue Coat Primary School in Wotton-Under-Edge
First I would put all the combinations together, then I'd line up all the double colour combinations together (red, red, white, white, green, green, blue, blue). After that I put all of the red tops
diagonally and the blue tops. I found it easier by spacing my cubes out, in my diagonal rows the saucer couldn't be the same as my double saucer, so I had to look really carefully according to the
rules. Finally I filled the rest in by always having only one of every colour in each row and column.
My tip is to look at the problem in a different way :).
Next, from Eliza at St. Joseph's School in Australia
To start with I drew a 4 x 4 grid. In the grid I placed 4 saucers of each colour so that It matched the criteria. When they were done I placed the tea cups on top so that there was only one green on
red and only one red on green and so on. Then I had four tea cups left and I places them on their own colour. Once I had placed the cups on their correct saucers I went through and check and double
checked. This is how I did tea cups.
Then, from Grace from Birdwood Primary in Australia
I used some counters and a grid that was four squares long and four squares wide. I placed counters for the saucers first and realised that I couldn't have the same colour counter on the diagonal. So
I placed different colour counters on the diagonals. Then I worked out where the remaining counters had to go. All of the counters for the saucers were placed and checked to see that there were no
counters the same colour in each row and column. I then placed the red cups on one diagonal and the white cups on the other diagonal. I then placed the blue and green cups. I then checked my work.
The picture seems to indicate that the coloured counters were replaced to give an interesting picture!
Next from Andrew from Dulwich College in Singapore
You would want to create the 2 main diagonals (the 4 square ones)matched up with the same teacup or saucer. Then once that's done, find the lower diagonal square of Position A (which is the upper
diagonal square is position F) and put in the same saucer in F as you did in A. Do the same thing for Position B and H. They can be random teacups for those 4 positions, but not for C,E,D and G.
Match the same saucer for the upper diagonal square for Position C (which the upper diagonal square is position E), but MAKE SURE THAT THE TEACUPS MATCH. Do exactly the same for positions D and G.
Then, do the remaining squares on the outside matching the saucers but at the same time making sure that the teacups match, as this is crucial. Once you have done the outside, this should be done and
if they are still mistakes, take out all the mistaken squares out and re-arrange them with diagonals so they match, although if you did everything carefully and accurately, there should be no need
for this bonus step. By the time you do all of this, you should have a beautiful, satisfying square of 16 saucers and teacups. Lovely, isn't it?
Next, here's Dylan's work sent in by the teacher from Dulwich College in Beijing China
Here is a solution from Dylan at an international school in Beijing. We spent two lessons exploring patterns and moved onto looking at 3 layers (including a plate). Dylan managed to discover the
patterns with rotating opposite cups and saucers to reach the combinations. He also ensured that the diagonal line had the same coloured cup and saucer as this was his
first strategy to solving the problem.
Finally Sam from Raymond Park Middle School in the USA
Well done all of you these are wonderful solutions.
Teachers' Resources
Why do this problem?
this problem
is an excellent way to work at problem solving with pupils in late primary and early secondary school. The problem lends itself for small group work, so that the learners have an opportunity to
decide on approaches. Then, having completed the challenge, groups can discuss the different solutions, and each method's strengths and weaknesses. If the introductory story is told to the full, it
is also a good problem for having pupils decide what information is useful and what is irrelevant.
Possible approach
Telling the story with little or no warning about why you're telling it is a good way to get learners engaged with this problem. Having set the scene, you could begin by giving the class a chance to
work in pairs to find the sixteen different combinations of cup/saucer.
It's a good idea to encourage children to work in pairs or small groups (perhaps up to four) when solving this challenge. Give them time to work on it and then gather the whole group together to have
a discussion about where they have got to so far. This sharing of ideas will help move everyone forward - you will find that different groups have approached the task in different ways (for example
some might set out the saucers first, others might keep the cup/saucer pairs together). Some might come up with relevant points which they think are important.
Once they have reached solutions, invite groups to share their approaches again. Is it possible to decide on a particuarly 'good' way of solving this challenge? (Learners themselves will be able to
articulate what they mean by 'good' in this context.) It would then be interesting to encourage each group to have another go at the problem using an alternative method of their choice.
The results of this investigation would make an engaging and attractive classroom display.
See also
this article
by a PGCE student about using this activity.
If you need to check solutions you might like to use this interactive.
(When you have a complete solution it will say Well Done!!)
Key questions
Tell me about this.
What do you think you have to do?
What have you tried so far?
Have you checked that all your cup and saucer combinations are different?
Possible extension
The problem of course enters into its own when questions are asked like, " I wonder what would happen if we ...?" For example, children might consider using three sets of cups and saucers; using
plates to go with the cups and saucers when you have three colours.
For more extension work
You can expect these pupils to look carefully at different solutions (as opposed to different ways of solving the challenge) and then compare them to sort out similarities and differences as well as
equivalences. Then the problem can be extended to include a third attribute, for example a plate, so that the cup, saucer, plate combination would use three different colours. Deciding how to record
solutions in this case is quite a challenge.
Is it possible to arrange the cups and saucers if the diagonals also have to be different? Is there a system for getting all the possible answers?
Possible support
For youngsters who have difficulties with colours you might want to use this image:
(A bigger version can be seen
Children have done this activity using a variety of different materials to help them - it can be made part of the challenge for them to decide on the materials they will use. You could start with
just three differently coloured cups/saucers to be arranged in a 3 by 3 grid so that the aim of the investigation is understood. (The
Teddy Town
problem is essentially the same investigation as this one, but starts at a simpler point.)
With children near the end of primary school, the activity can be approached in a different way although the challenges are essentially the same - it's all about using playing cards. The saucers
would be replaced by the suits and the cups by the value of the cards. So we have four suits and four different values of the cards. [I've used the Ace, King, Queen and Jack, although of course any
four will do.]
Suppose we are using playing cards. There are quite a few solutions, but I'll take just one particular kind.
The first solution here puts the Jack, Queen, King and Ace of differing suits in the central 2 by 2 square. Then, the outside ones in each row and column are gradually puzzled out - often through a
kind of logic. So here is a typical result.
The pupils can then be asked to tell you what they notice about this arrangement. Some will recall the rules and say that is what they notice. Others notice patterns. So, one way of opening it out
further is to concentrate on the patterns of the solutions found. Here is another solution based upon the same starting places in the centre square.
You can get some discussion going concerning the patterns they see. Some will see the Aces forming a pattern - since they stand out more, so you may ask, " Do you see any other patterns in the way
that the cards are placed.
Then there are the suits to look at, you can ask the pupils to describe, draw, record etc. what they've noticed.
Here, from the first solution above, is one of the many ways that they may come up with.
This shows how the suits form shapes, we have Green for Hearts, Blue for Spades, Grey for Clubs and Red for Diamonds.
This shows how the values of the cards link to form shapes, we have Green for Kings, Blue for Aces, Grey for Queens and Red for Jacks.
These two show patterns also derived from looking firstly at the suits and then the values.
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Science Students Life Hack: Easy Ways to Find the Centripetal Force of an Object - biomadam
Any movement on the curved road represents acceleration, and it requires a force directed towards the center of the road.
This force is called the centripetal force, which is why it is called the force “looking for the center.”
The following formula can calculate its value:
• Where m is the mass of the object
• Whereas r is the circular radius
• And v is the velocity with which the object is moving
You must have felt unpleasant when your car entered a steep turn. You must have experienced being thrown to the sidelines.
It means that a special force acted on you; this force is called centripetal force.
This force is very breathtaking in sharp turns when it presses you to the side of the car.
An outside observer will see everything differently; when the car makes a turn, the observer will imagine that the vehicle is merely continuing the rectilinear motion.
Anybody who is not affected by the external force will feel the same.
Observers will feel that the car doors are putting pressure on the persons inside the car.
However, in both of these views, the events are identical. The only difference will be the explanation of what is happening by the internal and external observer.
It should be noted that you must not confuse the term centripetal force with gravity and deformation forces.
The existence of the centripetal force is because of centripetal acceleration.
The physical cause of this centripetal force can vary based on different circumstances.
How to Calculate the Centripetal Force
If you have the value of mass, the velocity of the moving object, and the circular radius, you can collect the centripetal force by using the formula mentioned earlier.
You can also calculate this force by using a centripetal force calculator from the internet.
By entering the mass values in kg, radius in meter, and velocity in meter per second, you will get the desired output according to the input.
We got the value of the centripetal force 490 N by entering the following values:
Mass = 100 kg
Radius = 40 m
Velocity = 14 m/s
We also have the option to calculate the other variables like mass, radius, and velocity. So, you see, the calculation of the centripetal force is not very difficult.
You can do it manually or by using the tools available on the internet. Centripetal force is vital for an object for the sake of its circular motion.
How to find Centripetal force with Incomplete Information
If you don’t have the complete information to put in the formula or the online tool, then it looks like it is impossible to find the value of the centripetal force.
But you can work out what the force might be in this whole process.
For example, if you want to find the centripetal force acting on a particular planet exerted by the moon or a star, you might know that this type of force comes from gravity.
It means you can also find the force by the equation for gravitational force:
Where m1 and m2 are the masses, and r is the separation between these masses.
To calculate the value of force without radius, you may need the circumference whose formula is C=2πr.
You can also find this force by calculating just acceleration using the formula:
And this is also called Newton’s second law of motion, which can be calculated by entering the acceleration and mass of the object.
Do not confuse centripetal force with centrifugal force as both are counterparts of each other.
Examples of Centripetal Force
1. Example
As the earth moves around the sun, it changes its direction continuously. Its acceleration is further directed towards the sun.
Behind this acceleration, there exists a force according to Newton’s 1^st and 2^nd law of motion.
According to Newton’s law of motion, every action reacts. In this case, the force causing the acceleration is gravity.
2. Example
By swinging a ball on a string in a circular motion with constant speed, we will get the cause of acceleration in the form of tension in the string.
3. Example
Turning the car with constant speed is another example, where the source of force can be the friction between the road and the vehicle’s wheels.
Summing it up
Centripetal force can be observed at many moments in our daily lives, and it is also essential for the movement of objects in a circular path.
We have seen how it is easy to find the centripetal force if you have proper values.
Even if you don’t have values, you can calculate with other suitable formulas as we described earlier.
We hope this guide will be helpful when you look to find the centripetal force in your practical life.
Biomadam.com allows people to contribute to our platform with their writing. Contact us for more details.
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Trying to bridge the gap between WFC “Even Simpler Tiled Model” and CSP propositional rules
Photo by Ross Sneddon on Unsplash
“Even Simpler Tiled Model” (ESTM) leads to an interesting question about rules properties (article also posted on: https://dev.to/antigones/
Recently I tried to implement my own version of “Even Simpler Tiled Model” — a simplified version of Wave Function Collapse algorithm (source: https://github.com/antigones/py-wfc-estm) as explained
in this page.
“Even Simpler Tiled Model”, just like Wave Function Collapse, can be for example used to generate maps out of a set of rules. The idea is that it is possible to formalize those rules from a sample
map and then use them to generate a new one, “propagating” reasoning about admissible/unadmissible values from tile to tile.
Playing with my own implementation, a question arose: what defines a rule making a map difficult to collapse, when applying the Wave Function Collapse algorithm in its “Even Simpler Tiled Model”
(ESTM) simplified form?
Trying to answer this question, I ended up fallbacking the model to a propositional logic rules-related question and trying to formalize “Even Simpler Tiled Model” as a set of propositional logic
I took a sample map, defined as follows:
img = [
Analyzing the sample map, the following rules came out:
'L': {
(1, 0): {'L', 'C'},
(0, 1): {'L', 'C'},
(0, -1): {'L', 'C'},
(-1, 0): {'L'}
'C': {
(-1, 0): {'L'},
(0, -1): {'S', 'L', 'C'},
(1, 0): {'S'},
(0, 1): {'S', 'L', 'C'}
'S': {
(-1, 0): {'S', 'C'},
(0, -1): {'S', 'C'},
(1, 0): {'S'},
(0, 1): {'S', 'C'}}
The rules informally state the following (see map at the beginning of this post):
• a Land can have a Land at North, East, South, West; Coast in the East, West, South
• Sea can have Coast or Sea in the North, Sea in the South, Coast or Sea at East and West
• Coast must have Land in the North; can have Land or Coast and East and West; must have Sea at south
Let’s rewrite this set of rules as a set of logic propositions.
To accomplish this task and to be sure not to forget a rule, I wrote my own model checker using SymPy.
An 1x2 sized world
Since we have to rewrite the rules for each and every cell in the map, we first consider a 1x2 sized world.
In this world, we define:
L_00, “there is a land in (0,0)”
C_00, “there is a coast in (0,0)”
S_00, “there is sea in (0,0)”
L_10, “there is a land in (1,0)”
C_10, “there is a coast in (1,0)”
S_10, “there is a sea in (1,0)”
And the rules as:
C_00: S_10 & ~C_10 & ~L_10,
C_10: L_00 & ~C_00 & ~S_00,
L_00: ~S_10 & (C_10 | L_10), # a Land in (0,0) only admit a C or a L in (1,0), does not admit Sea
L_10: L_00 & ~C_00 & ~S_00,
S_00: S_10 & ~C_10 & ~L_10,
S_10: ~L_00 & (C_00 | S_00)
(C_00 | L_00 | S_00), # you have to choose at least a value
(C_10 | L_10 | S_10)
(where “:” stands for “if and only if”)
To have a “valid” world map (i.e. a map respecting all the defined rules) we can only choose a set of truth values assignment satisfying all the rules (a model).
The model checker prints all the assignments satisfying all the rules.
For this small world, we have the two following models:
Reading the first row in the truth table, we can see that choosing Land in (0,0) pushes a Coast or a Land in (1,0). This gives us two perfectly valid maps in just one row:
Reading the second row in the truth table, we can see that choosing a Sea or a Coast in (0,0) only pushes a Sea in (1,0):
leading to the two (both valid) maps:
A 2x2 map
We can extend reasoning for maps of size beyond 1x2. Let’s consider a 2x2 world map.
We can again write the propositional ruleset:
C_00: S_10 & ~C_10 & ~L_10 & (C_01 | L_01 | S_01),
C_01: S_11 & ~C_11 & ~L_11 & (C_00 | L_00 | S_00),
C_10: L_00 & ~C_00 & ~S_00 & (C_11 | L_11 | S_11),
C_11: L_01 & ~C_01 & ~S_01 & (C_10 | L_10 | S_10),
L_00: ~S_01 & ~S_10 & (C_01 | L_01) & (C_10 | L_10),
L_01: ~S_00 & ~S_11 & (C_00 | L_00) & (C_11 | L_11),
L_10: L_00 & ~C_00 & ~S_00 & ~S_11 & (C_11 | L_11),
L_11: L_01 & ~C_01 & ~S_01 & ~S_10 & (C_10 | L_10),
S_00: S_10 & ~C_10 & ~L_01 & ~L_10 & (C_01 | S_01),
S_01: S_11 & ~C_11 & ~L_00 & ~L_11 & (C_00 | S_00),
S_10: ~L_00 & ~L_11 & (C_00 | S_00) & (C_11 | S_11),
S_11: ~L_01 & ~L_10 & (C_01 | S_01) & (C_10 | S_10),
(C_00 | L_00 | S_00),
(C_01 | L_01 | S_01),
(C_10 | L_10 | S_10),
(C_11 | L_11 | S_11)
and use the model-checker to determine models:
We have four models here.
Let’s choose S in (1,1) — filtering all the models having S_11 to True:
The truth table suggests the following situation:
Let’s choose L in (0,0):
The truth table evolves the map to the following one:
which is valid.
If we take a step back and choose S in (0,0) then we have:
which leads to the following (equivalent) set of maps:
which can be “unraveled” as the following four maps:
Going up, 3x3 maps
Using the model checker to generate/check rules for a 3x3 map, we obtain the following models:
Let’s choose L for (2,2), to observe propagation as rule inference:
Leading to the map:
which shows propagation of Land values across the tiles!
Let’s choose L in (2,0) — and note that it is interchangeable with a Coast:
This gives two valid maps, where there is no difference in putting a Land or a Coast in (2,1):
A “very difficult” map
Let’s stress things and consider the following sample map:
img_hard = [
and again, let’s write out the rules as logic proposition:
B_00: B_10 & ~B_01 & ~K_01 & ~K_10 & ~S_01 & ~S_10,
B_01: B_11 & ~B_00 & ~K_11 & ~S_11 & (K_00 | S_00),
B_10: B_00 & ~B_11 & ~K_00 & ~K_11 & ~S_00 & ~S_11,
B_11: B_01 & ~B_10 & ~K_01 & ~S_01 & (K_10 | S_10),
K_00: B_01 & S_10 & ~B_10 & ~K_01 & ~K_10 & ~S_01,
K_01: S_11 & ~B_00 & ~B_11 & ~K_00 & ~K_11 & ~S_00,
K_10: B_11 & ~B_00 & ~K_00 & ~K_11 & ~S_00 & ~S_11,
K_11: ~B_01 & ~B_10 & ~K_01 & ~K_10 & ~S_01 & ~S_10,
S_00: B_01 & ~B_10 & ~K_01 & ~K_10 & ~S_01 & ~S_10,
S_01: ~B_00 & ~B_11 & ~K_00 & ~K_11 & ~S_00 & ~S_11,
S_10: B_11 & K_00 & ~B_00 & ~K_11 & ~S_00 & ~S_11,
S_11: K_01 & ~B_01 & ~B_10 & ~K_10 & ~S_01 & ~S_10,
(B_00 | K_00 | S_00),
(B_01 | K_01 | S_01),
(B_10 | K_10 | S_10),
(B_11 | K_11 | S_11)
The model checker only returns 1 model:
the only admissible solution.
For a 3x3 map with the “hard” rules, we obtain UNSAT and the map cannot collapse.
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Full Factorial
Design and Analysis of Catapult Full Factorial Experiment
Catapults are frequently used in Six-Sigma or Design of Experiments training. They are a powerful teaching tool and make the learning fun. If you have access to a catapult, we recommend that you
perform the actual experiment and use your own data. Of course, you can also follow along using the data provided. The response variable (Y) is distance, with the goal being to consistently hit a
target of 100 inches.
1. Click SigmaXL > Design of Experiments > 2-Level Factorial/Screening > 2-Level Factorial/Screening Designs.
2. The Number of X Factors can be 2 to 19. Using process knowledge, we will limit ourselves to 3 factors: Pull Back Angle, Stop Pin and Pin Height. Pull Back will be varied from 160 to 180 degrees,
Stop Pin will be positions 2 and 3 (count from the back), and Pin Height will be positions 2 and 3 (count from the bottom).
3. Select Number of Factors = 3.
4. The available designs are then given as: 4-Run, 2**(3-1), 1/2 Fraction, Res III and 8-Run, 2**3, Full-Factorial. If we had more than 5 factors, a Resolution III or Plackett-Burman Screening
design would typically be used. Here we will choose the 8-Run, 2**3, Full-Factorial design.
Notes: Design Generators and Aliasing of Effects will be reported for Fractional Factorial designs. When the Number of Factors is 9 or higher, Factor Name “I” is not used, in order to avoid
confusion with the Fractional Factorial Defining Relation “I”.
The Power Information is presented to assist the user with selection of number of runs and replicates for the design, so that one can see the trade-off between experimental cost and sensitivity
to detect Effects of interest.
Power (1-Beta) < 0.5 is considered as Very Low Power to detect Effect.
Power (1-Beta) >= 0.5 and < 0.8 is considered as Low Power to detect Effect.
Power (1-Beta) >= 0.8 and < 0.95 is considered as Medium Power to detect Effect.
Power (1-Beta) >= 0.95 and < 0.99 is considered as High Power to detect Effect.
Power (1-Beta) >= 0.99 is considered as Very High Power to detect Effect.
The power calculations require an estimate of experimental error, so when Replicates = 1 an assumption of 3 center points is used in order to give an approximate reference level of power
(regardless of the value for Number of Center Points per Block).
5. This design currently shows the following:
These power calculations assume 3 center points:
Very Low Power to detect Effect = 1*StDev (1-Beta < 0.5);
Very Low Power to detect Effect = 2*StDev (1-Beta < 0.5);
Medium Power to detect Effect = 3*StDev (0.8 <= 1-Beta < 0.95).
We would like to have medium power to detect an Effect = 2*StDev.
Change the Number of Replicates to 2. The Power Information is now:
Very Low Power to detect Effect = 1*StDev (1-Beta < 0.5);
Medium Power to detect Effect = 2*StDev (0.8 <= 1-Beta < 0.95);
Very High Power to detect Effect = 3*StDev (1-Beta >= 0.99).
We will therefore choose two replicates. The number of replicates will always be a tradeoff between the desired power and the cost of the experimental runs.
Specify 2 or more blocks if there are constraints such as the number of runs per day or some other known external “nuisance” variable (like 2 different catapults or 2 operators). Here we will
keep Blocks = 1 (i.e., no Blocking).
Center Points are useful to provide an estimate of experimental error with unreplicated designs, and allow detection of curvature. Typically 3 to 5 center points are used. Here we will not use
center points because we have replicated the design twice and do not expect significant curvature in the distance response. Furthermore, center points could not be set for Pin Height and Stop Pin
(without drilling additional holes!).
6. Complete the Factor Names, Level Settings and Response Name as shown:
7. Click OK. The following worksheet is produced:
8. You can enter information about the experiment in the fields provided. If you have access to a catapult, perform the experimental runs in the given randomized sequence, and enter the distance
values in the Distance column.
9. If you are not able to perform the catapult experiment, open the file Catapult DOE V6.xlsx.
10. Before we begin the regression analysis, we will have a quick look at the Main Effects and Interaction Plots. Click SigmaXL > Design of Experiments > 2 Level Factorial/Screening Main Effects &
Interaction Plots. The resulting plots are shown below:
Pull Back Angle is the dominant factor having the steepest slope. We can also see that the interaction terms are weak with the almost parallel lines.
11. Click SigmaXL > Design of Experiments > 2-Level Factorial/Screening > Analyze 2-Level Factorial/Screening Design.
12. We will use the default analyze settings (all terms in the model) to start. Click OK. The resulting Analysis report is shown:
13. The model looks very good with an R-Square value of 99.9%! The standard deviation (experimental error) is only 1.03 inches. Clearly Pull Back Angle is the most important predictor (X factor), but
all the main effects and two-way interaction are significant. However, the three-way interaction is not significant, so it should be removed from the model.
14. Click Recall Last Dialog (or press F3).
15. Remove the ABC interaction term as shown:
16. Click OK. The revised report is shown below:
17. All the terms in the model are now significant, and there is no evidence of lack of fit (P-value for lack-of-fit is 0.128 which is > .05).
18. Scroll down to view the Residual Plots. They also look very good, approximately normal, with no obvious patterns:
19. Scroll up to the Predicted Response Calculator. Enter the predicted values shown. These initial settings were determined by trial and error.
Note: CI is the 95% confidence interval for the long term mean and PI is the 95% prediction interval for individual values.
Note: The DOE Multiple Regression Model: Distance equation given above uses the coded coefficients, so if this equation is used to do predictions, the inputs must first be coded as done in the
Predicted Response Calculator.
20. Excel’s Solver may also be used to get a more exact solution:
21. The model prediction must then be confirmed with actual experimental runs at the given settings of Pull Back Angle = 179.5, Stop Pin = 2, and Pin Height = 2.
22. Alternative settings to achieve the target distance may be obtained with Contour/Surface Plots. Click SigmaXL > Design of Experiments > 2-Level Factorial/Screening > Contour/Surface Plots. Set
the Pin Height to 2 as shown (after clicking OK, you can use Recall SigmaXL Dialog to create another Contour/Surface plot with Pin Height set to 3):
23. Click OK. The following Contour and Surface Plots are displayed (with Pin Height = 2). Note the contour line with Catapult target distance = 100 inches. Although pin settings are discrete, they
appear as continuous, so this will be a constraint in our selection of alternative settings. In addition to Pull Back Angle = 179.5, Stop Pin = 2, Pin Height = 2, we see that Pull Back Angle
approx. = 171, Stop Pin = 3, Pin Height = 2 is also a valid setting. Alternative setting options are valuable in a designed experiment because they allow you to select lowest cost optimum
settings, or settings that are easier to control.
Tip: Use the contour/surface in conjunction with the predicted response calculator to determine optimal settings.
Analysis of Catapult Full Factorial Experiment with Advanced Multiple Regression
We will now redo the above analysis and optimization using Advanced Multiple Regression.
1. Open the file Catapult DOE Data for Adv MReg.xlsx. This is the Catapult DOE data copied into a workbook with A:, B: and C: removed from the Factor Names as they are not needed for Advanced
Multiple Regression.
2. Click Sheet 1 Tab. Click SigmaXL > Statistical Tools > Advanced Multiple Regression > Fit Multiple Regression Model. If necessary, click Use Entire Data Table, click Next.
3. Select Distance, click Numeric Response (Y) >>; select Pull Back Angle, Stop Pin, and Pin Height; click Continuous Predictors (X) >>. Check Standardize Continuous Predictors with option Coded:
Xmax = +1, Xmin = -1. Check Display Regression Equation with Unstandardized Coefficients. Use the default Confidence Level = 95.0%. Regular Residual Plots are checked by default. Check Main
Effects Plots and Interaction Plots. Leave Box-Cox Transformation unchecked.
□ Standardize Continuous Predictors with Coded: Xmax = +1, Xmin = -1 scales the continuous predictors so that Xmax is set to +1 and Xmin is set to -1. This is particularly useful for analyzing
data from a factorial design of experiments as we are doing here.
□ Display Regression Equation with Unstandardized Coefficients displays the prediction equation with unstandardized/uncoded coefficients but the Parameter Estimates table will still show the
standardized coefficients. This format is easier to interpret since there is only one coefficient value for each predictor.
4. Click Advanced Options. We will use the defaults as shown. Ensure that Stepwise/Best Subsets Regression is unchecked.
□ Term ANOVA Sum of Squares with Adjusted (Type III) provides a detailed ANOVA table for continuous and categorical predictors. Adjusted Type III is the reduction in the error sum of squares
(SS) when the term is added to a model that contains all the remaining terms.
□ R-Square Pareto Chart displays a Pareto chart of term R-Square values (100*SS[term]/SS[total]). A separate Pareto Chart is produced for Type III and Type I SS. If there is only one predictor
term, a Pareto Chart is not displayed.
□ Standardized Effect Pareto Chart displays a Pareto chart of term T values (=T.INV(1-P/2,df[error])). A separate Pareto Chart is produced for Type III and Type I SS. A significance reference
line is include (=T.INV(1-alpha/2,df[error])).
□ Saturated Model Pseudo Standard Error (Lenth’s PSE) is checked by default, but is not used here, as this is only applicable to saturated models with 0 error degrees of freedom.
5. Click OK. Using Term Generator, select ME + 2-Way Interactions. Click Select All >>. Include Constant is checked by default.
This matches the final model used in the original analysis for Distance. If we wanted to include the 3-Way Interaction, then ME + All Interactions would have been selected.
6. Click OK. The Advanced Multiple Regression report for Distance is given:
Note, the prediction equation is uncoded so the coefficients do not match the coded coefficients given in the Parameter Estimates table. If consistency is desired, one can always rerun the
analysis with Display Regression Equation with Unstandardized Coefficients unchecked. Blanks and special characters in the predictor names of the equation are converted to the underscore
character “_”.
The model summary statistics match the previous analysis. R-Square Predicted = 99.69%, also known as Leave-One-Out Cross-Validation, indicates how well a regression model predicts responses for
new observations and is typically less than R-Square Adjusted. This is also very good.
The Parameter Estimates and ANOVA match the previous analysis. The Pareto Chart of Standardized Effects for Distance with significance line is similar to the Pareto Chart of Abs(Coefficient) but
is based on the term T statistic.
Since this is an orthogonal design, Adjusted (Type III) Sum-of-Squares are the same as Sequential (Type I) Sum-of-Squares (not shown), so the Term R-Square Pareto shows the percent contribution
to variabity in the Distance and sums to R-Square = 99.9%.
7. The Durbin-Watson Test for Autocorrelation in Residuals table is:
The Durbin Watson (DW) test is used to detect the presence of positive or negative autocorrelation in the residuals at Lag 1. If either P-Value is < .05, then there is significant
autocorrelation. Here, there is no significant autocorrelation in the residuals, which is what we would expect in a randomized design of experiments.
8. The Breusch-Pagan Test for Constant Variance is:
There are two versions of the Breusch-Pagan (BP) test for Constant Variance: Normal and Koenker Studentized – Robust. SigmaXL applies an Anderson-Darling Normality test to the residuals in order
to automatically select which version to use. If the AD P-Value < 0.05, Koenker Studentized – Robust is used.
The report includes the test for All Terms and for individual predictors. All Terms denotes that all terms are in the model. This should be used to decide whether or not to take corrective
action. The individual predictor terms are evaluated one-at-a-time and provide supplementary information for diagnostic purposes. Note, this should always be used in conjunction with an
examination of the residual plots.
Here we see that the All Terms test is not significant, so we conclude that the variance is constant.
Tip: If the All Terms test is significant after model refinement, try a Box-Cox transformation. If that does not work, refit the model using Recall Last Dialog, click Advanced Options in the
Advanced Multiple Regression dialog, and uncheck Assume Constant Variance/No AC. SigmaXL will apply the White robust standard errors for non-constant variance. For details, see the Appendix:
Advanced Multiple Regression.
Tip: Lack of Constant Variance (a.k.a. Heteroskedasticity) is a nuisance for regression modelling but is also an opportunity. Examining the residual plots and individual predictors may yield
process knowledge that identifies variance reduction opportunities.
9. Click on Sheet MReg1 – Residuals to view the Residual Plots. Note, Sheet MReg# will increment every time a model is refitted.
The Residual Plots are similar to those in the previous analysis and look very good, approximately normal, with no obvious patterns.
Note: Residuals versus interaction terms are not plotted, but they can be manually created using the model design matrix to the right of the Residual Plots (use SigmaXL > Graphical Tools >
Scatter Plots).
10. Click on Sheet MReg1 – Plots. The Main Effects Plots and Interaction Plots for Overall Satisfaction are shown.
These are based on Fitted Means as predicted by the model, not Data Means as used in the previous analysis. Main Effects Plots with Fitted Means use the predicted value for the response versus
input predictor value, while holding all other variables at their respective means. Similarly for Interaction Plots, all predictors not being plotted are held at their respective means.
Pull Back Angle is the dominant factor having the steepest slope. We can also see that the interaction terms are weak with the almost parallel lines.
11. Click on Sheet MReg1 – Model. Scroll to the Predicted Response Calculator. Enter Pull Back Angle = 179.5, Stop Pin = 2, Pin Height = 2 to predict Overall Satisfaction with the 95% confidence
interval for the long term mean and 95% prediction interval for individual values:
Note the formula at cell L14 is an Excel formula.
This matches the previous initial settings. Here the full predictor names are used making it easier to use and interpret. The Coded Settings are calculated as part of the Excel formula. Also, the
prediction standard error SE is given.
12. Next, we will use SigmaXL’s built in Optimizer. Scroll to view the Optimize Options:
Here we can constrain the lower and upper bounds of the continuous predictors. (If there was a categorical predictor, e.g., different ball type, you could also specify a ball type to use for
optimization). Stop Pin and Pin Height are constrained to integers so these should be changed from 0 to 1 as shown.
The Optimizer will return only integer values for Stop Pin and Pin Height.
13. Scroll back to view the Goal setting and Optimize button. Specify Target = 100 as shown.
The optimizer uses Multistart Nelder-Mead Simplex to solve for the desired response goal with given constraints. For more information see the Appendix: Single Response Optimization.
14. Click Optimize. The response solution and prompt to paste values into the Input Settings of the Predicted Response Calculator is given:
15. Click Yes to paste the values.
16. Scroll to view the Optimize Options, and change the Stop Pin Lower Bound = 2, Upper Bound = 2; Pin Height Lower Bound = 2, Upper Bound = 2 as shown:
17. Click Optimize. The response solution and prompt to paste values into the Input Settings of the Predicted Response Calculator is given:
18. Click Yes to paste the values.
This now matches the Solver solution obtained in the previous analysis. Note however that the SE for the original SigmaXL solution is lower than the Solver solution. This is by design, when
multiple valid solutions are available, SigmaXL selects the one with the lowest prediction SE.
19. Next, we will create a Contour/Surface Plot. Click the Contour/Surface Plots button. Note that Stop Pin and Pin Height are not constrained to be integers in these plots.
20. A new sheet is created, MReg1 – Contour that displays the plots:
SigmaXL automatically creates a Contour/Surface Plot for each pairwise combination of continuous predictors. The plots on the left match those specified in the previous analysis.
Note that the table with the Hold Values gives the values used to hold a predictor constant if it is not in the plot.
Tip: The hold values are obtained from the Predicted Response Calculator settings, so if you wish to use different Hold Values, simply select the Model sheet, change the Enter Settings values and
recreate the plots.
Tip: Use the contour/surface plots in conjunction with the predicted response calculator to determine optimal settings.
Web Demos
Our CTO and Co-Founder, John Noguera, regularly hosts free Web Demos featuring SigmaXL and DiscoverSim
Click here to view some now!
Contact Us
Phone: 1.888.SigmaXL (744.6295)
Support: Support@SigmaXL.com
Sales: Sales@SigmaXL.com
Information: Information@SigmaXL.com
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Gödel's theorems -- transcending transcendence
It occurred to me that perhaps I should have preceded the previous post with this one, a discussion about the consequences of Gödel that would answer the question "so what?" So what if
Principia Mathematica
is never going to contain all of mathematical truth? What does that have to do with anything?
Logic is Inadequate
Gödel's Incompleteness Theorem exposes a glaring weakness in any formal system. It points at a strange disconnect between logic and truth. The beauty and austerity of formal logic become questionable
-- we can't know everything there is to know about the world using formal, deductive, logical reasoning!
I guess this is old news. But personal experience suggests that it is still easy to fall for the misconception that deductive reasoning is somehow "better" than other types of reasoning. Yes, it is
much cleaner, but heuristics, induction, abduction and analogical reasoning are so much more powerful, and play such an important role in intelligent behaviour.
Truth is Weird
Gödel's theorem shows that there are lots more subtleties in truth than we give it credit for. Perhaps the crux of Gödel's result come from our misunderstanding of what truth really mean in our
world. I guess the other question is: what exactly do mathematical truths mean in our world? Especially strange results like the
Banach-Tarski paradox
, so called the paradoxical decomposition of spheres...
Argument against strong AI
J.R. Lucas argued using Gödel's Incompleteness Theorem that machines can never be as intelligent as human beings. He argued that since machines are inevitably formal systems, Gödel's Incompleteness
Theorem applies. Thus, there must be some truth that machines ought not to be able to discover, but that humans can -- we can follow the construction from the last post to construct a statement that
means "this theorem is not provable by the machine". While we know that this statement is true, the machine would never be able to prove or disprove it. Thus humans will always have the upper hand.
Although this is an appealing argument, consensus is that it doesn't quite work. Two of the possible arguments against Lucas are:
1. In order for Gödel's theorem to hold, we must assume that the formal system is consistent. Computers need not be to be a consistent formal system. (Humans are quite inconsistent as well.)
2. There are paradoxical sentences that humans cannot assign truth value to; thus perhaps we are formal systems that are more powerful, but not something that cannot be surmounted by computers.
A common theme
Themes that come up in Gödel's theorem and its proof are quite ubiquitous. The futility of the formal system in attempt to "break out" of its bound of incompleteness is like Escher's dragon, below,
trying to break out of the 2-dimensionality of your screen.
There's a certain "zen"-ness to all this. Even the distinction between
is very much like the concept of the
versus the
. The existence of an unprovable statement is akin to the proposition that there will always be unknowable truths.
(Reminds me of this quote from Douglas Adams, the first one under "The Universe", for some strange reason)
The common theme here is, I believe, the desire to transcend, coupled with the inability to transcend. If right now, someone asked me to
bet on what the meaning of life is, then I would put on my "pretend to be Zen" hat and answer:
to transcend transcendence
End of Entry
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Fibonacci Series in Java using with Recursion, Scanner & For Loop
Fibonacci Series in Java using with Recursion, Scanner, For & While Loop
10 Sep 2024
753 Views
31 min read
Fibonacci Series in Java
Fibonacci Series in Java, It is the outcome of a series of numbers where the subsequent numbers build a relationship with the preceding numbers. This series will start from 0 to the numbers equal to
the sum of the two numbers that are previously defined. When the Java Fibonacci series program is given an integer input of N, it produces a Fibonacci Series of N numbers.
This Java tutorial will assist you in learning about the Fibonacci series in Java, how to generate the series using a scanner, how to implement the series using recursion, how to implement the series
using a for loop, how to implement the series using a while loop, how to implement the series using memoization, dynamic programming, iterative approach, and many other topics.
Fast-track your full stack development journey by enrolling in our Java Full Stack Developer Training now!
What is the Fibonacci Series?
Fibonacci Seriesis a sequence of numbers that start from 0 and 1; each increasing element is a result of adding the two preceding elements.The Fibonacci series looks like
The important point for the Fibonacci Series:
• The first number is 0.
• The second number is 1.
• The following number is equal to the sum of the two numbers that came before it.
• The Fibonacci sequence may be computed using the recursive formula F(n) = F(n-1) + F(n-2).
How can we use a Scanner to generate the Fibonacci Series?
The Scanner class in Java may be used to produce the Fibonacci seriesby allowing the user to enter the number of words. We follow some steps to generate the Fibonacci Series in that are:
• Step 1- we use to import thejava.util.Scanner class.
• Step 2- we define an Scanner class object that read user input.
• Step 3- Inquire as to how many Fibonacci series words the user wants to produce.
• Step 4- We use a loop and the user's input to create and print the Fibonacci series.
import java.util.Scanner;
public class FibonacciSeries {
public static void main(String[] args) {
// Create a Scanner object to read input
Scanner scanner = new Scanner(System.in);
System.out.print("Enter the number of terms: ");
int n = scanner.nextInt();
// Variables to store the first two terms
int firstTerm = 0, secondTerm = 1;
System.out.println("Fibonacci Series of " + n + " terms:");
// Loop to generate the Fibonacci series
for (int i = 1; i <= n; ++i) {
System.out.print(firstTerm + " ");
// Compute the next term
int nextTerm = firstTerm + secondTerm;
// Update the values for the next iteration
firstTerm = secondTerm;
secondTerm = nextTerm;
// Close the scanner object
Enter the number of terms: 5
Fibonacci Series of 5 terms:
• The program uses a Scanner to help the user with the number of terms in the Fibonacci series.
• It initializes the first two terms of the series as 0 and 1.
• A loop runs n times to generate and print each term of the Fibonacci series, updating the terms with each iteration.
• Finally, the Scanner object is closed to free up resources.
Fibonacci Series Program in Java Using Recursion
We can use the recursive method to produce the fibonacci series in Java. We can use it to design a Java method that calls itself to compute the Fibonacci number at a particular point.
import java.util.Scanner;
public class FibonacciRecursion {
public static int fibonacci(int n) {
if (n <= 1) {
return n;
return fibonacci(n - 1) + fibonacci(n - 2); // Recursive case
public static void main(String[] args) {
// Create a Scanner object to read input
Scanner scanner = new Scanner(System.in);
System.out.print("Enter the number of terms: ");
int n = scanner.nextInt();
System.out.println("Fibonacci Series up to " + n + " terms:");
for (int i = 0; i < n; i++) {
System.out.print(fibonacci(i) + " ");
// Close the scanner object
Enter the number of terms: 5
Fibonacci Series up to 5 terms:
• Use recursive method fibonacci(int n) to calculate each Fibonaccie lements, with base cases returning 0 and 1 and the recursive case adding the previous two numbers.
• User provides the number of terms, and a loop iteratively calls the fibonacci method to printteh series.
• We useScanner object to read user input and is closed at the end to free up resources.
Implementing Fibonacci Series in Java Using For Loop
The following is a basic iterative technique you may use with a for loop in Java to construct the Fibonacci sequence.
import java.util.Scanner;
public class FibonacciForLoop {
public static void main(String[] args) {
// Create a Scanner object to read input
Scanner scanner = new Scanner(System.in);
System.out.print("Enter the number of terms: ");
int n = scanner.nextInt();
// Variables to store the first two terms
int firstTerm = 0, secondTerm = 1;
System.out.println("Fibonacci Series of " + n + " terms:");
for (int i = 0; i < n; ++i) {
// Print the current term
System.out.print(firstTerm + " ");
// Compute the next term
int nextTerm = firstTerm + secondTerm;
// Update the terms for the next iteration
firstTerm = secondTerm;
secondTerm = nextTerm;
// Close the scanner object
Enter the number of terms: 5
Fibonacci Series of 5 terms:
• By using a Scanner object, The application assists the user in entering the Fibonacci series' term count.
• A for loop iterates n times, printing the current Fibonacci term and updating the terms for the next iteration.
• It gives another element by addingfirstTerm and secondTerm, updates firstTerm and secondTerm, and then prints the series.
Fibonacci Series in Java Using While Loop
Awhile loop helps to generate the fibonacci series in Java. This method is used to print the each element one after the other, until we could not get the desired output.
import java.util.Scanner;
public class FibonacciWhileLoop {
public static void main(String[] args) {
// Create a Scanner object to read input
Scanner scanner = new Scanner(System.in);
System.out.print("Enter the number of terms: ");
int n = scanner.nextInt();
// Variables to store the first two terms
int firstTerm = 0, secondTerm = 1;
System.out.println("Fibonacci Series up to " + n + " terms:");
int count = 0;
// Generate and print the Fibonacci series using a while loop
while (count < n) {
System.out.print(firstTerm + " "); // Print the current term
// Compute the next term
int nextTerm = firstTerm + secondTerm;
// Update the terms for the next iteration
firstTerm = secondTerm;
secondTerm = nextTerm;
// Increment the counter
// Close the scanner object
Enter the number of terms: 8
Fibonacci Series up to 8 terms:
• The program helps the user to enter the number of Fibonacci terms to generate and store this value in n.
• It uses a while loop to print each Fibonacci term up to the number specified, updating the terms and counter in each iteration.
• The next Fibonacci term is calculated as the sum of the current two terms firstTerm and secondTerm, and the loop continues until the desired number of terms is printed.
Fibonacci Series Using Memoization
Recursive algorithms are optimized using memorization, which prevents repetitive computations by retaining previously computed results. It improves performance for the Fibonacci sequence by caching
and reusing Fibonacci numbers.
import java.util.HashMap;
import java.util.Map;
public class FibonacciMemoization {
// Map to store previously computed Fibonacci numbers
private static Map<Integer, Integer> memo = new HashMap<>();
public static int fibonacci(int n) {
// Base cases
if (n <= 1) {
return n;
// Check if the value is already computed and stored in the map
if (memo.containsKey(n)) {
return memo.get(n);
// Compute the value and store it in the map
int result = fibonacci(n - 1) + fibonacci(n - 2);
memo.put(n, result);
return result;
public static void main(String[] args) {
int n = 10;
System.out.println("Fibonacci Series up to " + n + " terms:");
for (int i = 0; i < n; i++) {
System.out.print(fibonacci(i) + " ");
Fibonacci Series up to 10 terms:
• We use a HashMap to store and reuse previously calculated Fibonacci numbers that helps in speeding up the computation.
• the fibonacci technique first verifies the map before calculating each Fibonacci number iteratively.
• The main method prints the first 10 Fibonacci numbers by calling the memoized fibonacci method in a loop.
Fibonacci Series Using Dynamic Programming
For deviding a complex problems in the smaller problems, we use the dynamic approach to solve it. Every subproblem is solved by only one time and the result will be stored for the future use.
public class FibonacciDynamicProgramming {
public static void main(String[] args) {
int n = 6;
// Array to store Fibonacci numbers
int[] fib = new int[n];
// Base cases
if (n > 0) fib[0] = 0;
if (n > 1) fib[1] = 1;
// Fill the array using the dynamic programming approach
for (int i = 2; i < n; i++) {
fib[i] = fib[i - 1] + fib[i - 2];
System.out.println("Fibonacci sequence of " + n + " terms:");
for (int i = 0; i < n; i++) {
System.out.print(fib[i] + " ");
Fibonacci sequence of 6 terms:
• Define an array namefib to store the Fibonacci sequence, and create the first two elements (fib[0] and fib[1]) based on the base cases.
• Using the for loop to save the remaining of the array by using the Fibonacci formula fib[i] = fib[i - 1] + fib[i - 2].
• By iterating through the nth times, the result will be printed.
Fibonacci Series Using Iterative Approach
The Fibonacci sequence is computed iteratively by using a loop to calculate each term consecutively and without recursion. It is an easy and effective method to create the Fibonacci sequence by
thefixed number of terms.
public class FibonacciIterative {
public static void main(String[] args) {
int n = 9;
// Variables to store the first two terms
int firstTerm = 0, secondTerm = 1;
System.out.println("Fibonacci Series " + n + " terms:");
for (int i = 0; i < n; i++) {
System.out.print(firstTerm + " ");
// Compute the next term
int nextTerm = firstTerm + secondTerm;
// Update the terms for the next iteration
firstTerm = secondTerm;
secondTerm = nextTerm;
Fibonacci Series 9 terms:
• We start the first two numbers ob the Fibonacci series that arefirstTerm to 0 and secondTerm to 1.
• The for loop go throughn times, and it will printfirstTerm and then updating firstTerm and secondTerm to the following values in the sequence.
• Computes the next term as the sum of firstTerm and secondTerm, and updates the variables for the next iteration.
Fibonacci Series Using Static Variable
We use the static variable in Java to generate a Fibonacci series. This method becomes very useful when you want to manage a state across multiple method calls or instances of a class. This method
uses a static variable to record the current value or condition.
public class FibonacciStatic {
// Static variables to store the current and next terms
private static int firstTerm = 0;
private static int secondTerm = 1;
// Method to get the next Fibonacci number
public static int getNextFibonacci() {
int nextTerm = firstTerm + secondTerm;
// Update the terms for the next call
firstTerm = secondTerm;
secondTerm = nextTerm;
return firstTerm;
public static void main(String[] args) {
int n = 9; // generate the numbers of terms
System.out.println("Fibonacci Series up to " + n + " terms:");
for (int i = 0; i < n; i++) {
System.out.print(getNextFibonacci() + " ");
Fibonacci Series up to 9 terms:
• firstTerm and secondTerm are static variables used to keep track of the current and next Fibonacci terms across method calls.
• getNextFibonacci() computes the next Fibonacci number, updates the static variables, and returns the current term.
• The main method calls getNextFibonacci() in a loop to print the first n Fibonacci numbers, with n set to 9.
Fibonacci Series Using Direct Formula
By using Binet's formula, we don't need to use the recursion or iteration method. It helps us to generate the Fibonacci series in a quick way. The formula requires square roots and calculating powers
and is based on mathematical constants.
The nth Fibonacci number F(n) can be defined by using this formula:
public class FibonacciDirectFormula {
public static void main(String[] args) {
int n = 10; // Number of terms to generate
System.out.println("Fibonacci Series up to " + n + " terms:");
// Print Fibonacci series using the direct formula
for (int i = 0; i < n; i++) {
// Method to calculate the nth Fibonacci number using Binet's formula
public static long getFibonacci(int n) {
// Constants for Binet's formula
double sqrt5 = Math.sqrt(5);
double phi = (1 + sqrt5) / 2;
double psi = (1 - sqrt5) / 2;
// Binet's formula to calculate Fibonacci number
return Math.round((Math.pow(phi, n) - Math.pow(psi, n)) / sqrt5);
Fibonacci Series up to 10 terms:
• uses Binet's formula with constants phi and psi to compute the nth Fibonacci number directly.
• Math.pow() calculates powers, and Math.round() ensures the result is rounded to the nearest integer for accuracy.
• The main method prints the first n Fibonacci numbers by calling getFibonacci() in a loop.
In conclusion, we have discussed the Fibonacci series in Java, including different methods to generate the Fibonacci series. By selecting the best approach depending on the specifications of the
issue and performance factors, you may improve your comprehension of algorithm design and optimization in Java.
If you want to make your career as a Java developer, Scholorhat
provides you with a
Java Full Stack Course that you can join to make a brighter future.
The Fibonacci series has several applications in programming and computer science due to its mathematical properties and its use in various algorithms and problem-solving scenarios.
• Algorithm Design
• Computational Complexity
• Data Structure
• Optimization Problems
A Fibonacci time series is a sequence where each time step or interval follows the Fibonacci pattern, often used in financial markets to model periodic cycles or trends. It applies Fibonacci numbers
to time intervals, helping to analyze and forecast patterns in data.
Fibonacci heap is a data structure that provides efficient support for priority queue operations, including merging heaps and decreasing keys. It uses a collection of heap-ordered trees to optimize
performance in algorithms like Dijkstra's shortest path.
To write a Fibonacci series program in Java, you can use either iterative or recursive methods to compute the sequence. Initialize the first two terms and use loops or recursion to generate and print
subsequent terms.
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1- Easy illustration of the theory of pure bending.
Last Updated on October 12, 2024 by Maged kamel
The theory of pure bending.
Lecture objectives.
Our first item is the theory of pure bending assumptions and the relation. The second point is why we estimate the product of inertia. The next topic we will discuss is Mohr’s circle of inertia and
how it may be used to calculate the maximum and minimum values of the moment of inertia.
Assumptions to follow for the theory of pure bending.
First, we make some assumptions: when a beam is acted on by pure bending, it will be distorted.
But first, let’s look at the beam before deformation. We have two axes, X and Y, and we consider the beam’s section to be rectangular at the initial stage before deformation.
We have a section that is vertical and parallel to the neutral axis. The plain section that was parallel to the neutral axis before the deformation is thought to stay parallel to the neutral axis
after the deformation.
Following deformation, the beam will take on a curved shape and will no longer be vertical perpendicularly. Nonetheless, the section will remain perpendicular to the neutral axis upon deformation.
The second assumption is that bending causes minimal deformation.
The third assumption is that the beam is initially straight, with all longitudinal filaments bending into a circular arc with a shared center of curvature.
one means that all of these perpendicular sections are in one section, and if we assume another section, it will be a plain section perpendicular to the neutral axis. The third part will be
perpendicular to the neutral axis. If you expand all of these portions, they will also come together in the same place.
The line formed will be in the shape of a circular arc.
The fourth assumption of pure bending theory is that vertical deformation, or transverse strain (ฮตyy), can be omitted.
Compression is always followed by compression or transverse strain. In our situation, the transverse strain (ฮตyy=dh/h) will be negative. The longitudinal deformation will be equal to the expansion
of length divided by the initial length.
In ฮตyy, dl/l replaces length extension and is positive. We examine whether it has a small value and eliminate it.
The fifth assumption states that the radius of curvature is larger than the cross-sectional dimension. The sixth assumption states that Young’s modulus of elasticity in tension and compression will
be the same.
Pure bending: what is it? Pure bending results in a shear value of 0 when the beam experiences moments on both sides that are equal.
Why will there be no shear value? because there will be a pair produced by this bending moment on the right. The couple will be upward because of the anti-clockwise direction of the shear, which will
also cause a downward shear force on the right side. At the same time, an anti-clockwise resisting couple will be produced.
As a result, the shear forces will be opposed to one anotherโ downward for one and upward for the otherโ and they should cancel each other out.
For the bending moment diagram, the shear force will therefore equal zero for this reason. This is saggy if we wish to sketch it. Positively, value will be allocated to the appropriate assistance,
and this will be closed. That’s this, left to right.
The beam will experience bending with a positive value of equal bending. The left and right side moments will cause the beam to bend in the shape of an arc, and as the diagram shows, the enclosed
angle between the two perpendicular lines will be ฮฑ.
In the second scenario, which involves pure bending, you have a simply supported beam that is hinged and is being acted upon by two shear forces of equal values, P and P, which are operating on the
beam’s third point, L/3, and L/3, respectively.
In the shear forces figure, we have P, a positive shear force that rises and falls until it reaches B, and a negative shear that moves from the right side to the left and closes at point d.
As we have a positive bending moment (P*L/3) for the bending moment, the intermediate section will have a bending moment with no shear force between C and d = 0. The pure bending moment is present in
this second instance.
Stress formula for the case of pure bending.
The third supposition states that the beam is originally straight. As we can see, the section is rectangular, and the x- and y-axes were there before the deformation.
Consider a section where there is a small element separated by distance y before deformation, and the section’s apart distance is (dx).
Looking at the deformation, we observe that the beam would bend in this way, and the x, y, and perpendicular axis 3 all display the neutral axis, which was the horizontal neutral axis. Assuming that
an enclosed arc angle (dฮฑ) is to be studied and that the arc radius is R.
After equating, we will determine the strain value before deformation.
Following deformation, there was no strain.
This section, whose length is (dx), will eventually become Rdฮฑ. When the element at Y distance up is deformed, its length becomes (R-Y) multiplied by (dฮฑ), when it was initially dx.
Thus, the length of this red line was initially dx, or (R-y*dฮฑ). It’s moving toward the neutral axis if we compare it.
Since nothing has changed, we can interpret the dx as =(Rdฮฑ), where the deformation is either compression or tension.
We are above the neutral axis in this instance. Thus, the distortion will turn negative.
After subtracting +Rdฮฑ and -Rdฮฑ from (R-y)dฮฑ-Rdฮฑ, we are left with (-ydฮฑ). Given that a strain is a deformation across the origin length, we may calculate its derivative as (-ydฮฑ) from the
original length of R*dฮฑ. Consequently, we have (-y/R), and since stress/strain equals Young’s modulus of elasticity, our strain equals (-y)/R.
Since there is a strain, the strain * Young’s modulus of elasticity is given by the formula f= (e E). In other words, f=(-y/R)(E).
From assumption # 6, Young’s modulus of elasticity is the same for the case of tension and compression.
This is the pdf file used in the illustration of this post.
This is a link to a useful external resource by Jus.
This is a link to the following post: Post 2: 2-First moment of area and product of inertia at the CG.
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RIF Extensible Design Draft 2006-04-23 ~~~~~~~~~~~~~~~~~~~~~ Harold Boley (NRC), Michael Kifer (Stony Brook University), Jeff Pan (University of Aberdeen), Gerd Wagner (BTU Cottbus), Alex Kozlenkov
(Betfair), Jos de Bruijn (DERI), Mike Dean (BBN Technologies), Giorgos Stamou (NTUA) Summary This proposal starts with a language of conditions that can be used for positive rule bodies and then
extends it with negative literals, builtins, and constructs to query OWL and RDF (via SPARQL). The rationale for focusing on this sublanguage is that it can be shared among five different
sublanguages of interest to RIF. We then discuss various extensions of this sublanguage and propose an interoperation/interchange evaluation. A. RIF Condition Language This proposal develops a set of
fundamental concepts shared by the rule languages of interest to the RIF WG as outlined in the Design Roadmap (http://lists.w3.org/Archives/Public/public-rif-wg/2006Feb/0255.html). Here we focus on
the one part that is shared by Logic Programming rules, Production (Condition-Action) rules, Reactive (Event-Condition-Action) rules, Normative rules (Integrity Constraints), and queries. We call
this part 'Conditions' and the proposed sublanguage 'RIF Condition Language' (as a working title). The RIF Condition Language is common ground for specifying syntactic fragments in several different
dialects of RIF: - Rule bodies in a declarative logic programming dialect (LP) - Rule bodies in a first-order dialect (FO) - Conditions in the bodies in production rules dialects (PR) - The
non-action part of the rule bodies in reactive rules dialects (RR) - Integrity constraints (IC) - Queries in LP dialects (QY) Note that for rules this sublanguage is intended to be used only in the
bodies, not their heads. The various RIF dialects diverge in the way they specify rule heads and other parts of their rules. We believe that by focusing on the condition part of the rule bodies we
can achieve maximum syntactic (and some semantic) reuse among RIF dialects. Since different semantics are possible for syntactically identical rule sets, we propose that the semantics of a RIF rule
set be specified by a predefined attribute. We will elaborate on this proposal separately, but the idea is to organize the most important known semantic and syntactic features into a taxonomy (which
could be extended to accommodate new semantics/syntax), and the values of the aforementioned attribute would come from that taxonomy. A.1 Basis: Positive Conditions The basis of the language is
formed by what can appear in the bodies of Horn-like rules with equality -- conjunctions and disjunctions of atomic formulas and equations (such rules reduce to pure Horn). We later extend our
proposal to include builtins. As mentioned in the introduction, the motivation is that this sublanguage can be shared among the bodies of the rules expressed in the following RIF dialects: - FO
(first order) - LP (logic programming) - PR (production rules) - RR (reactive rules) This sublanguage can also be used to uniformly express: - IC (integrity constraints) - QY (queries) SYNTAX ------
Essential BNF for human-readable syntax: Data ::= value Ind ::= object Var ::= '?' name TERM ::= Data | Ind | Var | Expr Expr ::= Fun '(' TERM* ')' Atom ::= Rel '(' TERM* ')' | TERM '=' TERM LITFORM
::= Atom QUANTIF ::= 'Exists' Var+ '(' CONDIT ')' CONJ ::= 'And' '(' CONDIT* ')' DISJ ::= 'Or' '(' CONDIT* ')' CONDIT ::= LITFORM | QUANTIF | CONJ | DISJ Here LITFORM stands for Literal Formula and
anticipates the introduction of negated atoms later on. QUANTIF stands for Quantified Formula, which for Horn-like conditions can only be 'Exists' Formulas (Var+ variables should occur free in the
scoped CONDIT, so 'Exists' can quantify them; free variables are discussed below). More explicitly than in logic programming, CONJ expresses formula conjunctions, and DISJ expresses disjunctions.
Finally, CONDIT combines everything and defines RIF conditions, which can later be extended beyond LITFORM, QUANTIF, CONJ, or DISJ. We assume that all constants (Data and Ind) belong to two different
logical sorts: the sort of values and the sort of objects. This means that every constant carries with it the designation of the sort to which it belongs (indicating whether the constant is a value
or an object). Values (Data) can also optionally carry a designation of their datatype (e.g., TIME, Integer). Note that there are two uses of variables in the RIF condition language: free and
quantified. All quantified variables are quantified explicitly, existentially (and also universally, later). We adopt the usual scoping rules for quantification from first-order logic. Variables that
are not explicitly quantified are free. The free variables are needed because we are dealing with conditions that occur in rule bodies only. When a condition occurs in such a rule body, the free
variables in the condition are precisely those that also occur in the rule head. Such variables are quantified universally outside of the rule, and the scope of such quantification is the entire
rule. For instance, the variable ?X in the rule below is free in the condition that occurs in the rule body, but it is universally quantified outside of the rule. Condition with a free variable ?X:
... Exists ?Y (condition(..?X..?Y..)) ... Rule using the condition in its body: Forall ?X (head(...?X...) :- ... Exists ?Y (condition(..?X..?Y..)) ...) When conditions are used as queries, their free
variables are to be bound to carry the answer bindings back to the caller. The semantics of conditions that contain free variables is defined in the section SEMANTICS. Example 1 (A RIF condition in
human-readable syntax): In this condition, ?Buyer is quantified existentially, while ?Seller and ?Author are free: And ( Exists ?Buyer (purchase(?Buyer ?Seller book(?Author LeRif) $49)) ?Seller=?
Author ) This syntax is similar in style, and compatible to, the OWL Abstract Syntax (http://www.w3.org/TR/owl-semantics/syntax.html). An XML syntax can be obtained from the above BNF as follows. The
non-terminals in all-upercase such as CONDIT become XML entities, which act like macros and will not be visible in instance markups. The other non-terminals and symbols ('=', 'Exists', etc.) become
XML elements, which are adapted from RuleML as shown below. Optional attributes are added for finer distinctions. For example, an optional attribute in the Data element can point to an XML Schema
Datatype. - Data (value constant, including optional attribute for a datatype IRI) - Ind (object constant; empty element with attribute, iri, for oid; e.g. global: , local: ) - Var (logic variable;
optional type attribute, e.g. RDFS/OWL-class IRI) - Fun (n-ary function symbol; optional attribute designating interpreted [a.k.a. equation-defined] functions in contrast to uninterpreted [a.k.a.
free] functions) - Rel (n-ary relation symbol [a.k.a. predicate]) - Expr (expression formula) - Atom (atomic formula; optional attribute for external atoms) - Equal (prefix version of term equation '
=') - Exists (quantified formula for 'Exists') - And (conjunction; optional attribute for sequential conjunctions) - Or (disjunction; optional attribute for sequential disjunctions) Based on the FOL
RuleML (http://www.w3.org/Submission/FOL-RuleML) experience, this could be directly rewritten as a DTD or an XML Schema. Example 2 (A RIF condition in XML syntax): The condition formula in Example 1
can be serialized in XML as shown: Buyer purchase Buyer Seller book Author LeRif $49 Seller Author Note that the distinction between the sort of values and the sort of objects can be extended from
Data vs. Ind to variables (Var) by using an optional attribute. This corresponds to the so-called d-variables and i-variables in SWRL (http://www.w3.org/Submission/SWRL). Sorts can also be optionally
added to function symbols and predicates in order to support sorted languages. SEMANTICS --------- The semantics of a ruleset R is defined by a collection of intended models. In case of FOL, all
models of R are intended. In LP, only some models (e.g., well-founded or stable models of R) are intended. However, here we are talking only about condition parts of the rules. So, by semantics we
mean the notion of satisfaction of a formula in the interpretations of the various RIF dialects. For example, in FO, all first-order interpretations are appropriate. In LP, infinite Herbrand models
are typically used. In LP with the well-founded semantics, 3-valued Herbrand models are used. Stable model semantics uses only 2-valued interpretations. The notion of satisfaction in a model is the
same for all dialects of RIF, and in the 2-valued case does not depend on the semantic tags associated with the rulesets in which our conditions occur. (Well-founded interpretations, being 3-valued,
are slightly different.) Given a condition formula phi(X1,...,Xn) with free variables X1, ..., Xn and an interpretation M, define M(phi(X1,...,Xn)) as the set of all bindings (a1/X1,...,an/Xn) such
that M |= phi(a1,...,an). For instance, denoting the formula of Example 1 as phi(?Seller,?Author), if we use the following Herbrand interpretation: M = {purchase(Mary John book(Fred LeRif) $49),
purchase(Nina Fred book(Fred LeRif) $49) purchase(Alice Floob book(Floob LeRif) $49)} then M(phi(?Seller,?Author)) is: {(Fred/?Seller,Fred/?Author), (Floob/?Seller,Floob/?Author)} The notion of M(phi
(X1,...,Xn)) is independent of the semantics of the ruleset to which phi(X1,...,Xn) is posed, and it "plugs into" the model-theoretic semantics of the rulesets under FO, LP, and the operational
semantics of PR. Note that integrity constraints (which are closed formulas) are simply checked against the intended models of the ruleset. Therefore, the semantics of integrity constraints is fully
defined by the above notion of satisfaction in the models of rulesets. Thus, these constraints can be full first-order formulas even in case of LP rulesets. The same can be said about queries. While
semantics can be defined for very general classes of constraints and queries, this does not mean that the problem of constraint satisfaction/query answering is decidable or has efficient evaluation
techniques for such general classes. A.2 Extension: Negative Conditions This extension introduces two kinds of negation (Naf and Neg) to the RIF Condition Language, initially proposed for Phase 1
queries. Neg denotes classical negation as defined in first-order logic; special uses of it can be shared by FO and all dialects that support a form of classical negation (e.g., certain dialects of
LP; perhaps some PR dialects). Naf denotes negation as failure in one of its incarnations (well-founded or stable-model). The actual flavor of Naf is determined by inspecting the value of a semantic
tag associated with the ruleset. Naf is used in LP (and in queries and constraints over the intended models of LP); it can possibly be relevant to PR and RR. Alongside negation we can introduce
explicit universal quantification. SYNTAX ------ This extension introduces two negation symbols over atoms, and one legal nesting, for which we propose to use the notation Naf/Neg, as in RuleML:
LITFORM ::= Atom | 'Neg' Atom | 'Naf' Atom | 'Naf' 'Neg' Atom Note that in Phase 1 we are proposing to use negation only in queries. Phase 1 rules will all be Horn and their bodies will not have
negation. Phase 2 will extend RIF to allow negation in rules as well. The next step could be to allow universal quantification: QUANTIF ::= 'Exists' Var+ '(' CONDIT ')' | 'Forall' Var+ '(' CONDIT ')'
SEMANTICS --------- The semantics of atomic Neg in the models of the FO dialect is standard. For LP dialects supporting "classical" negation, Neg-negated predicates are interpreted by fresh
predicates that are appropriately axiomatized. For instance, Neg of p is associated with a fresh predicate, Neg_p, and Neg p(...) is said to be true in a model iff Neg_p(...) is. It is also required
that for any given tuple of ground arguments, the formulas p(...) and Neg_p(...) cannot be true in the same model. The following (Phase 2) dialects of RIF will support Neg and/or Naf: - LP/WF (the
well-founded semantics) generally supports only Naf, but some variants (e.g., Courteous LP) also support Neg. - LP/SMS (stable model semantics) supports both Neg and Naf (e.g., ERDF). - FO supports
Neg only. Queries and integrity constraints can use those forms of negation that are supported by the rulesets that they constrain or query. To distinguish the different possible semantics for the
same ruleset, RIF will define an attribute that will explicitly specify the semantics under which the corresponding ruleset is to be interpreted. A.3 Extension: Builtins in Conditions A builtin is a
relation or function for which there is a single, fixed interpretation. For instance, the relation "<" and the function "+" have fixed and unique interpretations over the domain of numbers. Although
builtin functions are not strictly needed, as they can be represented by builtin relations, a practical language needs both datatype relations and datatype functions. (The equality predicate can be
used to bind the result of a builtin function call to a logical variable.) Satisfaction of an atomic expression that is based on a builtin is done with respect to the concrete domain with which this
builtin is associated. Some builtins may be polymorphic and have several associated domains. For instance, in the domain of strings, "<" can be viewed as lexicographics ordering and "+" as string
concatenation. In that case, the proper interpretation of a builtin is determined by the arguments of the builtin predicate or function. RIF can support different sets of builtins. For instance, each
builtin can be identified by an IRI, in which case support for SWRL, SPARQL, and XQuery builtins is feasible. Note that, in general, satisfiability of a conjunction of builtin predicates may be
undecidable. In RIF we should probably strive to allow only those sets of builtins for which satisfiability of conjunctive formulas is decidable. A.4 Extension: Resource Conditions A special external
atom can be used to support SPARQL queries to RDF. RDF facts can be represented via a dedicated ternary predicate, and bNodes can be represented using existential variables. A.5 Extension: Ontology
Conditions A special external atom or an entire conjunction of such atoms can be used to achieve hybrid rule-ontology integration, allowing a query-like interface to OWL classes (including their
subClassOf subsumption) and properties (including subPropertyOf subsumption). Optional typing of RIF variables (e.g., for a Sorted Horn Logic) can be regarded as the special case of such an interface
for OWL-Lite classes. (Predefined classes, such as Integer, can be borrowed from XML Part 2, Datatypes.) The semantics of such integrations are different for RIF LP, FO, and PR. In case of FO, there
is no semantic mismatch and the overall semantics is first-order. In case of LP, recent work by Rosati (DL+Log, KR2006) provides a semantics for a very general class of rulesets that are tightly
integrated with DL-based ontologies. However, even more general, ad hoc interoperability between rules and ontologies might be needed. For PR and RR, interoperability with OWL is likely not to have
model-theoretic semantics as it will be ad hoc. B. Extension: RIF Horn Rule Language This extends Horn conditions A.1-A.5 to Horn rules by adding positive literals as rule heads (cf. Charter, Phase
1). For positive conjunctive queries, FO and LP semantics coincide. For queries that involve negation or universal quantification the semantics of FO and LP diverge. C. Extension: RIF Production and
Reaction Rule Languages This extends PR condition bodies A.1-A.5 to full rules by adding Production Rule conclusion actions and Reaction Rule events/actions (cf. Charter, Phase 2). D. Evaluation: Use
Cases First focus on a subset of the Working Draft Use Cases for A-C (http://www.w3.org/2005/rules/wg/ucr/draft-20060323.html, http://lists.w3.org/Archives/Public/public-rif-comments): D.1
Negotiating eBusiness Contracts Across Rule Platforms D.2 Negotiating eCommerce Transactions Through Disclosure of Buyer and Seller Policies and Preferences D.3 Collaborative Policy Development for
Dynamic Spectrum Access D.4 Access to Business Rules of Supply Chain Partners D.5 Managing Inter-Organizational Business Policies and Practices D.6 Ruleset Integration for Medical Decision Support
D.7 Interchanging Rule Extensions to OWL D.8 Vocabulary Mapping for Data Integration E. Evaluation: Interoperation/Interchange Experiments by RIF Participants on use cases D for languages A-C: E.1
SBU <-> NRC E.2 SBU <-> NRC <-> DERI E.3 BBN <-> SBU <-> NRC <-> DERI . . . F. Mappings to Other Languages Provide normative mappings to languages discussed in the Charter (order TBD). F.1 RDF F.2
OWL F.3 SWRL F.4 RuleML F.5 OMG PRR F.6 SBVR F.7 ISO Common Logic F.8 SPARQL
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Abdus Salam
When I was writing in August about physicist Sheldon Glashow’s objection to Abdus Salam being awarded a share of the 1979 physics Nobel Prize, I learnt that it was because Salam had derived a theory
that Glashow had derived as well, taking a different route, but ultimately the final product was non-renormalisable. A year or so later, Steven Weinberg derived the same theory but this time also
ensured that it was renormalisable. Glashow said Salam shouldn’t have won the prize because Salam hadn’t brought anything new to the table, whereas Glashow had derived the initial theory and Weinberg
had made it renormalisable.
His objections aside, the episode brought to my mind the work of Kenneth Wilson, who made important contributions to the renormalisation toolkit. Specifically, using these tools, physicists ensure
that the equations that they’re using to model reality don’t get out of hand and predict impossible values. An equation might be useful to solve problems in 99 scenarios but in one, it might predict
an infinity (i.e. the value of a physical variable approaches a very large number), rendering the equation useless. In such cases, physicists use renormalisation techniques to ensure the equation
works in the 100th scenario as well, without predicting infinities. (This is a simplistic description that I will finesse throughout this post.)
In 2013, when Kenneth Wilson died, I wrote about the “Indian idea of infiniteness” – including how scholars in ancient India had contemplated very large numbers and their origins, only for this
knowledge to have all but disappeared from the public imagination today because of the country’s failure to preserve it. In both instances, I never quite fully understood what renormalisation really
entailed. The following post is an attempt to fix this gap.
You know electrons. Electrons have mass. Not all this mass is implicit mass per se. Some of it is the mass of the particle itself, sometimes called the shell mass. The electron also has an electric
charge and casts a small electromagnetic field around itself. This field has some energy. According to the mass-energy equivalence (E = mc^2, approx.), the energy should correspond to some mass. This
is called the electron’s electromagnetic mass.
Now, there is an equation to calculate how much a particle’s electromagnetic mass will be – and this equation shows that this mass is inversely proportional to the particle’s radius. That is, smaller
the particle, the more its electromagnetic mass. This is why the mass of a single proton, which is larger than the electron, has a lower contribution from its electromagnetic mass.
So far so good – but quickly a problem arises. As the particle becomes smaller, according to the equation, its electromagnetic mass will increase. In technical terms, as the particle radius
approaches zero, its mass will approach infinity. If its mass approaches infinity, the particle will be harder to move from rest, or accelerate, because a very large and increasing amount of energy
will be required to do so. So the equation predicts that smaller charged particles, like quarks, should be nearly impossible to move around. Yet this is not what we see in experiments, where these
particles do move around.
In the first decade of the 20th century (when the equation existed but quarks had not yet been discovered), Max Abraham and Hendrik Lorentz resolved this problem by assuming that the shell mass of
the particle is negative. It was the earliest (recorded) instance of such a tweak – so that the equations we use to model reality don’t lose touch with that reality – and was called renormalisation.
Assuming the shell mass is negative is silly, of course, but it doesn’t affect the final result in a way that breaks the theory. To renormalise, in this context, assumes that our mathematical
knowledge of the event to be modelled is not complete enough, or that introducing such completeness would make the majority of other problems intractable.
There is another route physicists take to make sure equations and reality match, called regularisation. This is arguably more intuitive. Here, the physicist modifies the equation to include a ‘cutoff
factor’ that represents what the physicist assumes is their incomplete knowledge of the phenomenon to which the equation is being applied. By applying a modified equation in this way, the physicist
argues that some ‘new physics’ will be discovered in future that will complete the theory and the equation to perfectly account for the mass.
(I personally prefer regularisation because it seems more modest, but this is an aesthetic choice that has nothing to do with the physics itself and is thus moot.)
It is sometimes the case that once a problem is solved by regularisation, the cutoff factor disappears from the final answer – so effectively it helped with solving the problem in a way that its
presence or absence doesn’t affect the answer.
This brings to mind the famous folk tale of the goat negotiation problem, doesn’t it? A fellow in a village dies and bequeaths his 17 goats to three sons thus: the eldest gets half, the middle gets a
third and the youngest gets one-ninth. Obviously the sons get into a fight: the eldest claims nine instead of 8.5 goats, the middle claims six instead of 5.67 and the youngest claims two instead of
1.89. But then a wise old woman turns up and figures it out. She adds one of her own goats to the father’s 17 to make up a total of 18. Now, the eldest son gets nine goats, the middle son gets six
goats and the youngest son gets two goats. Problem solved? When the sons tally up the goats they received, the realise that the total is still 17. The old woman’s goat is left, which she then takes
back and gets on her way. The one additional goat was the cutoff factor here: you add it to the problem, solve it, get a solution and move on.
The example of the electron was suitable but also convenient: the need to renormalise particle masses originally arose in the context of classical electrodynamics – the first theory developed to
study the behaviour of charged particles. Theories that physicists developed later, in each case to account for some phenomena that other theories couldn’t, also required renormalisation in different
contexts, but for the same purpose: to keep the equations from predicting infinities. Infinity is a strange number that compromises our ability to make sense of the natural universe because it
spreads itself like an omnipresent screen, obstructing our view of the things beyond. To get to them, you must scale an unscaleable barrier.
While the purpose of renormalisation has stayed the same, it took on new forms in different contexts. For example, quantum electrodynamics (QED) studies the behaviour of charged particles using the
rules of quantum physics – as opposed to classical electrodynamics, which is an extension of Newtonian physics. In QED, the charge of an electron actually comes out to be infinite. This is because
QED doesn’t have a way to explain why the force exerted by a charged particle decreases as you move away. But in reality electrons and protons have finite charges. How do we fix the discrepancy?
The path of renormalisation here is as follows: Physicists assume that any empty space is not really empty. There may be no matter there, sure, but at the microscopic scale, the vacuum is said to be
teeming with virtual particles. These are pairs of particles that pop in and out of existence over very short time scales. The energy that produces them, and the energy that they release when they
annihilate each other and vanish, is what physicists assume to be the energy inherent to space itself.
Now, say an electron-positron pair, called ‘e’ and ‘p’, pops up near an independent electron, ‘E’. The positron is the antiparticle of the electron and has a positive charge, so it will move closer
to E. As a result, the electromagnetic force exerted by E’s electric charge becomes screened at a certain distance away, and the reduced force implies a lower effective charge. As the virtual
particle pairs constantly flicker around the electron, QED says that we can observe only the effects of its screened charge.
By the 1960s, physicists had found several fundamental particles and were trying to group them in a way that made sense – i.e. that said something about why these were the fundamental particles and
not others, and whether an incomplete pattern might suggest the presence of particles still to be discovered. Subsequently, in 1964, two physicists working independently – George Zweig and Murray
Gell-Mann – proposed that protons and neutrons were not fundamental particles but were made up of smaller particles called quarks and gluons. They also said that there were three kinds of quarks and
that the quarks could bind together using the gluons (thus the name). Each of these particles had an electric charge and a spin, just like electrons.
Within a year, Oscar Greenberg proposed that the quarks would also have an additional ‘color charge’ to explain why they don’t violate Pauli’s exclusion principle. (The term ‘colour’ has nothing to
do with colours; it is just the label that unamiginative physicists selected when they were looking for one.) Around the same time, James Bjorken and Sheldon Glashow also proposed that there would
have to be a fourth kind of quark, because then the new quark-gluon model could explain three more unsolved problems at the time. In 1968, physicists discovered the first evidence for quarks and
gluons in experiments, proving that Zweig, Gell-Mann, Glashow, Bjorken, Greenberg, etc. were right. But as usual, there was a problem.
Quantum chromodynamics (QCD) is the study of quarks and gluons. In QED, if an electron and a positron interact at higher energies, their coupling will be stronger. But physicists who designed
experiments in which they could observe the presence of quarks found the opposite was true: at higher energies, the quarks in a bound state behaved more and more like individual particles, but at
lower energies, the effects of the individual quarks didn’t show, only that of the bound state. Seen another way, if you move an electron and a positron apart, the force between them gradually drops
off to zero. But if you move two quarks apart, the force between them will increase for short distance before falling off to zero. It seemed that QCD would defy QED renormalisation.
A breakthrough came in 1973. If a quark ‘Q’ is surrounded by virtual quark-antiquark pairs ‘q’ and ‘q*’, then q* would move closer to Q and screen Q’s colour charge. However, the gluons have the
dubious distinction of being their own antiparticles. So some of these virtual pairs are also gluon-gluon pairs. And gluons also carry colour charge. When the two quarks are moved apart, the space in
between is occupied by gluon-gluon pairs that bring in more and more colour charge, leading to the counterintuitive effect.
However, QCD has had need of renormalisation in other areas, such as with the quark self-energy. Recall the electron and its electromagnetic mass in classical electrodynamics? This mass was the
product of the electromagnetic energy field that the electron cast around itself. This energy is called self-energy. Similarly, quarks bear an electric charge as well as a colour charge and cast a
chromo-electric field around themselves. The resulting self-energy, like in the classical electron example, threatens to reach an extremely high value – at odds with reality, where quarks have a
relatively lower, certainly finite, self-energy.
However, the simple addition of virtual particles wouldn’t solve the problem either, because of the counterintuitive effects of the colour charge and the presence of gluons. So physicists are forced
to adopt a more convoluted path in which they use both renormalisation and regularisation, as well as ensure that the latter turns out like the goats – where a new factor introduced into the
equations doesn’t remain in the ultimate solution. The mathematics of QCD is a lot more complicated than that of QED (they are notoriously hard even for specially trained physicists), so the
renormalisation and regularisation process is also correspondingly inaccessible to non-physicists. More than anything, it is steeped in mathematical techniques.
All this said, renormalisation is obviously quite inelegant. The famous British physicist Paul A.M. Dirac, who pioneered its use in particle physics, called it “ugly”. This attitude changed the most
due to the work of Kenneth Wilson. (By the way, his PhD supervisor was Gell-Mann.)
Quarks and gluons together make up protons and neutrons. Protons, neutrons and electrons, plus the forces between them, make up atoms. Atoms make up molecules, molecules make up compounds and many
compounds together, in various quantities, make up the objects we see all around us.
This description encompasses three broad scales: the microscopic, the mesoscopic and the macroscopic. Wilson developed a theory to act like a bridge – between the forces that quarks experience at the
microscopic scale and the forces that cause larger objects to undergo phase transitions (i.e. go from solid to liquid or liquid to vapour, etc.). When a quark enters or leaves a bound state or if it
is acted on by other particles, its energy changes, which is also what happens in phase transitions: objects gain or lose energy, and reorganise themselves (liquid –> vapour) to hold or shed that
By establishing this relationship, Wilson could bring to bear insights gleaned from one scale to difficult problems at a different scale, and thus make corrections that were more streamlined and more
elegant. This is quite clever because even renormalisation is the act of substituting what we are modelling with what we are able to observe, and which Wilson improved on by dropping the direct
substitution in favour of something more mathematically robust. After this point in history, physicists adopted renormalisation as a tool more widely across several branches of physics. As physicist
Leo Kadanoff wrote in his obituary for Wilson in Nature, “It could … be said that Wilson has provided scientists with the single most relevant tool for understanding the basis of physics.”
This said, however, the importance of renormalisation – or anything like it that compensates for the shortcomings of observation-based theories – was known earlier as well, so much so that physicists
considered a theory that couldn’t be renormalised to be inferior to one that could be. This was responsible for at least a part of Sheldon Glashow’s objection to Abdus Salam winning a share of the
physics Nobel Prize.
The question of Abdus Salam ‘deserving’ his Nobel
Peter Woit has blogged about an oral history interview with theoretical physicist Sheldon Glashow published in 2020 by the American Institute of Physics. (They have a great oral history of physics
series you should check out if you’re interested.) Woit zeroed in on a portion in which Glashow talks about his faltering friendship with Steven Weinberg and his issues with Abdus Salam’s nomination
for the physics Nobel Prize.
Glashow, Weinberg and Salam together won this prize in 1979, for their work on the work on electroweak theory, which describes the behaviour of two fundamental forces, the electromagnetic force and
the weak force. Glashow recalls that his and Weinberg’s friendship – having studied and worked together for many years – deteriorated in the 1970s, a time in which both scientists were aware that
they were due a Nobel Prize. According to Glashow, however, Weinberg wanted the prize to be awarded only to himself and Salam.
This is presumably because of how the prize-winning work came to be: with Glashow’s mathematical-physical model published in 1960, Weinberg building on it seven years later, with Salam’s two relevant
papers appeared a couple years after Glashow’s paper and a year after Weinberg’s. Glashow recalls that Salam’s work was not original, that each of his two papers respectively echoed findings already
published in Glashow’s and Weinberg’s papers. Instead, Glashow continues, Salam received the Nobel Prize probably because he had encouraged his peers and his colleagues to nominate him a very large
number of times and because he set up the International Centre for Theoretical Physics (ICTP) in Trieste.
This impression, of Salam being undeserving from a contribution-to-physics point of view in Glashow’s telling, is very at odds with the impression of Salam based on reading letters and comments by
Weinberg and Pervez Hoodbhoy and by watching the documentary Salam – The First ****** Nobel Laureate.
The topic of Salam being a Nobel laureate was never uncomplicated, to begin with: he was an Ahmadi Muslim who enjoyed the Pakistan government’s support until he didn’t, when he was forced to flee the
country; his intentions with the ICTP – to give scholars from developing countries a way to study physics without having to contend with often-crippling resource constrains – were also noble.
Hoodbhoy has also written about the significance of Salam’s work as a physicist and the tragedy of his name and the memories of his contributions having been erased from all the prominent research
centres in Pakistan.
Finally, one of Salam’s nominees for a Nobel Prize was the notable British physicist and Nobel laureate Paul A.M. Dirac, and it seems strange that Dirac would endorse Salam if he didn’t believe
Salam’s work deserved it.
Bearing these facts in mind, Glashow’s contention appears to be limited to the originality of Salam’s work. But to my mind, even if Salam’s work was really derivative, it was at par with that of
Glashow and Weinberg. More importantly, while I believe the Nobel Prizes deserve to be abrogated, the prize-giving committee did more good than it might have realised by including Salam among its
winners: in the words of Weinberg, “Salam sacrificed a lot of possible scientific productivity by taking on that responsibility [to set up ICTP]. It’s a sacrifice I would not make.”
Glashow may not feel very well about Salam’s inclusion for the 1979 prize and the Nobel Prizes as we know are only happy to overlook anything other than the scientific work itself, but if the
committee really screwed up, then they screwed up to do a good thing.
Then again, even though Glashow wasn’t alone (he was joined by Martinus J.G. Veltman on his opinions against Salam), the physicists’ community at large doesn’t share his views. Glashow also cites an
infamous 2014 paper by Norman Dombey, in which Dombey concluded that Salam didn’t deserve his share of the prize, but the paper’s reputation itself is iffy at best.
In fact, this is all ultimately a pointless debate: there are just too many people who deserve a Nobel Prize but don’t win it while a deeper dive into the modern history of physics should reveal a
near-constant stream of complaints against Nobel laureates and their work by their peers. It should be clear today that both winning a prize and not winning a prize ought to mean nothing to the
practice of science.
The other remarkable thing about Glashow’s comments in the interview (as cited by Woit) is what I like to think of as the seemingly eternal relevance of Brian Keating’s change of mind. Brian Keating
is an astrophysicist who was at the forefront of the infamous announcement that his team had discovered evidence of cosmic inflation, an epoch of the early universe in which it is believed to have
expanded suddenly and greatly, in March 2014. There were many problems leading up to the announcement but there was little doubt at the time, and Keating also admitted later, that its rapidity was
motivated by the temptation to secure a Nobel Prize.
Many journalists, scientists and others observers of the practice of science routinely and significantly underestimate the effect the Nobel Prizes exert on scientific research. The prospect of
winning the prize for supposedly discovering evidence of cosmic inflation caused Keating et al. to not wait for additional, confirmatory data before making their announcement. When such data did
arrive, from the Planck telescope collaboration, Keating et al. suffered for it with their reputation and prospects.
Similarly, Weinberg and Glashow fell out because, according to Glashow, Weinberg didn’t wish Glashow to give a talk in 1979 discussing possible alternatives to the work of Weinberg and Salam because
Weinberg thought doing such a thing would undermine his and Salam’s chances of being awarded a Nobel Prize. Eventually it didn’t, but that’s beside the point: this little episode in history is as
good an illustration as any of how the Nobel Prizes and their implied promises of laurels and prestige render otherwise smart scientists insecure, petty and elbows-out competitive – in exchange for
sustaining an absurd and unjust picture of the scientific enterprise.
All of this goes obviously against the spirit of science.
The ignoble president and the Nobel Prize
What is the collective noun for a group of Nobel laureates? I’m considering ballast. A ballast of Nobel laureates is appealing because these people, especially if they are all white and male, often
tend to take themselves too seriously and are taken so by others as well. I’m not saying they tend to say meaningless things but only that they – and we, speaking generally – overestimate the import
of their words, mistaking them for substance when more often than not they are just air (often as a result of being dragged into, or being compelled to comment on, matters in which they may not have
been involved if they hadn’t received Nobel Prizes).
On September 7, Physics World reported that 81 Nobel laureates had “voiced their support” for Joe Biden ahead of the impending presidential elections in the US. This move, so to speak, echoed a
letter authored by a group of 150 or so well-known Indian scientists ahead of the 2019 Lok Sabha elections in India. The Indian group had asked that the people “vote wisely”, principally in order to
protect constitutional safeguards and against those who violated or would abrogate them. This was sage advice – even though the Bharatiya Janata Party, whose actions and policies in its first term in
power from 2014 to 2019 had dismissed just this wisdom, won a thumping majority – but it was jarring on one count.
The letter’s authors taken together constituted an important subset of the national community of scientists – a community that had stayed largely silent through a spate of horrific incidents of
violence, harassment and subversion of institutions and people alike for five years or so. Though it was courageous to have spoken up at a crucial moment (even if the letter didn’t directly name the
party or those political candidates whose ideologies were evidently opposed to the ethos the letter’s authors advocated), there was a nagging feeling that perhaps it was too little too late. And in a
way, it was.
In addition, the advantages of scientists grouping together as such wasn’t clear – if only to me. Scientists are members of society just as much as most other people are. While it makes sense to come
together as scientists, especially as scientists in the same field, to oppose or support an idea that defies or benefits that field, to accumulate as scientists to offer advice on a matter that they
haven’t spoken up about before and on which they have as much authority as non-scientists sounds like a plea – apart from broadcasting their support for Biden or saying “vote wisely” or whatever – to
defer to their especial authority as scientists, in particular as ‘leading scientists’, as they say, or as Nobel laureates, with emphases on the ‘leading’ and ‘Nobel’.
Otherwise, what does a group of scientists really mean? The Nobel laureates who have spoken up now in favour of Biden offer a similarly confusing proposition. The citizens of a democratic country
coming together to vote means they are governing themselves. They are engaging in a specifically defined activity part of a suite of processes the traversal of which gives rise to effective,
politically legitimate governments. The employees of a factory coming together to protest their wages (while forsaking them) means they are striving to uphold their rights as labourers. Twenty-two
people coming onto a large, grassy field to kick one ball around according to a prefixed and predetermined rule-set means they are playing football. What does a group of Nobel laureates coming
together to endorse a presidential candidate mean – other than the moment being crafted to attract the press’s attention?
The fact of their being Nobel laureates does not qualify their endorsement for any different or higher recognition than, say, the endorsement of a businessperson, a badminton champion or a poet, or
in fact the many, many scientists who are being good in ways that no award can measure. (Whether the laureates themselves aspire to such relevance is moot.) And this is true from both the electoral
and civilian perspectives: neither Trump nor Modi are going to rue the lack of scientists’ support if only because, ironically, the scientists would rather wait to organise among themselves on rare
occasions instead of speaking up as often as is necessary or even diffuse into the superseding community of ‘protestors’ to oppose injustice. Just as much as a biologist needs to have studied
evolution as well as have successfully demonstrated their proficiency in order to be acknowledged as an expert on evolution – so that they may then dispense thoughts and ideas on the topic that could
be taken seriously – good political guidance needs to emerge from a similar enterprise, grounded in knowledge of public affairs and civic engagement.
This also means speaking up once in a while can only influence one’s audience so much. Generic appeals to “vote wisely” are well-taken but, considered in context, their potential to change minds is
bound to be awfully finite, or even patronising, depending on the context. For example, Physics World quoted Bill Foster, the Democratic representative from Illinois, reportedly the “only physicist
in Congress” and the person who canvassed the laureates, saying:
[Asking the laureates to back Biden] was like pushing at an open door. … there was a lot of enthusiasm because of the difference [the laureates] perceive in the scientific understanding [between
Biden and Trump]. … They recognise the harm being done by ignoring science in public policy. And it’s not only science; it’s logic and integrity. The scientific community wants to get to a
situation in which they trust people’s word. … The only reason we’re in a position to develop vaccines rapidly is decades of scientific research. This may be an opportunity for the scientific
community to remind everyone about long-term investment in science.
Foster’s effort is clearly aimed at hitting the limelight – which it did; getting 81 Nobel laureates is more glamorous than getting 81 well-regarded principal investigators, scientist-communicators,
lecturers or postdocs. However, the extent to which such an exercise will be able to sway public opinion is hard to say. I personally can’t imagine, assuming for a moment that we are all Americans as
well, that I, my father (a libertarian of sorts), my mother (a devout Hindu), one of her brothers (a staunch BJP supporter) or my father’s brother-in-law (a seemingly committed centrist) would ever
think, “Oh, a Nobel laureate has vouched for Biden (or a group of scientists have recommended against Narendra Modi). I should think about whether or not I wish to vote for him (or not for the
And while I don’t presume to know why each of these laureates endorsed Biden’s candidature, their combined support – as compiled on Foster’s initiative – together with Foster’s words indicate that
the community of laureates is simply looking out for itself, just as much as every other community is, but in its case wielding the COVID-19 pandemic and its ‘pre-approval’ of anything scientific to
press its point that Trump, in its view, is not fit to be president. It is easy to agree that Trump should go but impossible to agree that he must go because science is not in charge! Science is not
supposed to be in charge; this – whether the US or India – is a democracy, not a scientocracy, and the government constituted by the performance of democratic rights and responsibilities must not
function to the exclusion of disciplines, considerations or even knowledge other than those with a scientific basis. The Physics World article also quotes Carol Greider, a 2009 medicine laureate,
[She] asserted that elected leaders “should be making decisions based on facts and science,” adding that she “strongly endorses” Biden, in particular because of his “commitment to putting public
health professionals, not politicians, back in charge”.
This is a dangerously sweeping statement. The pandemic has temporarily legitimised a heightened alertness to the prescriptions of science, at pain of death in many cases, but it can be no excuse –
even if pandemics are expected to become more common and/or more dangerous – to substitute broad-based decision-making to that based only on “facts and science”, nor to substitute politicians with
public health professionals as the people in charge. In fact, and ultimately, if the laureates’ endorsement this year parallels Greider’s thoughts, it would seem there is an opinion among some of
these scientists that “facts and science” ought to constitute the foundations of all political decision-making. Many Indian scientists are already of this view.
The social scientist Prakash Kashwan discussed a similar issue in the context of climate geoengineering in The Wire in December 2018; his conclusions, outlined in the short excerpt below, apply just
as well to the pandemic:
Decisions about which unresolved questions of geoengineering deserve public investment can’t be left only to the scientists and policymakers. The community of climate engineering scientists tends
to frame geoengineering in certain ways over other equally valid alternatives. This includes considering the global average surface temperature as the central climate impact indicator and
ignoring vested interests linked to capital-intensive geoengineering infrastructure. This could bias future R&D trajectories in this area. And these priorities, together with the assessments
produced by eminent scientific bodies, have contributed to the rise of a de facto form of governance. In other words, some ‘high-level’ scientific pronouncements have assumed stewardship of
climate geoengineering in the absence of other agents. Such technocratic modes of governance don’t enjoy broad-based social or political legitimacy.
Yes, pseudoscience during the pandemic is bad; denying the reality of climate change is bad. But speaking out solely against these ills – in much the same way the ‘Marches for Science’ in India have
seemed to do, by drawing scientists out onto streets to (rightfully) demand better pay for scientists and more respect for scientific prescriptions even as the community of scientists has not
featured prominently in protests against other excesses by the Government of India, especially the persecution of Muslims and Dalits – suggests a refusal to see oneself as citizen first, as being
committed to the accomplishment of one’s goals as a scientist ahead of being concerned with direr issues that predominantly affect the minority or, more worryingly, to see in science alone the
solutions to all of our social ills. (Young scientists have been the exception by far.)
Disabusing those who cling to this view and persuading them of the reality that their authority is a dream-state maintained by science’s privileged relationship with the modern state and capitalism’s
exploitative relationship with the scientific enterprise is a monumental task, requiring decades of sustained interrogation, dialogue and reflection. Thankfully, we have a cheap and very-short-term
substitute in our midst for now: on September 9, news reports emerged that a Norwegian politician named Christian Tybring-Gjedde had nominated Trump for the 2021 Nobel Peace Prize, allegedly for
brokering the peace agreement between Israel and the UAE. I really hope the prize-awarding committee takes the nomination seriously and that Trump receives the prize. Irrespective of what
consequences such an event will have on American politics, it will be a golden opportunity for the world – and especially India – to see that the Nobel Prizes are a deeply human and therefore
uniquely flawed enterprise, as much as any other award or recognition, that they are capable of being wrong or even just plain stupid.
The Pakistani physicist Abdus Salam did the Nobel Prizes a big favour when he received the physics prize in 1979. But by and large, motivated by Henry Kissinger winning the peace prize six years
earlier and Barack Obama doing so in 2009, the popular perception of these prizes has only become increasingly irredeemable since. I have full confidence in His Laureateship Donald J. Trump being
able to tear down this false edifice – by winning it, and then endorsing himself.
Review: ‘Salam – The First ****** Nobel Laureate’ (2018)
Awards are elevated by their winners. For all of the Nobel Prizes’ flaws and shortcomings, they are redeemed by what its laureates choose to do with them. To this end, the Pakistani physicist and
activist Abdus Salam (1926-1996) elevates the prize a great deal.
Salam – The First ****** Nobel Laureate is a documentary on Netflix about Salam’s life and work. The stars in the title stand for ‘Muslim’. The label has been censored because Salam belonged to the
Ahmadiya sect, whose members are forbidden by law in Pakistan to call themselves Muslims.
After riots against this sect broke out in Lahore in 1953, Salam was forced to leave Pakistan, and he settled in the UK. His departure weighed heavily on him even though he could do very little to
prevent it. He would return only in the early 1970s to assist Zulfiqar Ali Bhutto with building Pakistan’s first nuclear bomb. However, Bhutto would soon let the Pakistani government legislate
against the Ahmadiya sect to appease his supporters. It’s not clear what surprised Salam more: the timing of India’s underground nuclear test or the loss of Bhutto’s support, both within months of
each other, that had demoted him to a second-class citizen in his home country.
In response, Salam became more radical and reasserted his Muslim identity with more vehemence than he had before. He resigned from his position as scientific advisor to the president of Pakistan,
took a break from physics and focused his efforts on protesting the construction of nuclear weapons everywhere.
It makes sense to think that he was involved. Someone will know. Whether we will ever get convincing evidence… who knows? If the Ahmadiyyas had not been declared a heretical sect, we might have
found out by now. Now it is in no one’s interest to say he was involved – either his side or the government’s side. “We did it on our own, you know. We didn’t need him.”
Tariq Ali
Whether or not it makes sense, Salam himself believed he wouldn’t have solved the problems he did that won him the Nobel Prize if he hadn’t identified as Muslim.
If you’re a particle physicist, you would like to have just one fundamental force and not four. … If you’re a Muslim particle physicist, of course you’ll believe in this very, very strongly,
because unity is an idea which is very attractive to you, culturally. I would never have started to work on the subject if I was not a Muslim.
Abdus Salam
This conviction unified at least in his mind the effects of the scientific, cultural and political forces acting on him: to use science as a means to inspire the Pakistani youth, and Muslim youth in
general, to shed their inferiority complex, and his own longstanding desire to do something for Pakistan. His idea of success included the creation of more Muslim scientists and their presence in the
ranks of the world’s best.
[Weinberg] How proud he was, he said, to be the first Muslim Nobel laureate. … [Isham] He was very aware of himself as coming from Pakistan, a Muslim. Salam was very ambitious. That’s why I think
he worked so hard. You couldn’t really work for 15 hours a day unless you had something driving you, really. His work always hadn’t been appreciated, shall we say, by the Western world. He was
different, he looked different. And maybe that also was the reason why he was so keen to get the Nobel Prize, to show them that … to be a Pakistani or a Muslim didn’t mean that you were inferior,
that you were as good as anybody else.
The documentary isn’t much concerned with Salam’s work as a physicist, and for that I’m grateful because the film instead offers a view of his life that his identity as a figure of science often
sidelines. By examining Pakistan’s choices through Salam’s eyes, we get a glimpse of a prominent scientist’s political and religious views as well – something that so many of us have become more
reluctant to acknowledge.
Like with Srinivasa Ramanujan, one of whose theorems was incidentally the subject of Salam’s first paper, physicists saw a genius in Salam but couldn’t tell where he was getting his ideas from. Salam
himself – like Ramanujan – attributed his prowess as a physicist to the almighty.
It’s possible the production was conceived to focus on the political and religious sides of a science Nobel laureate, but it puts itself at some risk of whitewashing his personality by consigning the
opinions of most of the women and subordinates in his life to the very end of its 75-minute runtime. Perhaps it bears noting that Salam was known to be impatient and dismissive, sometimes even
manipulative. He would get angry if he wasn’t being understood. His singular focus on his work forced his first wife to bear the burden of all household responsibilities, and he had difficulty
apologising for his mistakes.
The physicist Chris Isham says in the documentary that Salam was always brimming with ideas, most of them bizarre, and that Salam could never tell the good ideas apart from the sillier ones. Michael
Duff continues that being Salam’s student was a mixed blessing because 90% of his ideas were nonsensical and 10% were Nobel-Prize-class. Then, the producers show Salam onscreen talking about how
physicists intend to understand the rules that all inanimate matter abides by:
To do this, what we shall most certainly need [is] a complete break from the past and a sort of new and audacious idea of the type which Einstein has had in the beginning of this century.
Abdus Salam
This echoes interesting but not uncommon themes in the reality of India since 2014: the insistence on certainty, the attacks on doubt and the declining freedom to be wrong. There are of course
financial requirements that must be fulfilled (and Salam taught at Cambridge) but ultimately there must also be a political maturity to accommodate not just ‘unapplied’ research but also research
that is unsure of itself.
With the exception of maybe North Korea, it would be safe to say no country has thus far stopped theoretical physicists from working on what they wished. (Benito Mussolini in fact setup a centre that
supported such research in the late-1920s and Enrico Fermi worked there for a time.) However, notwithstanding an assurance I once received from a student at JNCASR that theoretical physicists need
only a pen and paper to work, explicit prohibition may not be the way to go. Some scientists have expressed anxiety that the day will come if the Hindutvawadis have their way when even the fruits of
honest, well-directed efforts are ridden with guilt, and non-applied research becomes implicitly disfavoured and discouraged.
Salam got his first shot at winning a Nobel Prize when he thought to question an idea that many physicists until then took for granted. He would eventually be vindicated but only after he had been
rebuffed by Wolfgang Pauli, forcing him to drop his line of inquiry. It was then taken up and to its logical conclusion by two Chinese physicists, Tsung-Dao Lee and Chen-Ning Yang, who won the Nobel
Prize for physics in 1957 for their efforts.
Whenever you have a good idea, don’t send it for approval to a big man. He may have more power to keep it back. If it’s a good idea, let it be published.
Abdus Salam
Salam would eventually win a Nobel Prize in 1979, together with Steven Weinberg and Sheldon Glashow – the same year in which Gen. Zia-ul-Haq had Bhutto hung to death after a controversial trial and
set Pakistan on the road to Islamisation, hardening its stance against the Ahmadiya sect. But since the general was soon set to court the US against its conflict with the Russians in Afghanistan, he
attempted to cast himself as a liberal figure by decorating Salam with the government’s Nishan-e-Imtiaz award.
Such political opportunism contrived until the end to keep Salam out of Pakistan even if, according to one of his sons, it “never stopped communicating with him”. This seems like an odd place to be
in for a scientist of Salam’s stature, who – if not for the turmoil – could have been Pakistan’s Abdul Kalam, helping direct national efforts towards technological progress while also striving to be
close to the needs of the people. Instead, as Pervez Hoodbhoy remarks in the documentary:
Salam is nowhere to be found in children’s books. There is no building named after him. There is no institution except for a small one in Lahore. Only a few have heard of his name.
Pervez Hoodbhoy
In fact, the most prominent institute named for him is the one he set up in Trieste, Italy, in 1964 (when he was 38): the Abdus Salam International Centre for Theoretical Physics. Salam had wished to
create such an institution after the first time he had been forced to leave Pakistan because he wanted to support scientists from developing countries.
Salam sacrificed a lot of possible scientific productivity by taking on that responsibility. It’s a sacrifice I would not make.
Steven Weinberg
He also wanted the scientists to have access to such a centre because “USA, USSR, UK, France, Germany – all the rich countries of the world” couldn’t understand why such access was important, so
refused to provide it.
When I was teaching in Pakistan, it became quite clear to me that either I must leave my country, or leave physics. And since then I resolved that if I could help it, I would try to make it
possible for others in my situation that they are able to work in their own countries while still [having] access to the newest ideas. … What Trieste is trying to provide is the possibility that
the man can still remain in his own country, work there the bulk of the year, come to Trieste for three months, attend one of the workshops or research sessions, meet the people in his subject.
He had to go back charged with a mission to try to change the image of science and technology in his own country.
In India, almost everyone has heard of Rabindranath Tagore, C.V. Raman, Amartya Sen and Kailash Satyarthi. One reason our memories are so robust is that Jawaharlal Nehru – and “his insistence on
scientific temper” – was independent India’s first prime minister. Another is that India has mostly had a stable government for the last seven decades. We also keep remembering those Nobel laureates
because of what we think of the Nobel Prizes themselves. This perception is ill-founded at least as it currently stands: of the prizes as the ultimate purpose of human endeavour and as an institution
in and of itself – when in fact it is just one recognition, a signifier of importance sustained by a bunch of Swedish men that has been as susceptible to bias and oversight as any other historically
significant award has been.
However, as Salam (the documentary) so effectively reminds us, the Nobel Prize is also why we remember Abdus Salam, and not the many, many other Ahmadi Muslim scientists that Pakistan has disowned
over the years, has never communicated with again and to whom it has never awarded the Nishan-e-Imtiaz. If Salam hadn’t won the Nobel Prize, would we think to recall the work of any of these
scientists? Or – to adopt a more cynical view – would we have focused so much of our attention on Salam instead of distributing it evenly between all disenfranchised Ahmadi Muslim scholars?
One way or another, I’m glad Salam won a Nobel Prize. And one way or another, the Nobel Committee should be glad it picked Salam, too, for he elevated the prize to a higher place.
Note: The headline originally indicated the documentary was released in 2019. It was actually released in 2018. I fixed the mistake on October 6, 2019, at 8.45 am.
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I have been
with maps that are a byproduct of my systematising a
cubed sphere grid
. I thought it would give a better perspective on the distribution of surface stations and their gaps, especially with the poles. So here are plots of the stations, land and sea, which have reported
April 2017 data, as used in TempLS. The ERSST data has already undergone some
It shows the areas in proportion. However, it shows multiple Antarctica's etc, which exaggerates the impression of bare spots, so you have to allow for that. One could try a different projection -
here is one focussing on a strip including the America's:
So now there are too many Africa's. However, between them you get a picture of coverage good and bad. Of course, then the question is to quantify the effect of the gaps.
In my last post, I showed an equal area world map projection that was a by-product of the cubed sphere gridding of the Earth's surface. It was an outline plot, which makes it a bit harder to read.
Producing a colored plot was tricky, because the coloring process in R requires an intact loop, which ends where it started, and the process of unfolding the cube onto which the map is initially
projected makes cuts.
So I fiddled more with that, and eventually got it working. I'll show the result below. You'll notice more clearly the local distortion near California and Victoria. And it clarifies how stuff gets
split up by the cuts marked by blue lines. I haven't shown the lat/lon lines this time; they are much as before.
This post follows on from the previous post, which described the cubed sphere mapping which preserves areas in taking a surface grid from cube to sphere. I should apologise here for messing up the
links for the associated WebGL plot for that post. I had linked to a local file version of the master JS file, so while it worked for me, I now realise that it wouldn't work elsewhere. I've fixed
If you have an area preserving plot onto the flat surfaces of a (paper) cube, then you only have to unfold the cube to get an equal-area map of the world on a page. It necessarily has distortion, and
of course the cuts you make in taking apart the cube. But the area preserving aspect is interesting. So I'll show here how it works.
I've repeated the top and bottom of the cube, so you see multiple poles. Red lines are latitudes, green longitudes. The blue lines indicate the cuts in unfolding the cube, and you should try to not
let your eye wander across them, because there is confusing duplication. And there is necessarily distortion near the ends of the lines. But it is an equal area map.
Well, almost. I'm using the single parameter tan() mapping from the previous post. I have been spending far too much time developing almost perfectly 1:1 area mappings. But I doubt they would make a
noticeable difference. I may write about that soon, but it is rather geekish stuff.
I wrote back in 2015 about an improvement on standard latitude/longitude gridding for fields on Earth. That is essentially representing the earth on a cylinder, with big problems at the poles. It is
much better to look to a more sphere-like shape, like a platonic solid. I described there a mesh derived from a cube. Even more promising is the icosahedron, and I wrote about that more recently,
here and here.
I should review why and when gridding is needed. The original use was in mapping, so you could refer to a square where some feature might be found. The uniform lat/lon grid has a big merit - it is
easy to decide which cell a place belongs in (just rounding). That needs to be preserved in any other scheme. Another use is in graphics, where shading or contouring is done. This is a variant of
interpolation. If you know some values in a grid cell, you can estimate other places in the cell.
A variant of interpolation is averaging, or integration. You calculate cell averages, then add up to get the global. For this, the cell should be small enough that behaviour within it can be regarded
as homogeneous. One sample point is reasonably representative of the whole. Then they are added according to area. Of course, the problem is that "small enough" may mean that many cells have no data.
A more demanding use still is in solution of partial differential equations, as in structural engineering or CFD, including climate GCMs. For that, you need to not only know about the cell, but its
A cubed sphere is just a regular rectangular grid (think Rubik) on the cube projected, maybe after re-mapping on the cube, onto the sphere. I was interested to see that this is now catching on in
the world of GCMs. Here is one paper written to support its use in the GFDL model. Here is an early and explanatory paper. The cube grid has all the required merits. It's easy enough to find the
cell that a given place belongs in, provided you have the mapping. And the regularity means that, with some fiddly bits, you can pick out the neighbors. That supported the application that I wrote
about in 2015, which resolved empty cells by using neighboring information. As described there, the resulting scheme is one of the best, giving results closely comparable with the triangular mesh
and spherical harmonics methods. I called it enhanced infilling.
I say "easy enough", but I want to make it my routine basis (instead of lat/lon), so that needs support. Fortunately, the grids are generic; they don't depend on problem type. So I decided to make
an R structure for standard meshes made by bisection. First the undivided cube, then 4 squares on each face, then 16, and so on. I stopped at 64, which gives 24576 cells. That is the same number of
cells as in a 1.6° square mesh, but the lat/lon grid has some cells larger. You have to go to 1.4° to get equatorial cells of the same size.
I'll give more details in an appendix, with a link to where I have posted it. It has a unique cell numbering, with an area of each cell (for weighting), coordinated of the corners on the sphere, a
neighbor structure, and I also give the cell numbers of all the measurement points that TempLS uses. There are also functions for doing the various conversions, from 3d coordinates on sphere to
cube, and to cell numbering.
There is also a WebGL depiction of the tesselated sphere, with outline world map, and the underlying cube with and without remapping.
As with TempLS, GISS showed May unchanged from April, at 0.88°C. Although that is down from the extreme warmth of Feb-Mar, it is still very warm historically. In fact, it isn't far behind the 0.93°C
of May 2016. June looks like being cooler, which reduces the likelihood of 2017 exceeding 2016 overall.
The overall pattern was similar to that in TempLS. A big warm band from N of China to Morocco (hot), with warmth in Europe, and cold in NW Russia. Wark Alaska, coolish Arctic and Antarctica mixed.
As usual, I will compare the GISS and previous TempLS plots below the jump.
I've been intermittently commenting on a thread on the long-quiet Climate Audit site. Nic Lewis was showing some interesting analysis on the effect of interpolation length in GISS, using the Python
version of GISS code that he has running. So the talk turned to numerical integration, with the usual grumblers saying that it is all too complicated to be done by any but a trusted few (who actually
don't seem to know how it is done). Never enough data etc.
So Olof chipped in with an interesting observation that with the published UAH 2.5x2.5° grid data (lower troposphere), an 18 point subset was sufficient to give quite good results. I must say that I
was surprised at so few, but he gave this convincing plot:
He made it last year, so it runs to 2015. There was much scepticism there, and some aspersions, so I set out to emulate it, and of course, it was right. My plots and code are here, and the graph
alone is here.
So I wondered how this would work with GISS. It isn't as smooth as UAH, and the 250 km less smooth than 1200km interpolation. So while 18 nodes (6x3) isn't quite enough, 108 nodes (12x9) is pretty
good. Here are the plots:
I should add that this is the very simplest grid integration, with no use of enlightened infilling, which would help considerably. The code is here.
Of course, when you look at a statistic over a longer period, even this small noise fades. Here are the GISS trends over 50 years:
1967-2016 trend C/Cen Full mesh 108 points 18 points
250km 1.658 1.703 1.754
1200km 1.754 1.743 1.768
This is a somewhat different problem from my intermittent search for a 60-station subset. There has already been smoothing in gridding. But it shows that the spatial and temporal fluctuations that we
focus on in individual maps are much diminished when aggregated over time or space.
TempLS mesh was virtually unchanged , from 0.722°C to 0.725°C. This follows the smallish rise of 0.06°C in the NCEP/NCAR index, and larger rises in the satellite indices. The May temperature is still
warm, in fact, not much less than May 2016 (0.763°C). But it puts 2017 to date now a little below the annual average for 2016.
The main interest is at the poles, where Antarctica was warm, and the Arctic rather cold, which may help retain the ice. There was a band of warmth running from Mongolia to Morocco, and cold in NW
Russia.. Here is the map:
So far in 2017, in the Moyhu NCEP/NCAR index, January to March were very warm, but April was a lot cooler. May recovered a little, rising from 0.34 to 0.4°C, on the 1994-2013 anomaly base. This is
still warm by historic standards, ahead of all annual averages before 2016, but it diminishes the likelihood that 2017 will be warmer than 2016.
There were few notable patterns of hot and cold - cold in central Russia and US, but warm in western US, etc. The Arctic was fairly neutral, which may explain the fairly slow melting of the ice..
Update - UAH lower troposphere V6 ;rose considerably, from 0.27°C to 0.45°C in May.
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NumPy's max() and maximum(): Find Extreme Values in Arrays – Real Python
The NumPy library supports expressive, efficient numerical programming in Python. Finding extreme values is a very common requirement in data analysis. The NumPy max() and maximum() functions are two
examples of how NumPy lets you combine the coding comfort offered by Python with the runtime efficiency you’d expect from C.
In this tutorial, you’ll learn how to:
• Use the NumPy max() function
• Use the NumPy maximum() function and understand why it’s different from max()
• Solve practical problems with these functions
• Handle missing values in your data
• Apply the same concepts to finding minimum values
This tutorial includes a very short introduction to NumPy, so even if you’ve never used NumPy before, you should be able to jump right in. With the background provided here, you’ll be ready to
continue exploring the wealth of functionality to be found in the NumPy library.
NumPy: Numerical Python
NumPy is short for Numerical Python. It’s an open source Python library that enables a wide range of applications in the fields of science, statistics, and data analytics through its support of fast,
parallelized computations on multidimensional arrays of numbers. Many of the most popular numerical packages use NumPy as their base library.
Introducing NumPy
The NumPy library is built around a class named np.ndarray and a set of methods and functions that leverage Python syntax for defining and manipulating arrays of any shape or size.
NumPy’s core code for array manipulation is written in C. You can use functions and methods directly on an ndarray as NumPy’s C-based code efficiently loops over all the array elements in the
background. NumPy’s high-level syntax means that you can simply and elegantly express complex programs and execute them at high speeds.
You can use a regular Python list to represent an array. However, NumPy arrays are far more efficient than lists, and they’re supported by a huge library of methods and functions. These include
mathematical and logical operations, sorting, Fourier transforms, linear algebra, array reshaping, and much more.
Today, NumPy is in widespread use in fields as diverse as astronomy, quantum computing, bioinformatics, and all kinds of engineering.
NumPy is used under the hood as the numerical engine for many other libraries, such as pandas and SciPy. It also integrates easily with visualization libraries like Matplotlib and seaborn.
NumPy is easy to install with your package manager, for example pip or conda. For detailed instructions plus a more extensive introduction to NumPy and its capabilities, take a look at NumPy
Tutorial: Your First Steps Into Data Science in Python or the NumPy Absolute Beginner’s Guide.
In this tutorial, you’ll learn how to take your very first steps in using NumPy. You’ll then explore NumPy’s max() and maximum() commands.
Creating and Using NumPy Arrays
You’ll start your investigation with a quick overview of NumPy arrays, the flexible data structure that gives NumPy its versatility and power.
The fundamental building block for any NumPy program is the ndarray. An ndarray is a Python object wrapping an array of numbers. It may, in principle, have any number of dimensions of any size. You
can declare an array in several ways. The most straightforward method starts from a regular Python list or tuple:
>>> import numpy as np
>>> A = np.array([3, 7, 2, 4, 5])
>>> A
array([3, 7, 2, 4, 5])
>>> B = np.array(((1, 4), (1, 5), (9, 2)))
>>> B
array([[1, 4],
[1, 5],
[9, 2]])
You’ve imported numpy under the alias np. This is a standard, widespread convention, so you’ll see it in most tutorials and programs. In this example, A is a one-dimensional array of numbers, while B
is two-dimensional.
Notice that the np.array() factory function expects a Python list or tuple as its first parameter, so the list or tuple must therefore be wrapped in its own set of brackets or parentheses,
respectively. Just throwing in an unwrapped bunch of numbers won’t work:
>>> np.array(3, 7, 2, 4, 5)
Traceback (most recent call last):
TypeError: array() takes from 1 to 2 positional arguments but 5 were given
With this syntax, the interpreter sees five separate positional arguments, so it’s confused.
In your constructor for array B, the nested tuple argument needs an extra pair of parentheses to identify it, in its entirety, as the first parameter of np.array().
Addressing the array elements is straightforward. NumPy’s indices start at zero, like all Python sequences. By convention, a two-dimensional array is displayed so that the first index refers to the
row, and the second index refers to the column. So A[0] is the first element of the one-dimensional array A, and B[2, 1] is the second element in the third row of the two-dimensional array B:
>>> A[0] # First element of A
>>> A[4] # Fifth and last element of A
>>> A[-1] # Last element of A, same as above
>>> A[5] # This won't work because A doesn't have a sixth element
Traceback (most recent call last):
IndexError: index 5 is out of bounds for axis 0 with size 5
>>> B[2, 1] # Second element in third row of B
So far, it seems that you’ve simply done a little extra typing to create arrays that look very similar to Python lists. But looks can be deceptive! Each ndarray object has approximately a hundred
built-in properties and methods, and you can pass it to hundreds more functions in the NumPy library.
Almost anything that you can imagine doing to an array can be achieved in a few lines of code. In this tutorial, you’ll only be using a few functions, but you can explore the full power of arrays in
the NumPy API documentation.
Creating Arrays in Other Ways
You’ve already created some NumPy arrays from Python sequences. But arrays can be created in many other ways. One of the simplest is np.arange(), which behaves rather like a souped-up version of
Python’s built-in range() function:
>>> np.arange(10)
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> np.arange(2, 3, 0.1)
array([ 2., 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9])
In the first example above, you only specified the upper limit of 10. NumPy follows the standard Python convention for ranges and returns an ndarray containing the integers 0 to 9. The second example
specifies a starting value of 2, an upper limit of 3, and an increment of 0.1. Unlike Python’s standard range() function, np.arange() can handle non-integer increments, and it automatically generates
an array with np.float elements in this case.
NumPy’s arrays may also be read from disk, synthesized from data returned by APIs, or constructed from buffers or other arrays.
NumPy arrays can contain various types of integers, floating-point numbers, and complex numbers, but all the elements in an array must be of the same type.
You’ll start by using built-in ndarray properties to understand the arrays A and B:
>>> A.size
>>> A.shape
>>> B.size
>>> B.shape
(3, 2)
The .size attribute counts the elements in the array, and the .shape attribute contains an ordered tuple of dimensions, which NumPy calls axes. A is a one-dimensional array with one row containing
five elements. Because A has only one axis, A.shape returns a one-element tuple.
By convention, in a two-dimensional matrix, axis 0 corresponds to the rows, and axis 1 corresponds to the columns, so the output of B.shape tells you that B has three rows and two columns.
Python strings and lists have a very handy feature known as slicing, which allows you to select sections of a string or list by specifying indices or ranges of indices. This idea generalizes very
naturally to NumPy arrays. For example, you can extract just the parts you need from B, without affecting the original array:
>>> B[2, 0]
>>> B[1, :]
array([1, 5])
In the first example above, you picked out the single element in row 2 and column 0 using B[2, 0]. The second example uses a slice to pick out a sub-array. Here, the index 1 in B[1, :] selects row 1
of B. The : in the second index position selects all the elements in that row. As a result, the expression B[1, :] returns an array with one row and two columns, containing all the elements from row
1 of B.
If you need to work with matrices having three or more dimensions, then NumPy has you covered. The syntax is flexible enough to cover any case. In this tutorial, though, you’ll only deal with one-
and two-dimensional arrays.
If you have any questions as you play with NumPy, the official NumPy docs are thorough and well-written. You’ll find them indispensable if you do serious development using NumPy.
NumPy’s max(): The Maximum Element in an Array
In this section, you’ll become familiar with np.max(), a versatile tool for finding maximum values in various circumstances.
Note: NumPy has both a package-level function and an ndarray method named max(). They work in the same way, though the package function np.max() requires the target array name as its first parameter.
In what follows, you’ll be using the function and the method interchangeably.
Python also has a built-in max() function that can calculate maximum values of iterables. You can use this built-in max() to find the maximum element in a one-dimensional NumPy array, but it has no
support for arrays with more dimensions. When dealing with NumPy arrays, you should stick to NumPy’s own maximum functions and methods. For the rest of this tutorial, max() will always refer to the
NumPy version.
np.max() is the tool that you need for finding the maximum value or values in a single array. Ready to give it a go?
Using max()
To illustrate the max() function, you’re going to create an array named n_scores containing the test scores obtained by the students in Professor Newton’s linear algebra class.
Each row represents one student, and each column contains the scores on a particular test. So column 0 contains all the student scores for the first test, column 1 contains the scores for the second
test, and so on. Here’s the n_scores array:
>>> import numpy as np
>>> n_scores = np.array([
... [63, 72, 75, 51, 83],
... [44, 53, 57, 56, 48],
... [71, 77, 82, 91, 76],
... [67, 56, 82, 33, 74],
... [64, 76, 72, 63, 76],
... [47, 56, 49, 53, 42],
... [91, 93, 90, 88, 96],
... [61, 56, 77, 74, 74],
... ])
You can copy and paste this code into your Python console if you want to follow along. To simplify the formatting before copying, click >>> at the top right of the code block. You can do the same
with any of the Python code in the examples. Once you’ve done that, the n_scores array is in memory. You can ask the interpreter for some of its attributes:
>>> n_scores.size
>>> n_scores.shape
(8, 5)
The .shape and .size attributes, as above, confirm that you have 8 rows representing students and 5 columns representing tests, for a total of 40 test scores.
Suppose now that you want to find the top score achieved by any student on any test. For Professor Newton’s little linear algebra class, you could find the top score fairly quickly just by examining
the data. But there’s a quicker method that’ll show its worth when you’re dealing with much larger datasets, containing perhaps thousands of rows and columns.
Try using the array’s .max() method:
The .max() method has scanned the whole array and returned the largest element. Using this method is exactly equivalent to calling np.max(n_scores).
But perhaps you want some more detailed information. What was the top score for each test? Here you can use the axis parameter:
>>> n_scores.max(axis=0)
array([91, 93, 90, 91, 96])
The new parameter axis=0 tells NumPy to find the largest value out of all the rows. Since n_scores has five columns, NumPy does this for each column independently. This produces five numbers, each of
which is the maximum value in that column. The axis parameter uses the standard convention for indexing dimensions. So axis=0 refers to the rows of an array, and axis=1 refers to the columns.
The top score for each student is just as easy to find:
>>> n_scores.max(axis=1)
array([83, 57, 91, 82, 76, 56, 96, 77])
This time, NumPy has returned an array with eight elements, one per student. The n_scores array contains one row per student. The parameter axis=1 told NumPy to find the maximum value for each
student, across the columns. Therefore, each element of the output contains the highest score attained by the corresponding student.
Perhaps you want the top scores per student, but you’ve decided to exclude the first and last tests. Slicing does the trick:
>>> filtered_scores = n_scores[:, 1:-1]
>>> filtered_scores.shape
(8, 3)
>>> filtered_scores
array([72, 75, 51],
[53, 57, 56],
[77, 82, 91],
[56, 82, 33],
[76, 72, 63],
[56, 49, 53],
[93, 90, 88],
[56, 77, 74]])
>>> filtered_scores.max(axis=1)
array([75, 57, 91, 82, 76, 56, 93, 77])
You can understand the slice notation n_scores[:, 1:-1] as follows. The first index range, represented by the lone :, selects all the rows in the slice. The second index range after the comma, 1:-1,
tells NumPy to take the columns, starting at column 1 and ending 1 column before the last. The result of the slice is stored in a new array named filtered_scores.
With a bit of practice, you’ll learn to do array slicing on the fly, so you won’t need to create the intermediate array filtered_scores explicitly:
>>> n_scores[:, 1:-1].max(axis=1)
array([75, 57, 91, 82, 76, 56, 93, 77])
Here you’ve performed the slice and the method call in a single line, but the result is the same. NumPy returns the per-student set of maximum n_scores for the restricted set of tests.
Handling Missing Values in np.max()
So now you know how to find maximum values in any completely filled array. But what happens when a few array values are missing? This is pretty common with real-world data.
To illustrate, you’ll create a small array containing a week’s worth of daily temperature readings, in Celsius, from a digital thermometer, starting on Monday:
>>> temperatures_week_1 = np.array([7.1, 7.7, 8.1, 8.0, 9.2, np.nan, 8.4])
>>> temperatures_week_1.size
It seems the thermometer had a malfunction on Saturday, and the corresponding temperature value is missing, a situation indicated by the np.nan value. This is the special value Not a Number, which is
commonly used to mark missing values in real-world data applications.
So far, so good. But a problem arises if you innocently try to apply .max() to this array:
>>> temperatures_week_1.max()
Since np.nan reports a missing value, NumPy’s default behavior is to flag this by reporting that the maximum, too, is unknown. For some applications, this makes perfect sense. But for your
application, perhaps you’d find it more useful to ignore the Saturday problem and get a maximum value from the remaining, valid readings. NumPy has provided the np.nanmax() function to take care of
such situations:
>>> np.nanmax(temperatures_week_1)
This function ignores any nan values and returns the largest numerical value, as expected. Notice that np.nanmax() is a function in the NumPy library, not a method of the ndarray object.
Exploring Related Maximum Functions
You’ve now seen the most common examples of NumPy’s maximum-finding capabilities for single arrays. But there are a few more NumPy functions related to maximum values that are worth knowing about.
For example, instead the maximum values in an array, you might want the indices of the maximum values. Let’s say you want to use your n_scores array to identify the student who did best on each test.
The .argmax() method is your friend here:
>>> n_scores.argmax(axis=0)
array([6, 6, 6, 2, 6])
It appears that student 6 obtained the top score on every test but one. Student 2 did best on the fourth test.
You’ll recall that you can also apply np.max() as a function of the NumPy package, rather than as a method of a NumPy array. In this case, the array must be supplied as the first argument of the
function. For historical reasons, the package-level function np.max() has an alias, np.amax(), which is identical in every respect apart from the name:
>>> n_scores.max(axis=1)
array([83, 57, 91, 82, 76, 56, 96, 77])
>>> np.max(n_scores, axis=1)
array([83, 57, 91, 82, 76, 56, 96, 77])
>>> np.amax(n_scores, axis=1)
array([83, 57, 91, 82, 76, 56, 96, 77])
In the code above, you’ve called .max() as a method of the n_scores object, and as a stand-alone library function with n_scores as its first parameter. You’ve also called the alias np.amax() in the
same way. All three calls produce exactly the same results.
Now you’ve seen how to use np.max(), np.amax(), or .max() to find maximum values for an array along various axes. You’ve also used np.nanmax() to find the maximum values while ignoring nan values, as
well as np.argmax() or .argmax() to find the indices of the maximum values.
You won’t be surprised to learn that NumPy has an equivalent set of minimum functions: np.min(), np.amin(), .min(), np.nanmin(), np.argmin(), and .argmin(). You won’t deal with those here, but they
behave exactly like their maximum cousins.
NumPy’s maximum(): Maximum Elements Across Arrays
Another common task in data science involves comparing two similar arrays. NumPy’s maximum() function is the tool of choice for finding maximum values across arrays. Since maximum() always involves
two input arrays, there’s no corresponding method. The np.maximum() function expects the input arrays as its first two parameters.
Using np.maximum()
Continuing with the previous example involving class scores, suppose that Professor Newton’s colleague—and archrival—Professor Leibniz is also running a linear algebra class with eight students.
Construct a new array with the values for Leibniz’s class:
>>> l_scores = np.array([
... [87, 73, 71, 59, 67],
... [60, 53, 82, 80, 58],
... [92, 85, 60, 79, 77],
... [67, 79, 71, 69, 87],
... [86, 91, 92, 73, 61],
... [70, 66, 60, 79, 57],
... [83, 51, 64, 63, 58],
... [89, 51, 72, 56, 49],
... ])
>>> l_scores.shape
(8, 5)
The new array, l_scores, has the same shape as n_scores.
You’d like to compare the two classes, student by student and test by test, to find the higher score in each case. NumPy has a function, np.maximum(), specifically designed for comparing two arrays
in an element-by-element manner. Check it out in action:
>>> np.maximum(n_scores, l_scores)
array([[87, 73, 75, 59, 83],
[60, 53, 82, 80, 58],
[92, 85, 82, 91, 77],
[67, 79, 82, 69, 87],
[86, 91, 92, 73, 76],
[70, 66, 60, 79, 57],
[91, 93, 90, 88, 96],
[89, 56, 77, 74, 74]])
If you visually check the arrays n_scores and l_scores, then you’ll see that np.maximum() has indeed picked out the higher of the two scores for each [row, column] pair of indices.
What if you only want to compare the best test results in each class? You can combine np.max() and np.maximum() to get that effect:
>>> best_n = n_scores.max(axis=0)
>>> best_n
array([91, 93, 90, 91, 96])
>>> best_l = l_scores.max(axis=0)
>>> best_l
array([92, 91, 92, 80, 87])
>>> np.maximum(best_n, best_l)
array([92, 93, 92, 91, 96])
As before, each call to .max() returns an array of maximum scores for all the students in the relevant class, one element for each test. But this time, you’re feeding those returned arrays into the
maximum() function, which compares the two arrays and returns the higher score for each test across the arrays.
You can combine those operations into one by dispensing with the intermediate arrays, best_n and best_l:
>>> np.maximum(n_scores.max(axis=0), l_scores.max(axis=0))
array([91, 93, 90, 91, 96])
This gives the same result as before, but with less typing. You can choose whichever method you prefer.
Handling Missing Values in np.maximum()
Remember the temperatures_week_1 array from an earlier example? If you use a second week’s temperature records with the maximum() function, you may spot a familiar problem.
First, you’ll create a new array to hold the new temperatures:
>>> temperatures_week_2 = np.array(
... [7.3, 7.9, np.nan, 8.1, np.nan, np.nan, 10.2]
... )
There are missing values in the temperatures_week_2 data, too. Now see what happens if you apply the np.maximum function to these two temperature arrays:
>>> np.maximum(temperatures_week_1, temperatures_week_2)
array([ 7.3, 7.9, nan, 8.1, nan, nan, 10.2])
All the nan values in both arrays have popped up as missing values in the output. There’s a good reason for NumPy’s approach to propagating nan. Often it’s important for the integrity of your results
that you keep track of the missing values, rather than brushing them under the rug. But here, you just want to get the best view of the weekly maximum values. The solution, in this case, is another
NumPy package function, np.fmax():
>>> np.fmax(temperatures_week_1, temperatures_week_2)
array([ 7.3, 7.9, 8.1, 8.1, 9.2, nan, 10.2])
Now, two of the missing values have simply been ignored, and the remaining floating-point value at that index has been taken as the maximum. But the Saturday temperature can’t be fixed in that way,
because both source values are missing. Since there’s no reasonable value to insert here, np.fmax() just leaves it as a nan.
Just as np.max() and np.nanmax() have the parallel minimum functions np.min() and np.nanmin(), so too do np.maximum() and np.fmax() have corresponding functions, np.minimum() and np.fmin(), that
mirror their functionality for minimum values.
Advanced Usage
You’ve now seen examples of all the basic use cases for NumPy’s max() and maximum(), plus a few related functions. Now you’ll investigate some of the more obscure optional parameters to these
functions and find out when they can be useful.
Reusing Memory
When you call a function in Python, a value or object is returned. You can use that result immediately by printing it or writing it to disk, or by feeding it directly into another function as an
input parameter. You can also save it to a new variable for future reference.
If you call the function in the Python REPL but don’t use it in one of those ways, then the REPL prints out the return value on the console so that you’re aware that something has been returned. All
of this is standard Python stuff, and not specific to NumPy.
NumPy’s array functions are designed to handle huge inputs, and they often produce huge outputs. If you call such a function many hundreds or thousands of times, then you’ll be allocating very large
amounts of memory. This can slow your program down and, in an extreme case, might even cause a memory or stack overflow.
This problem can be avoided by using the out parameter, which is available for both np.max() and np.maximum(), as well as for many other NumPy functions. The idea is to pre-allocate a suitable array
to hold the function result, and keep reusing that same chunk of memory in subsequent calls.
You can revisit the temperature problem to create an example of using the out parameter with the np.max() function. You’ll also use the dtype parameter to control the type of the returned array:
>>> temperature_buffer = np.empty(7, dtype=np.float32)
>>> temperature_buffer.shape
>>> np.maximum(temperatures_week_1, temperatures_week_2, out=temperature_buffer)
array([ 7.3, 7.9, nan, 8.1, nan, nan, 10.2], dtype=float32)
The initial values in temperature_buffer don’t matter, since they’ll be overwritten. But the array’s shape is important in that it must match the output shape. The displayed result looks like the
output that you received from the original np.maximum() example. So what’s changed? The difference is that you now have the same data stored in temperature_buffer:
>>> temperature_buffer
array([ 7.3, 7.9, nan, 8.1, nan, nan, 10.2], dtype=float32)
The np.maximum() return value has been stored in the temperature_buffer variable, which you previously created with the right shape to accept that return value. Since you also specified dtype=
np.float32 when you declared this buffer, NumPy will do its best to convert the output data to that type.
Remember to use the buffer contents before they’re overwritten by the next call to this function.
Filtering Arrays
Another parameter that’s occasionally useful is where. This applies a filter to the input array or arrays, so that only those values for which the where condition is True will be included in the
comparison. The other values will be ignored, and the corresponding elements of the output array will be left unaltered. In most cases, this will leave them holding arbitrary values.
For the sake of the example, suppose you’ve decided, for whatever reason, to ignore all scores less than 60 for calculating the per-student maximum values in Professor Newton’s class. Your first
attempt might go like this:
>>> n_scores
array([[63, 72, 75, 51, 83],
[44, 53, 57, 56, 48],
[71, 77, 82, 91, 76],
[67, 56, 82, 33, 74],
[64, 76, 72, 63, 76],
[47, 56, 49, 53, 42],
[91, 93, 90, 88, 96],
[61, 56, 77, 74, 74]])
>>> n_scores.max(axis=1, where=(n_scores >= 60))
ValueError: reduction operation 'maximum' does not have an identity,
so to use a where mask one has to specify 'initial'
The problem here is that NumPy doesn’t know what to do with the students in rows 1 and 5, who didn’t achieve a single test score of 60 or better. The solution is to provide an initial parameter:
>>> n_scores.max(axis=1, where=(n_scores >= 60), initial=60)
array([83, 60, 91, 82, 76, 60, 96, 77])
With the two new parameters, where and initial, n_scores.max() considers only the elements greater than or equal to 60. For the rows where there is no such element, it returns the initial value of 60
instead. So the lucky students at indices 1 and 5 got their best score boosted to 60 by this operation! The original n_scores array is untouched.
Comparing Differently Shaped Arrays With Broadcasting
You’ve learned how to use np.maximum() to compare arrays with identical shapes. But it turns out that this function, along with many others in the NumPy library, is much more versatile than that.
NumPy has a concept called broadcasting that provides a very useful extension to the behavior of most functions involving two arrays, including np.maximum().
Whenever you call a NumPy function that operates on two arrays, A and B, it checks their .shape properties to see if they’re compatible. If they have exactly the same .shape, then NumPy just matches
the arrays element by element, pairing up the element at A[i, j] with the element at B[i, j]. np.maximum() works like this too.
Broadcasting enables NumPy to operate on two arrays with different shapes, provided there’s still a sensible way to match up pairs of elements. The simplest example of this is to broadcast a single
element over an entire array. You’ll explore broadcasting by continuing the example of Professor Newton and his linear algebra class. Suppose he asks you to ensure that none of his students receives
a score below 75. Here’s how you might do it:
>>> np.maximum(n_scores, 75)
array([[75, 75, 75, 75, 83],
[75, 75, 75, 75, 75],
[75, 77, 82, 91, 76],
[75, 75, 82, 75, 75],
[75, 76, 75, 75, 76],
[75, 75, 75, 75, 75],
[91, 93, 90, 88, 96],
[75, 75, 77, 75, 75]])
You’ve applied the np.maximum() function to two arguments: n_scores, whose .shape is (8, 5), and the single scalar parameter 75. You can think of this second parameter as a 1 × 1 array that’ll be
stretched inside the function to cover eight rows and five columns. The stretched array can then be compared element by element with n_scores, and the pairwise maximum can be returned for each
element of the result.
The result is the same as if you had compared n_scores with an array of its own shape, (8, 5), but with the value 75 in each element. This stretching is just conceptual—NumPy is smart enough to do
all this without actually creating the stretched array. So you get the notational convenience of this example without compromising efficiency.
You can do much more with broadcasting. Professor Leibniz has noticed Newton’s skulduggery with his best_n_scores array, and decides to engage in a little data manipulation of her own.
Leibniz’s plan is to artificially boost all her students’ scores to be at least equal to the average score for a particular test. This will have the effect of increasing all the below-average
scores—and thus produce some quite misleading results! How can you help the professor achieve her somewhat nefarious ends?
Your first step is to use the array’s .mean() method to create a one-dimensional array of means per test. Then you can use np.maximum() and broadcast this array over the entire l_scores matrix:
>>> mean_l_scores = l_scores.mean(axis=0, dtype=np.integer)
>>> mean_l_scores
array([79, 68, 71, 69, 64])
>>> np.maximum(mean_l_scores, l_scores)
array([[87, 73, 71, 69, 67],
[79, 68, 82, 80, 64],
[92, 85, 71, 79, 77],
[79, 79, 71, 69, 87],
[86, 91, 92, 73, 64],
[79, 68, 71, 79, 64],
[83, 68, 71, 69, 64],
[89, 68, 72, 69, 64]])
The broadcasting happens in the highlighted function call. The one-dimensional mean_l_scores array has been conceptually stretched to match the two-dimensional l_scores array. The output array has
the same .shape as the larger of the two input arrays, l_scores.
Following Broadcasting Rules
So, what are the rules for broadcasting? A great many NumPy functions accept two array arguments. np.maximum() is just one of these. Arrays that can be used together in such functions are termed
compatible, and their compatibility depends on the number and size of their dimensions—that is, on their .shape.
The simplest case occurs if the two arrays, say A and B, have identical shapes. Each element in A is matched, for the function’s purposes, to the element at the same index address in B.
Broadcasting rules get more interesting when A and B have different shapes. The elements of compatible arrays must somehow be unambiguously paired together so that each element of the larger array
can interact with an element of the smaller array. The output array will have the .shape of the larger of the two input arrays. So compatible arrays must follow these rules:
1. If one array has fewer dimensions than the other, only the trailing dimensions are matched for compatibility. The trailing dimensions are those that are present in the .shape of both arrays,
counting from the right. So if A.shape is (99, 99, 2, 3) and B.shape is (2, 3), then A and B are compatible because (2, 3) are the trailing dimensions of each. You can completely ignore the two
leftmost dimensions of A.
2. Even if the trailing dimensions aren’t equal, the arrays are still compatible if one of those dimensions is equal to 1 in either array. So if A.shape is (99, 99, 2, 3) as before and B.shape is
(1, 99, 1, 3) or (1, 3) or (1, 2, 1) or (1, 1), then B is still compatible with A in each case.
You can get a feel for the broadcasting rules by playing around in the Python REPL. You’ll be creating some toy arrays to illustrate how broadcasting works and how the output array is generated:
>>> A = np.arange(24).reshape(2, 3, 4)
>>> A
array([[[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]],
[[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]])
>>> A.shape
(2, 3, 4)
>>> B = np.array(
... [
... [[-7, 11, 10, 2], [-6, 7, -2, 14], [ 7, 4, 4, -1]],
... [[18, 5, 22, 7], [25, 8, 15, 24], [31, 15, 19, 24]],
... ]
... )
>>> B.shape
(2, 3, 4)
>>> np.maximum(A, B)
array([[[ 0, 11, 10, 3], [ 4, 7, 6, 14], [ 8, 9, 10, 11]],
[[18, 13, 22, 15], [25, 17, 18, 24], [31, 21, 22, 24]]])
There’s nothing really new to see here yet. You’ve created two arrays of identical .shape and applied the np.maximum() operation to them. Notice that the handy .reshape() method lets you build arrays
of any shape. You can verify that the result is the element-by-element maximum of the two inputs.
The fun starts when you experiment with comparing two arrays of different shapes. Try slicing B to make a new array, C:
>>> C = B[:, :1, :]
>>> C
array([[[-7, 11, 10, 2]],
[[18, 5, 22, 7]]])
>>> C.shape
(2, 1, 4)
>>> np.maximum(A, C)
array([[[ 0, 11, 10, 3], [ 4, 11, 10, 7], [ 8, 11, 10, 11]],
[[18, 13, 22, 15], [18, 17, 22, 19], [20, 21, 22, 23]]]))
The two arrays, A and C, are compatible because the new array’s second dimension is 1, and the other dimensions match. Notice that the .shape of the result of the maximum() operation is the same as
A.shape. That’s because C, the smaller array, is being broadcast over A. The result of a broadcast operation between arrays will always have the .shape of the larger array.
Now you can try an even more radical slicing of B:
>>> D = B[:, :1, :1]
>>> D
>>> D.shape
(2, 1, 1)
>>> np.maximum(A, D)
array([[[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]],
[[18, 18, 18, 18], [18, 18, 18, 19], [20, 21, 22, 23]]])
Once again, the trailing dimensions of A and D are all either equal or 1, so the arrays are compatible and the broadcast works. The result has the same .shape as A.
Perhaps the most extreme type of broadcasting occurs when one of the array parameters is passed as a scalar:
>>> np.maximum(A, 10)
array([[[10, 10, 10, 10], [10, 10, 10, 10], [10, 10, 10, 11]],
[[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]])
NumPy automatically converts the second parameter, 10, to an array([10]) with .shape (1,), determines that this converted parameter is compatible with the first, and duly broadcasts it over the
entire 2 × 3 × 4 array A.
Finally, here’s a case where broadcasting fails:
>>> E = B[:, 1:, :]
>>> E
array([[[-6, 7, -2, 14], [ 7, 4, 4, -1]],
[[25, 8, 15, 24], [31, 15, 19, 24]]])
>>> E.shape
(2, 2, 4)
>>> np.maximum(A, E)
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3,4) (2,2,4)
If you refer back to the broadcasting rules above, you’ll see the problem: the second dimensions of A and E don’t match, and neither is equal to 1, so the two arrays are incompatible.
You can read more about broadcasting in Look Ma, No for Loops: Array Programming With NumPy. There’s also a good description of the rules in the NumPy docs.
The broadcasting rules can be confusing, so it’s a good idea to play around with some toy arrays until you get a feel for how it works!
In this tutorial, you’ve explored the NumPy library’s max() and maximum() operations to find the maximum values within or across arrays.
Here’s what you’ve learned:
• Why NumPy has its own max() function, and how you can use it
• How the maximum() function differs from max(), and when it’s needed
• Which practical applications exist for each function
• How you can handle missing data so your results make sense
• How you can apply your knowledge to the complementary task of finding minimum values
Along the way, you’ve learned or refreshed your knowledge of the basics of NumPy syntax. NumPy is a hugely popular library because of its powerful support for array operations.
Now that you’ve mastered the details of NumPy’s max() and maximum(), you’re ready to use them in your applications, or continue learning about more of the hundreds of array functions supported by
If you’re interested in using NumPy for data science, then you’ll also want to investigate pandas, a very popular data-science library built on top of NumPy. You can learn about it in The Pandas
DataFrame: Make Working With Data Delightful. And if you want to produce compelling images from data, take a look at Python Plotting With Matplotlib (Guide).
The applications of NumPy are limitless. Wherever your NumPy adventure takes you next, go forth and matrix-multiply!
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5.6 Break – Even Point for a single product
Finding the break-even point
A company breaks even for a given period when sales revenue and costs charged to that period are equal. Thus, the break-even point is that level of operations at which a company realizes no net
income or loss.
A company may express a break-even point in dollars of sales revenue or number of units produced or sold. No matter how a company expresses its break-even point, it is still the point of zero income
or loss. To illustrate the calculation of a break-even point watch the following video and then we will work with the previous company, Video Productions.
Before we can begin, we need two things from the previous page: Contribution Margin per unit and Contribution Margin RATIO. These formulas are:
Contribution Margin per unit = Sales Price – Variable Cost per Unit
Contribution Margin Ratio = Contribution margin (Sales – Variable Cost)
Break-even in units
Recall that Video Productions produces DVDs selling for $20 per unit. Fixed costs per period total $40,000, while variable cost is $12 per unit. We compute the break-even point in units as:
BE Units = Fixed Costs
Contribution Margin per unit
Video Productions contribution margin per unit is $ 8 ($ 20 selling price per unit – $ 12 variable cost per unit). The break even point in units would be calculated as:
BE Units = Fixed Costs $40,000 = 5,000 units
Contribution Margin per unit $8
The result tells us that Video Productions breaks even at a volume of 5,000 units per month. We can prove that to be true by computing the revenue and total costs at a volume of 5,000 units. Revenue
= (5,000 units X $20 sales price per unit) $100,000. Total costs = $100,000 ($40,000 fixed costs + $60,000 variable costs calculated as $12 per unit X 5,000 units).
Look at the cost-volume-profit chart and note that the revenue and total cost lines cross at 5,000 units—the break-even point. Video Productions has net income at volumes greater than 5,000, but it
has losses at volumes less than 5,000 units.
Break-even in sales dollars Companies frequently think of volume in sales dollars instead of units. For a company such as GM that makes Cadillacs and certain small components, it makes no sense to
think of a break-even point in units. GM breaks even in sales dollars.
The formula to compute the break-even point in sales dollars looks a lot like the formula to compute the break-even in units, except we divide fixed costs by the contribution margin ratio instead of
the contribution margin per unit.
The contribution margin ratio expresses the contribution margin as a percentage of sales. To calculate this ratio, divide the contribution margin per unit by the selling price per unit, or total
contribution margin by total revenues. Video Production’s contribution margin ratio is:
Contribution Margin Ratio = Contribution margin $8 = 0.4 or 40%
Sales $20
Or, referring to the income statement in which Video Productions had a total contribution margin of $48,000 on revenues of $ 120,000, we compute the contribution margin ratio as contribution margin
$48,000 / Revenues $120,000 = 0.40 or 40%.
That is, for each dollar of sales, there is a $ 0.40 left over after variable costs to contribute to covering fixed costs and generating net income.
Using this contribution margin ratio, we calculate Video Production’s break-even point in sales dollars as:
BE in Sales Dollars = Fixed Costs $40,000 = $100,000
Contribution Margin RATIO 0.40
The break-even volume of sales is $ 100,000 (can also be calculated as break even point in units 5,000 units x sales price $ 20 per unit). At this level of sales, fixed costs plus variable costs
equal sales revenue, as shown here:
Revenue $ 100,000 (5,000 units x $20 per unit)
Less: variable costs 60,000 (5,000 units x $12 per unit)
Contribution margin 40,000 (100,000 – 60,000)
Less: Fixed costs 40,000
Net Income $ 0
Margin of Safety
If a company’s current sales are more than its break-even point, it has a margin of safety equal to current sales minus break-even sales. The margin of safety is the amount by which sales can
decrease before the company incurs a loss. For example, assume Video Productions currently has sales of $120,000 and its break-even sales are $ 100,000. The margin of safety is $ 20,000, computed as
Margin of safety = Current sales – Break even sales
Margin of safety = $ 120,000 – $ 100,000 = $ 20,000
Sometimes people express the margin of safety as a percentage, called the margin of safety rate or just margin of safety percentage. The margin of safety rate is equal to
Margin of Safety Percent = Current Sales – Break even Sales
Current Sales
Using the data just presented, we compute the margin of safety rate is $20,000 / 120,000 = 16.67 %
This means that sales volume could drop by 16.67 percent before the company would incur a loss.
Targeted Profit or Income
You can also use this same type of analysis to determine how many sales units or sales dollars you would need to make a specific profit (very helpful!). The good news is you have already learned the
basic formula, we are just changing it slightly. The formulas we will need are:
Units at Target Profit = Fixed Costs + Target Income
Contribution Margin per unit
Sales Dollars for Target Profit = Fixed Costs + Target Income
Contribution Margin RATIO
These look familiar (or they should!). These are the same formulas we used for break even analysis but this time we have added target income. If you think about it, it IS the same formula because
at break even our target income is ZERO.
Let’s look at another example. The management of a major airline wishes to know how many seats must be sold on Flight 529 to make $8,000 in profit. To solve this problem, management must identify and
separate costs into fixed and variable categories.
The fixed costs of Flight 529 are the same regardless of the number of seats filled. Fixed costs include the fuel required to fly the plane and crew (with no passengers) to its destination;
depreciation on the plane used on the flight; and salaries of required crew members, gate attendants, and maintenance and refueling personnel. Fixed costs are $12,000.
The variable costs vary directly with the number of passengers. Variable costs include snacks and beverages provided to passengers, baggage handling costs, and the cost of the additional fuel
required to fly the plane with passengers to its destination. Management would express each variable cost on a per passenger basis. Variable costs are $25 per passenger.
Tickets are sold for $125 each. The contribution margin is $100 ($125 sales – $25 variable) and the contribution margin ratio is 80% ($100 contribution margin /$125 sales). We can calculate the
units and sales dollar required to make $8,000 in profit by:
Units at Target Profit = Fixed Costs + Target Income = 12,000 + 8,000 = $20,000 = 200 tickets
Contribution Margin per unit $100 $100
The sales dollars required could be calculated as break even units of 200 tickets x $125 sales price per ticket = $25,000 or by using the following formula:
Sales Dollars for Target Profit = Fixed Costs + Target Income = 12,000 + 8,000 = $20,000 = $25,000
Contribution Margin RATIO 0.80 0.80
Management can also use its knowledge of cost-volume-profit relationships to determine whether to increase sales promotion costs in an effort to increase sales volume or to accept an order at a
lower-than-usual price. In general, the careful study of cost behavior helps management plan future courses of action.
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Re: st: AW: Mata MP
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: st: AW: Mata MP
From Sergiy Radyakin <[email protected]>
To [email protected]
Subject Re: st: AW: Mata MP
Date Thu, 26 Nov 2009 14:30:26 -0500
On Thu, Nov 26, 2009 at 3:06 AM, Martin Weiss <[email protected]> wrote:
> <>
> " I believe that all estimation commands should benefit from multiple CPUs"
> As http://www.stata.com/statamp/report.pdf says, there are two reasons why
> no performance improvement is achieved: " ...either because (the commands)
> are inherently sequential ... or because no effort was made to parallelize
> them". For instance, -xtmixed- does not run faster, either, but certainly
> belongs to the group of estimation commands as well. The fact that no effort
> was made itself is probably grounded in the insight that the performance
> improvement to be expected is not all that great.
Martin, this is precisely the question: is it not implemented because
there is no more efficient algorithm, or because the developers didn't
have time to implement it.
Here is a comparison of Mata's matrix multiplication (looks good) and
matrix determinant computation (doesn't look good):
or here if the above link doesn't work:
Computing a determinant in Stata (as opposed to Mata) is ~4-5 times
slower, but also not sensitive to the number of CPUs.
We know that computing a determinant is highly-parallelizable. (see
e.g. here: http://cjtcs.cs.uchicago.edu/articles/1997/5/cj97-05.pdf
and references within).
So the fact that it is not sensitive to the number of CPUs in Mata
tells me that most probably "no effort was made" not because there
would be no effect, but because it is still work in progress.
It would be great if there were explicit parallelization instructions
in Mata (like it is done in most modern languages), so that we could
implement parallel algorithms ourself.
Best, Sergiy Radyakin
> HTH
> Martin
> -----Ursprüngliche Nachricht-----
> Von: [email protected]
> [mailto:[email protected]] Im Auftrag von Sergiy Radyakin
> Gesendet: Donnerstag, 26. November 2009 05:12
> An: [email protected]
> Betreff: st: Mata MP
> Dear All,
> 1) I have just bumped into a phrase: "since ghk2() is written Mata it
> does not benefit from multiple processors in Stata/MP".
> Here: http://www.cgdev.org/files/1421516_file_Roodman_cmp_FINAL.pdf
> I wonder why is that?
> 2) mprobit is not benefitting from multiple CPUs. Is it a fundamental
> problem and nothing can be done? or is it just
> "not-yet-implemented"? (It is surprising, since I believe that all
> estimation commands should benefit from multiple CPUs).
> Thank you,
> Sergiy Radyakin
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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On the cohomology of certain non-compact Shimura varieties (AM-173)
This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the
Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can
be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology. Morel
uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of
applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the
group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)
-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear
Original language English (US)
Publisher Princeton University Press
Number of pages 217
ISBN (Electronic) 9781400835393
ISBN (Print) 9780691142937
State Published - Jan 4 2010
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'On the cohomology of certain non-compact Shimura varieties (AM-173)'. Together they form a unique fingerprint.
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Syntax Algebra
Note the syntax, it can constitute an algebra.
The idea of a syntax algebra, is that, a computer language whose syntax is similar to that of mathematical formalism, that it is just like equations or formulas, that one can manipulate them
meaningfully. (somewhat similar to the idea of term rewriting system)
Contrast it with the ad hoc syntax of haskell or ocaml, whose syntax are so ad hoc and non-systematic — total garbage. 〔see OCaml Syntax Sucks〕
by the way, the picture is from:
〔A Galois Connection Calculus for Abstract Interpretation By Patrick Cousot And Radhia Cousot. At [S:https://www.di.ens.fr/~rcousot/publications.www/CousotCousot-POPL14-ACM-p2-3-2014.pdf:S] ,
accessed on 2016-09-03〕 [local copy CousotCousot-POPL14-ACM-p2-3-2014.pdf]
abstract: We introduce a Galois connection calculus for language independent specification of abstract interpretations used in programming language semantics, formal verification, and static
I don't know what's “Galois connection calculus”.
[via https://x.com/whitequark/status/771851320374366208]
here's a easy introduction. 〔Order Theory, Galois Connections and Abstract Interpretation By David Henriques Carnegie Mellon University. At http://www.cs.cmu.edu/~emc/15414-f11/lecture/
lec19_GaloisConnections.pdf [local copy lec19_GaloisConnections.pdf〕〕
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Tiered Reinforcement Learning: Pessimism in the Face of Uncertainty and Constant Regret
Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Jiawei Huang, Li Zhao, Tao Qin, Wei Chen, Nan Jiang, Tie-Yan Liu
We propose a new learning framework that captures the tiered structure of many real-world user-interaction applications, where the users can be divided into two groups based on their different
tolerance on exploration risks and should be treated separately. In this setting, we simultaneously maintain two policies $\pi^{\text{O}}$ and $\pi^{\text{E}}$: $\pi^{\text{O}}$ (``O'' for
``online'') interacts with more risk-tolerant users from the first tier and minimizes regret by balancing exploration and exploitation as usual, while $\pi^{\text{E}}$ (``E'' for ``exploit'')
exclusively focuses on exploitation for risk-averse users from the second tier utilizing the data collected so far. An important question is whether such a separation yields advantages over the
standard online setting (i.e., $\pi^{\text{E}}=\pi^{\text{O}}$) for the risk-averse users. We individually consider the gap-independent vs.~gap-dependent settings. For the former, we prove that the
separation is indeed not beneficial from a minimax perspective. For the latter, we show that if choosing Pessimistic Value Iteration as the exploitation algorithm to produce $\pi^{\text{E}}$, we can
achieve a constant regret for risk-averse users independent of the number of episodes $K$, which is in sharp contrast to the $\Omega(\log K)$ regret for any online RL algorithms in the same setting,
while the regret of $\pi^{\text{O}}$ (almost) maintains its online regret optimality and does not need to compromise for the success of $\pi^{\text{E}}$.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 49
For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or
present on all sides, an all-encompassing atmosphere.” And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.”
Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. Today’s post will focus on
the introductory overview for Grade 1. Note that the Common Core Standards will appear in blue, followed by an ambient translation.
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition and subtraction within 20; (2) developing understanding of whole number relationship and
place value, including grouping of tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and
composing and decomposing geometric shapes.
In my Catholic school Grade 1 picture, I am one of sixty(!) students. There are twenty seven boys standing around the perimeter in white shirts and ties, and thirty three girls seated at desks in
white puffed sleeve, peter pan collar blouses and pleated navy blue jumpers. It was orderly! It had to be with one little nun in charge. I learned a lot from the good sisters, up though Grade 12,
but I did not ever learn to love (or even like) math.
Math and movement must go together, especially in the early grades. Math wants to sing and dance and play, and it is in this context that I will strive to “ambientify” it. In this blog, all posts
up through Grade 4 will be close in content and style to the Math By Hand curriculum, which is mainly Waldorf-inspired (along with other alternative approaches).
Some major differences distinguish Waldorf from mainstream education, such as: (1) simple, one-digit multiplication and division are taught alongside addition and subtraction; (2) place value waits
until Grade 2; (3) formal time and measurement does not appear until Grade 3; (4) geometric shapes are not directly addressed but are thoroughly covered with form drawing.
(1) Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based
models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to
develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on
two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and
subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.
Discrete objects are good tools for learning how to count and calculate, but if the objects used are natural rather than man-made, they can better serve more than one purpose. Cute, colorful,
plastic teddy bear counters serve the purpose but may also be distracting. And any opportunity to connect children back to nature should be taken, because “nature deficit disorder” is real, and
Re plastic connecting cubes, yes these and other materials like them are widely used by many innovative math programs, including Montessori. But their use may abstract something that perhaps should
not be abstracted: the depth and dimensionality of number and math. So if their purpose is to impart meaning to math functions, meaning may be more likely to be met with Waldorf methods such as
attributing different personalities to each of the 4 processes.
Increasingly sophisticated strategies such as “making tens” may be too cumbersome and clumsy for now. I saw examples of this online (i.e., finding the total of 8 + 6 by making ten trades to arrive
at the answer using 10 + 4 instead), and found it to be a circuitous method, frustrating for both children and their parents. Children crying over hours of homework, and parents wringing their hands
over not understanding the method themselves, does not make a pretty picture. Comparing a variety of solution strategies may happen in a more integrated way as the deeper relationships between the
processes are understood.
(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop
understanding of and solve problems involving their relative sizes, they think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed
of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes.
Again, multiplication and division are added into the mix along with addition and subtraction, so that a working relationship involving equivalency and proofs may be in place from the very beginning.
The concepts of “greater / less-than” can be learned in a concrete, pictorial way, as can counting the teens to 20, or the differences between the cardinal and ordinal numbers.
(3) Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with
equal-sized units) and the transitivity.
(It’s footnoted that, “Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this technical term.” (Whew! I’d better hone my
understanding of it as well . . .) Though measurement and time will not be formally introduced until Grade 3, a solid foundation will be built in Grade 1 with various experiences and activities.
(4) Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding of part-whole relationships as well as the properties of
the original and composite shapes. As they combine shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and
different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry.
Geometry is not made conscious and executed with instruments until Grade 6 in the Waldorf method. But the foundations are laid, strong and deep, with form drawing. Starting with a month of main
lessons in Grade 1, and continuing once a week from Grades 1 – 5, forms are explored and compared in every possible combination and relationship.
True knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal. On to Grade 1
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@misc{316, keywords = {Plasma Physics (physics.plasm-ph), FOS: Physical sciences}, author = {Allen Boozer}, title = {Required Toroidal Confinement for Fusion and Omnigeneity}, abstract = {
Deuterium-tritium (DT) burning requires a long energy confinement times compared to collision times, so the particle distribution functions must approximate local-Maxwellians. Non-equilibrium
thermodynamics is applicable, which gives relations among transport, entropy production, the collision frequency, and the deviation from a Maxwellian. The distribution functions are given by the
Fokker-Planck equation, which is an advection-diffusion equation. A large hyperbolic operator, the Vlasov operator with the particle trajectories as its characteristics, equals a small diffusive
operator, the collision operator. The collisionless particle trajectories would be chaotic in stellarators without careful optimization. This would lead to rapid entropy production and transport --
far beyond what is consistent with a self-sustaining DT burn. Omnigeneity is the weakest general condition that is consistent with a sufficiently small entropy production associated with the thermal
particle trajectories. Omnigeneity requires that the contours of constant magnetic field strength be unbounded in at least one of the two angular coordinates in magnetic surfaces and that there be a
symmetry in the field-strength wells along the field lines. Even in omnigenous plasmas, fluctuations due to microturbulence can produce chaotic particle trajectories and the gyro-Bohm transport seen
in many stellarator and tokamak experiments. The higher the plasma temperature above 10~keV, the smaller the transport must be compared to gyro-Bohm for a self-sustaining DT burn. The hot alphas of
DT fusion heat the electrons. When the ion-electron equilibration time is long compared to the ion energy confinement time, a self-sustaining DT burn is not possible, which sets a limit on the
electron temperature.
}, year = {2022}, month = {08/2022}, publisher = {arXiv}, url = {https://arxiv.org/abs/2208.02391}, doi = {10.48550/ARXIV.2208.02391}, }
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JavaScript – How to find all subsets of a set in JavaScript? (Powerset of array)
To find all subsets of a set (also known as the power set) in JavaScript, you can use a backtracking algorithm. Here’s an example code snippet that generates all subsets of an array step by step:
function findSubsets(nums) {
const subsets = [];
backtrack(nums, [], 0);
function backtrack(nums, currentSubset, startIndex) {
// Add the current subset to the result
// Generate new subsets by adding one element at a time
for (let i = startIndex; i < nums.length; i++) {
// Add current element to the current subset
// Generate new subsets starting from the next index
backtrack(nums, currentSubset, i + 1);
// Remove the current element from the current subset to try a new one
return subsets;
// Example usage
const nums = [1, 2, 3];
const subsets = findSubsets(nums);
Here’s a step-by-step breakdown of the code:
1. The function `findSubsets` takes an array `nums` as a parameter and creates an empty `subsets` array to store the result.
2. The function calls the `backtrack` function with the initial parameters `nums`, an empty array for the `currentSubset`, and a `startIndex` of 0.
3. Inside the `backtrack` function, the current subset is added to the `subsets` array using `.slice()` to make a copy of the subset (rather than a reference).
4. Next, a loop is used to generate new subsets by adding one element at a time. The loop starts from the `startIndex` to avoid duplications, as the previous elements are already considered in
previous iterations.
5. In each iteration of the loop, the current element (`nums[i]`) is added to the `currentSubset` using `.push()`.
6. After adding the current element, the `backtrack` function is recursively called with the updated `currentSubset` and the next `startIndex` (i + 1).
7. This recursive call continues the generation of new subsets, exploring different combinations by appending each remaining element to the `currentSubset`.
8. After all recursive calls are completed for a particular element, the current element is removed from the `currentSubset` using `.pop()`.
9. Once the loop is finished, all subsets have been generated and the `subsets` array is returned.
10. Finally, the `findSubsets` function is called with an example input array `[1, 2, 3]`, and the resulting `subsets` array is logged to the console.
Example output:
[1, 2],
[1, 2, 3],
[1, 3],
[2, 3],
The output shows all possible subsets of the input array `[1, 2, 3]`, including the empty subset `[]` and the complete set `[1, 2, 3]`.
Was this article helpful?
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Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations | Quanta Magazine
For more than 250 years, mathematicians have been trying to “blow up” some of the most important equations in physics: those that describe how fluids flow. If they succeed, then they will have
discovered a scenario in which those equations break down — a vortex that spins infinitely fast, perhaps, or a current that abruptly stops and starts, or a particle that whips past its neighbors
infinitely quickly. Beyond that point of blowup — the “singularity” — the equations will no longer have solutions. They will fail to describe even an idealized version of the world we live in, and
mathematicians will have reason to wonder just how universally dependable they are as models of fluid behavior.
But singularities can be as slippery as the fluids they’re meant to describe. To find one, mathematicians often take the equations that govern fluid flow, feed them into a computer, and run digital
simulations. They start with a set of initial conditions, then watch until the value of some quantity — velocity, say, or vorticity (a measure of rotation) — begins to grow wildly, seemingly on track
to blow up.
Yet computers can’t definitively spot a singularity, for the simple reason that they cannot work with infinite values. If a singularity exists, computer models might get close to the point where the
equations blow up, but they can never see it directly. Indeed, apparent singularities have vanished when probed with more powerful computational methods.
Such approximations are still important, however. With one in hand, mathematicians can use a technique called a computer-assisted proof to show that a true singularity exists close by. They’ve
already done it for a simplified, one-dimensional version of the problem.
Now, in a preprint posted online earlier this year, a team of mathematicians and geoscientists has uncovered an entirely new way to approximate singularities — one that harnesses a recently developed
form of deep learning. Using this approach, they were able to peer at the singularity directly. They are also using it to search for singularities that have eluded traditional methods, in hopes of
showing that the equations aren’t as infallible as they might seem.
The work has launched a race to blow up the fluid equations: on one side, the deep learning team; on the other, mathematicians who have been working with more established techniques for years.
Regardless of who might win the race — if anyone is indeed able to reach the finish line — the result showcases how neural networks could help transform the search for new solutions to scores of
different problems.
The Disappearing Blowup
The equations at the center of the new work were written down by Leonhard Euler in 1757 to describe the motion of an ideal, incompressible fluid — a fluid that has no viscosity, or internal friction,
and that cannot be squeezed into a smaller volume. (Fluids that do have viscosity, like many of those found in nature, are modeled instead by the Navier-Stokes equations; blowing those up would earn
a $1 million Millennium Prize from the Clay Mathematics Institute.) Given the velocity of each particle in the fluid at some starting point, the Euler equations should predict the flow of the fluid
for all time.
But mathematicians want to know whether in some situations — even though nothing might seem amiss at first — the equations could eventually run into trouble. (There’s reason to suspect this might be
the case: The ideal fluids they model don’t behave anything like real fluids that are just the slightest bit viscous. The formation of a singularity in the Euler equations could explain this
In 2013, a pair of mathematicians proposed just such a scenario. Since the dynamics of a full three-dimensional fluid flow can get impossibly complicated, Thomas Hou, a mathematician at the
California Institute of Technology, and Guo Luo, now at the Hang Seng University of Hong Kong, considered flows that obey a certain symmetry.
In their simulations, a fluid rotates inside a cylindrical cup. The fluid in the top half of the cup swirls clockwise, while the bottom half swirls counterclockwise. The opposing flows lead to the
formation of other complicated currents that cycle up and down. Soon enough, at a point along the boundary where the opposing flows meet, the fluid’s vorticity explodes.
Merrill Sherman/Quanta Magazine
While this demonstration provided compelling evidence of a singularity, without a proof it was impossible to know for sure that it was one. Before Hou and Luo’s work, many simulations proposed
potential singularities, but most of them disappeared when tested later on a more powerful computer. “You think there is one,” said Vladimir Sverak, a mathematician at the University of Minnesota.
“Then you put it on a bigger computer with much better resolution, and somehow what seemed like a good singularity scenario just turns out to not really be the case.”
That’s because these solutions can be finicky. They’re vulnerable to small, seemingly trivial errors that can accumulate with each time step in a simulation. “It’s a subtle art to try to do a good
simulation on a computer of the Euler equation,” said Charlie Fefferman, a mathematician at Princeton University. “The equation is so sensitive to tiny, tiny errors in the 38th decimal place of the
Still, Hou and Luo’s approximate solution for a singularity has held up against every test thrown at it so far, and it has inspired a great deal of related work, including full proofs of blowup for
weaker versions of the problem. “It’s by far the best scenario for singularity formation,” Sverak said. “Many people, including myself, believe that this time it’s a real singularity.”
To fully prove blowup, mathematicians need to show that, given the approximate singularity, a true one exists nearby. They can rewrite that statement — that a real solution lives in a sufficiently
close neighborhood of the approximation — in precise mathematical terms, and then show that it’s true if certain properties can be verified. Verifying those properties, however, requires a computer
once again: this time, to perform a series of computations (which involve the approximate solution), and to carefully control the errors that might accumulate in the process.
Hou and his graduate student Jiajie Chen have been working toward a computer-assisted proof for several years now. They’ve refined the approximate solution from 2013 (in an intermediate result they
have not yet made public), and are now using that approximation as the foundation for their new proof. They’ve also shown that this general strategy can work for problems that are easier to solve
than the Euler equations.
Now another group has joined the hunt. They’ve found an approximation of their own — one that closely resembles Hou and Luo’s result — using a completely different approach. They’re currently using
it to write their own computer-assisted proof. To obtain their approximation, though, they first needed to turn to a new form of deep learning.
Glacial Neural Networks
Tristan Buckmaster, a mathematician at Princeton who is currently a visiting scholar at the Institute for Advanced Study, encountered this new approach purely by chance. Last year, Charlie
Cowen-Breen, an undergraduate in his department, asked him to sign off on a project. Cowen-Breen had been studying ice sheet dynamics in Antarctica under the supervision of the Princeton geophysicist
Ching-Yao Lai. Using satellite imagery and other observations, they were trying to infer the viscosity of the ice and predict its future flow. But to do that, they relied on a deep learning approach
that Buckmaster hadn’t seen before.
Unlike traditional neural networks, which get trained on lots of data in order to make predictions, a “physics-informed neural network,” or PINN, must satisfy a set of underlying physical constraints
as well. These might include laws of motion, energy conservation, thermodynamics — whatever scientists might need to encode for the particular problem they’re trying to solve.
Injecting physics into the neural network serves several purposes. For one, it allows the network to answer questions when very little data is available. It also enables the PINN to infer unknown
parameters in the original equations. In a lot of physical problems, “we know roughly how the equations should look like, but we don’t know what the coefficients of [certain] terms should be,” said
Yongji Wang, a postdoctoral researcher in Lai’s lab and one of the new paper’s co-authors. That was the case for the parameter that Lai and Cowen-Breen were trying to determine.
“We call it hidden fluid mechanics,” said George Karniadakis, an applied mathematician at Brown University who developed the first PINNs in 2017.
Cowen-Breen’s request got Buckmaster thinking. The classical methods for solving the Euler equations with a cylindrical boundary — as Hou, Luo and Chen had done — involved painstaking progressions
through time. But because of that dependence on time, they could only get very close to the singularity without ever reaching it: As they crept closer and closer to something that might look like
infinity, the computer’s calculations would get more and more unreliable, so that they couldn’t actually look at the point of blowup itself.
But the Euler equations can be represented with another set of equations that, through a technical trick, sweep time aside. Hou and Luo’s 2013 result wasn’t just notable for pinning down a very
precise approximate solution; the solution they found also seemed to have a particular kind of “self-similar” structure. That meant that as the model evolved through time, its solution followed a
certain pattern: Its shape at a later time looked a lot like its original shape, only bigger.
That feature meant that mathematicians could focus on a time before the singularity occurred. If they zoomed in on that snapshot at the right rate — as if they were looking at it under a microscope
with an ever-adjusting magnification setting — they could model what would happen later, right up to the point of the singularity itself. Meanwhile, if they re-scaled things in this way, nothing
would actually go terribly wrong in this new system, and they could eliminate any need to deal with infinite values. “It’s just approaching some nice limit,” Fefferman said, and that limit represents
the occurrence of the blowup in the time-dependent version of the equations.
“It’s easier to model these [re-scaled] functions,” Sverak said. “And so, if you can describe a singularity by using a [self-similar] function, it’s a big advantage.”
The problem is that for this to work, the mathematicians don’t just have to solve the equations (now written in self-similar coordinates) for the usual parameters, such as velocity and vorticity. The
equations themselves also have an unknown parameter: the variable that governs the rate of magnification. Its value has to be just right to ensure that the solution to the equations corresponds to a
blowup solution in the original version of the problem.
The mathematicians would have to solve the equations forward and backward simultaneously — a difficult if not impossible task to achieve using traditional methods.
But finding those kinds of solutions is exactly what PINNs were designed for.
The Road to Blowup
In retrospect, Buckmaster said, “it seems like an obvious thing to do.”
He, Lai, Wang and Javier Gómez-Serrano, a mathematician at Brown University and the University of Barcelona, established a set of physical constraints to help guide their PINN: conditions related to
symmetry and other properties, as well as the equations they wanted to solve (they used a set of 2D equations, rewritten using self-similar coordinates, that are known to be equivalent to the 3D
Euler equations at points approaching the cylindrical boundary).
They then trained the neural network to search for solutions — and for the self-similar parameter — that satisfied those constraints. “This method is very flexible,” Lai said. “You can always find a
solution as long as you impose the correct constraints.” (In fact, the group showcased that flexibility by testing the method on other problems.)
The team’s answer looked a lot like the solution that Hou and Luo had arrived at in 2013. But the mathematicians hope that their approximation paints a more detailed picture of what’s happening,
since it marks the first direct calculation of a self-similar solution for this problem. “The new result specifies more precisely how the singularity is formed,” Sverak said — how certain values will
blow up, and how the equations will collapse.
“You’re really extracting the essence of the singularity,” Buckmaster said. “It was very difficult to show this without neural networks. It’s clear as night and day that it’s a much easier approach
than traditional methods.”
Gómez-Serrano agrees. “This is going to be part of the standard toolboxes that people are going to have at hand in the future,” he said.
Once again, PINNs have revealed what Karniadakis called “hidden fluid mechanics” — only this time, they made headway on a far more theoretical problem than the ones PINNs are usually used for. “I
haven’t seen anybody use PINNs for that,” Karniadakis said.
That’s not the only reason mathematicians are excited. PINNs might also be perfectly situated to find another type of singularity that’s all but invisible to traditional numerical methods. These
“unstable” singularities might be the only ones that exist for certain models of fluid dynamics, including the Euler equations without a cylindrical boundary (which are already much more complicated
to solve) and the Navier-Stokes equations. “Unstable things do exist. So why not find them?” said Peter Constantin, a mathematician at Princeton.
But even for the stable singularities that classical techniques can handle, the solution the PINN provided for the Euler equations with a cylindrical boundary “is quantitative and precise and has a
much better chance of being made rigorous,” Fefferman said. “Now there’s a road map [toward a proof]. It will take a lot of work. It will take a lot of skill. I imagine it will take some originality.
But I don’t see that it will take genius. I think it’s doable.”
Buckmaster’s group is now racing against Hou and Chen to get to the finish line first. Hou and Chen have a head start: According to Hou, they have made substantial progress over the past couple years
toward improving their approximate solution and completing a proof — and he suspects that Buckmaster and his colleagues will have to refine their approximate solution before they will get their own
proof to work. “There’s very little margin for error,” Hou said.
That said, many experts hope that the 250-year quest to blow up the Euler equations is nearly at an end. “Conceptually, I think … that all the important parts are in place,” said Sverak. “It’s just
very hard to nail down the details.”
Update: April 12, 2022
This article was updated to include Tristan Buckmaster’s affiliation with the Institute for Advanced Study.
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Language GANs Falling Short - ShortScience.org
Summary by CodyWild 5 years ago
This paper’s high-level goal is to evaluate how well GAN-type structures for generating text are performing, compared to more traditional maximum likelihood methods. In the process, it zooms into the ways that the current set of metrics for comparing text generation fail to give a well-rounded picture of how models are performing.
In the old paradigm, of maximum likelihood estimation, models were both trained and evaluated on a maximizing the likelihood of each word, given the prior words in a sequence. That is, models were good when they assigned high probability to true tokens, conditioned on past tokens. However, GANs work in a fundamentally new framework, in that they aren’t trained to increase the likelihood of the next (ground truth) word in a sequence, but to generate a word that will make a discriminator more likely to see the sentence as realistic. Since GANs don’t directly model the probability of token t, given prior tokens, you can’t evaluate them using this maximum likelihood framework.
This paper surveys a range of prior work that has evaluated GANs and MLE models on two broad categories of metrics, occasionally showing GANs to perform better on one or the other, but not really giving a way to trade off between the two.
- The first type of metric, shorthanded as “quality”, measures how aligned the generated text is with some reference corpus of text: to what extent your generated text seems to “come from the same distribution” as the original. BLEU, a heuristic frequently used in translation, and also leveraged here, measures how frequently certain sets of n-grams occur in the reference text, relative to the generated text. N typically goes up to 4, and so in addition to comparing the distributions of single tokens in the reference and generated, BLEU also compares shared bigrams, trigrams, and quadgrams (?) to measure more precise similarity of text.
- The second metric, shorthanded as “diversity” measures how different generated sentences are from one another. If you want to design a model to generate text, you presumably want it to be able to generate a diverse range of text - in probability terms, you want to fully sample from the distribution, rather than just taking the expected or mean value. Linguistically, this would be show up as a generator that just generates the same sentence over and over again. This sentence can be highly representative of the original text, but lacks diversity. One metric used for this is the same kind of BLEU score, but for each generated sentence against a corpus of prior generated sentences, and, here, the goal is for the overlap to be as low as possible
The trouble with these two metrics is that, in their raw state, they’re pretty incommensurable, and hard to trade off against one another. The authors of this paper try to address this by observing that all models trade off diversity and quality to some extent, just by modifying the entropy of the conditional token distribution they learn. If a distribution is high entropy, that is, if it spreads probability out onto more tokens, it’s likelier to bounce off into a random place, which increases diversity, but also can make the sentence more incoherent. By contrast, if a distribution is too low entropy, only ever putting probability on one or two words, then it will be only ever capable of carving out a small number of distinct paths through word space.
The below table shows a good example of what language generation can look like at high and low levels of entropy
The entropy of a softmax distribution be modified, without changing the underlying model, by changing the *temperature* of the softmax calculation. So, the authors do this, and, as a result, they can chart out that model’s curve on the quality/diversity axis. Conceptually, this is asking “at a range of different quality thresholds, how good is this model’s diversity,” and vice versa. I mentally analogize this to a ROC curve, where it’s not really possible to compare, say, precision of models that use different thresholds, and so you instead need to compare the curve over a range of different thresholds, and compare models on that.
When they do this for GANs and MLEs, they find that, while GANs might dominate on a single metric at a time, when you modulate the temperature of MLE models, they’re able to achieve superior quality when you tune them to commensurate levels of diversity.
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MathFiction: The Finan-seer (Edward L. Locke)
Contributed by Vijay Fafat
I have to admit that this particular story blew me away for multiple reasons. It is one of the most mathematical of tales ever to appear in pulp magazines, and pound-for-pound in terms of length (so
to speak), it gives a run for the money to many recent mathfiction tales. The writing is crisp, with biting sarcasm which applies surprisingly well to this day, as related to economists,
mathematicians and Wall Street professionals. And the title of the story itself is a delightful spelling pun.
The Trustees of Trent University were convinced by “Wall Street Wolves” that the university’s endowment fund should be invested not just in safe fixed income securities but also in the stock market.
Things were initially all chirpy-happy but then, the markets took a decisive turn and now the university finances are in dire straits. In particular, the mess was partly the fault of their own
Economics department, which had “analyzed the past performance of the market by applying the theory of Fourier Series, and then extrapolated the result.", with rather unfortunate results...
In desperation, the Trustees wondered whether the esteemed professors of the university, the so-called “Long Hairs”, could help with their “profound thoughts”. Of course they can! A wise
mathematician, Newcombe, gets into a back and forth with a JP Morgan-like figure from the Economics department:
(quoted from The Finan-seer)
"I must say that T am disappointed that my friends in Economics thought that Fourier Series were appropriate to this sort of problem. 1 had the impression that Economics was not an exact science and
that the mathematical ,techniques used were rudimentary, but it does shock me a bit that they were that naive." The mathematician had addressed his remarks to Professor Johnsrud of Economics.
“Newcomb, I believe that someone, a mathematician no doubt, once claimed that God must have been a mathematician. You mathematicians have never forgotten that. Let me remind you that the converse of
that proposition is not true.”
“My dear Johnsrud, I did not intend to hurt your feelings. It docs seem to me that you might at least have used the Theory of Stationary Time Series. In the last war the theory was well developed in
connection with the problem of predicting the future position of enemy aircraft, when the data on its present position was distorted by extraneous disturbances. [...]. I want to add that since then
we have acquired an even more potent attack on just such problems. I am referring to the book by Morgenstern and von Neumann, “Theory of Games and Economic Behavior”. It seems to me that we should
exploit the possibilities of this viewpoint.
From there, the story dives deep into an exposition of Game Theory, probabilities, expectation values and decision-making matrices before resurfacing to show how the Professors use the university's
new computers and the new-found discipline of Game Theory to successfully take on Wall Street (were the world so simple!). The author devotes a couple of pages to explain game theory, including a toy
example of a two-player, 3-by-3 payoff grid. The explanation of the toy example is very good and its translation into a stock-market problem, though glossed over, seems plausible (esp. in 1949).
In the first round, even as newspapers declare that the professors would be slaughtered by the “Wolves”, the professors start making an average gain of 200,000 dollars a day.
After a while, Prof Johnsrud starts getting nervous because his autocorrelation analysis indicates their streak is about to break massively. But Newcombe retains unshakeable faith in his models...
and as we have seen repeatedly, this faith gets crushed by the market. Round 2 is lost to Wall Street...
That is when Mr. James, professor of Physics, realizes from a quantum mechanical perspective that their operations had become too big for the markets and they could no longer ignore the
back-reaction, harking back to the original (incorrect) explanation of the Heisenberg Uncertainty Principle... when this is corrected for, Round 3 is won by the professors handily.
But by then, they also come to a sage realization that they must stop. As it always happens, their success is getting copied, a race for faster computer and program trading is on, and it is best to
go back to the ivory towers...
I found it remarkable that the author took the trouble to explain that at one point, the strategy starts losing money as the market wises up and starts using the same counter-techniques - a hallmark
of the "Efficient Markets" theory popularized after the 70s. There is speculation at the end of the story about how game theory deployed with high-tech surveillance methods can be used to win at the
casino, again a nice touch. There is a final implicit challenge to the reader to figure out one particular feature of the 3-by-3 grid example.
The story comes across as a cavalier romp but aside from the simplification necessary to fit the story in a few pages, is very cleanly done. A delightful tale and a great example of mathfiction.
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How to Solve Time and Work Problems #3 Fast Method
This time, our topic of discussion will be those problems in which a number of workers i.e. men and women or men and boys are given a task and we have to determine how much time they will take to
perform the given task. The type of questions as you already know, are asked in many competitive exams and other exams.
Suppose we are given a problem like given below
Q: If 4 men or 6 boys can do a task in 12 days then 10 men and 3 boys will do that task in how many days?
Now carefully read the question. In the first line, 4 men or 6 boys are given but the question is asked with 10 men and 3 boys means men and boys combined.
If you do this question by its normal computational method, then it will take time of about 2-3 minutes to solve. But I am giving you a trick here to solve this type of question that will solve this
question in seconds. You just have to put the formula and you will get the answer. Is not it so simple.
So we should learn the rule for this type of questions.
Rule:- if m1 men or b1 boys does a task in d1 days then m2 men and b2 boys will do it in
=(m1b1d1)/(m1b2+m2b1) days
So the solution to the above problem will be
10 men and 3 boys will do the work in
=(4*6*12)/(4*3+10*6) days
=24*12/(12+60) days
=4 days
So 10 men and 3 boys will do the same task in just 4 days. This is the shortest trick you can find for this type of question.
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Back to Papers Home Back to Papers of School of Physics
Paper IPM / P / 6523
School of Physics
Title: Geometric Phases, Bundle Classification, and Group Representation
Author(s): A. Mostafazadeh
Status: Published
Journal: J. Math. Phys.
No.: 27
Vol.: 37
Year: 1996
Pages: 1218-1233
Supported by: IPM
The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the
representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for
establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semi-simple dynamical Lie
groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also
addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the
Riemannian structure on the parameter space and Berry's connection. The results about the fiber bundles and gorup theory are used to introduce a procedure to reduce the problem of the nonadiabatic
(geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-Abelian monopoles is
pointed out.
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How to record time spent in a specified zone
How to record time spent in a specified zone post and replies
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How to record time spent in a specified zone
Hello, what I need to measure is how long a snake's head spends inside a certain zone. The zone in this case is a prey trail (a few inches wide), so I want to see how long the snake's head
Mark is within the confines of this trail (i.e. how interested is the snake in the prey trail). When the head is outside the specified zone, that time doesn't count. I need to count how many
Teshera seconds the head is within the prey trail. Can Tracker do this? I've been playing around with it but can't seem to figure it out. Thank you.
2 Posts
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Re: How to record time spent in a specified zone -
A logic statement like that within Tracker sounds tricky. But it would be easy if you export the coordinates to Excel.
The statement in Excel would look like:
If the X coordinate is in the B column and the Y coordinate is in the C column, this defines a box where X = (0,2], and Y = (0,1] and puts a 1 in the column wherever the coordinate is in
Paul Nord that range.
29 Posts
Simply sum the column to find the number of timesteps where the condition statement is true.
Attached sample excel file.
Attached File: rangecountinglogic.xlsx
Re: How to record time spent in a specified zone -
Mark Teshera Ok thank you I will try this
2 Posts
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One afternoon each week I volunteer in my kid’s classroom teaching math. They are working on standard model multiplication and occasionally trip over number places. It seems so intuitive to me, but
of course it’s something that must be learned. Happily this has given me some experience with trying to code a decimal system display.
One of their worksheets require placing all the tiles of the digits 0-9 into places in some mathematical functions, like a sort of arithmetic Sudoku and one ah-ha moment I often see is the
realization that the “0” can be placed to the left of other digits without changing the amount. Computers are of course not native base-10 counters either, so in order to emulate a mechanical
display, they need a little help understanding place values as well.
Here’s the bit of script I landed on to ensure that the display always shows 4 digits, regardless of the count.
local countString = tostring(count)
-- generate leading 0's
if count < 1000 then
countString = "0" .. countString
if count < 100 then
countString = "0" .. countString
if count < 10 then
countString = "0" .. countString
That takes care of ensuring there’s always 4 digits even when the count is under 1,000, but rather than simply have the display update instantly when changing it, I plan to use script to animate them
moving like the tumblers on a real tally counter. To accomplish this, I’ll need to be able to move each digit place independently, instead of drawing the entire string at once.
I can extract each character as a substring (since it is a string read left-to-right thousands are index 1, and ones are index 4), then I just need to tell it to draw those characters individually.
After a long time working with visual scripting and editors, my hand is itching to drag elements around on the screen to place them, but of course it doesn’t work that way. I have to instruct the
Playdate exactly what to draw where every frame. Still an interesting challenge, though!
To draw them, and to draw any digits on the “tumbler” that are incoming from underneath (when increasing) or above (when decreasing), I at first tried merely adding or subtracting 1 from that place
digit, but of course Lua views them as integers instead of single digits, so I needed to describe using decimal place values, i.e. that counting above 9 or below 0 loops the digit around again and
affects the next highest place value as well. Since the font is precisely 100 pixels wide, I can position it horizontally on the screen by subtracting 1 from the index and multiplying that by 100
(thousands at index 1 draw at x=0, and ones at index 4 draw at x=300).
if playdate.buttonIsPressed("down") then
-- Iterate through the number places from right to left
for i = 4, 1, -1 do
if isDigitChanging then
-- if it's counting down past zero, make the oncoming number a 9
if (string.sub (countString, i, i)) - 1 < 0 then
numberFont:drawText("0",((i-1)* 100), 100 + verticalOffsets[i])
numberFont:drawText("9",((i-1)* 100), verticalOffsets[i])
-- otherwise make the oncoming number 1 less and indicate this is the last digit to change
numberFont:drawText((string.sub (countString, i, i)),((i-1)* 100), 100 + verticalOffsets[i])
numberFont:drawText((string.sub (countString, i, i)) - 1,((i-1)* 100), verticalOffsets[i])
isDigitChanging = false
-- if this digit should not change, just draw it
numberFont:drawText((string.sub (countString, i, i)),((i-1)* 100),100)
updateVerticalOffset( i )
The array of vertical offsets are what drives the animation. On my mechanical counter, presing the plunger a little moves the tumbler a little, and past a certain point, the tumbler snaps into the
new position. On the Playdate, the buttons have no travel, so I can’t replicate this feeling exactly, but having the action of changing the count divided into button pressed/held/released events can
still imitate a certain amount of mechanical “chunkiness”.
So instead, when the button is pressed and held, it will move the character in the appropriate direction up to a maximum amount (accelerating using Euler’s number, which is a satisfying go-to
constant for such things). Releasing it instantly changes the value and centers it in the window again. If I had curves at my disposal I might add a slight bounce to it, but I think coding up a curve
is a little more than I’m willing to tackle at the moment, and this happens quickly enough to feel pretty good.
function updateVerticalOffset(i)
local multiplier = 2.71828
if verticalOffsets[i] * multiplier >= verticalOffsetMax then
verticalOffsets[i] = verticalOffsetMax
verticalOffsets[i] = verticalOffsets[i] * multiplier
The result when incrementing the count using a button press.
I also wish I knew how to make animated GIFs from the Playdate simulator, but I think you get the idea.
After that, I just need to hide the vertically offset characters, so that it appears one is viewing these tumblers through a window. I will admit this is probably no the most optimal way to do it,
but I simply drew a big white rectangle over the top and bottom of the count display. In a commercial project, I’d probably get some help with a way to display this stuff in a more performant manner,
but it should be sufficient for my purposes.
Now my next challenge will be to get a similar animated behavior when using the crank to change the numbers!
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GreeneMath.com | Ace your next Math Test!
Arc Length on a Circle
Additional Resources:
In this lesson, we will learn how to find the length of an arc of a circle. The formula we will use will come directly from the definition of an angle θ in radians. Our formula for the length s of
the arc intercepted on a circle with a radius r by a central angle of measure θ radians is given as s = rθ, where θ is in radians. Additionally, we will learn how to find the area of a sector of a
circle. The area of a sector of a circle of radius r and central angle θ is given by:
area = (r^2θ) / (2).
+ Show More +
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Networks are present in our lives in numerous different environments: to name just a few, networks can model social relationships, they can model the Internet and links between web pages, they might
model the spread of a virus infection between people, and they might represent computer processors/sensors that have to exchange information. This project aims to obtain new insights into the
behaviour of networks, which are studied from a geometric and computational perspective. Thereto, the project brings together researchers from different areas such as computational geometry, discrete
mathematics, graph drawing, and probability. Among of the topics of research are enumerative problems on geometric networks, crossing numbers, random networks, imprecise models of data, restricted
orientation geometry. Combinatorial approaches are combined with algorithms. Algorithmic applications of networks are also studied in the context of unmanned aerial vehicles (UAVs) and in the context
of musical information retrieval (MIR). The project contains the work packages: “Geometric networks”, "Stochastic Geometry and Networks", “Restricted orientation geometry”, “Graph-based algorithms
for UAVs and for MIR”, and “Dissemination and gender equality promotion”. The project connects researchers from 14 universities located in Austria, Belgium, Canada, Chile, Czech Republic, Italy,
Mexico, and Spain, who will collaborate and share their different expertise in order to obtain new knowledge on the combinatorics of networks and applications.
ComPoSe — Combinatorics of Point Sets and Arrangements of Objects
This CRP focuses on combinatorial properties of discrete sets of points and other simple geometric objects primarily in the plane. In general, geometric graphs are a central topic in discrete and
computational geometry, and many important questions in mathematics and computer science can be formulated as problems on geometric graphs. In the current context, several families of geometric
graphs, such as proximity and skeletal structures, constitute useful abstractions for the study of combinatorial properties of the point sets on which they are defined. For arrangements of other
objects, such as lines or convex sets, their combinatorial properties are usually also described via an underlying graph structure.
The following four tasks are well-known hard problems in this area and will form the backbone of the current project. We will consider the intriguing class of Erdős-Szekeres type problems, variants
of graph problems with colored vertices, counting and enumeration problems for specific classes of geometric graphs, and generalizations of order types as a versatile tool to investigate the
combinatorics of point sets. All these problems are combinatorial problems on geometric graphs and are interrelated in the sense that approaches developed for one of them will also be useful for the
others. Moreover, progress in one direction might provide a better understanding for related questions. Our main objective is to gain deeper insight into the structure of this type of problems and to
contribute major steps towards their final solution.
Erdős-Szekeres problems. We will investigate specific variants of this famous group of problems, such as colored versions, and use newly developed techniques, such as a recent generalized notion of
convexity, to progress on this topic. A typical example is the convex monochromatic quadrilateral problem in Section (iv) of the Call for Outline Proposals: Prove or disprove that every (sufficiently
large) bichromatic point set contains an empty convex monochromatic quadrilateral. Recent progress on this and other Erdős-Szekeres type problems has been made by the PIs Aichholzer, Hurtado, Pach,
Valtr, and Welzl.
Colored point sets. An interesting family of questions is the existence of constrained colorings of point sets. We may consider, for instance, the problem of coloring a set of points in a way such
that any unit disk with sufficiently many points contains all colors. Also, colored versions of classical Helly-type results continue to be a source of fundamental problems, requiring the use of
combinatorial and topological tools. In particular we are interested in colored versions of Tverberg-type results and their generalization of Tverberg-Vrećica-type. Pach founded the class of
‘covering colored sets’ problems and will cooperate on these problems with Cardinal and Felsner in particular, but also with all other PIs.
Counting, enumerating, and sampling of crossing-free configurations. Planar graphs are a core topic in abstract graph theory. Their counterpart in geometric graph theory are crossing-free (plane)
graphs. Interesting questions arise from considering specific classes of plane graphs, such as triangulations, spanning cycles, spanning trees, and matchings. For example, the flip-graph of the set
of all graphs of a given class allows a fast enumeration of all elements from this class and even efficient optimization with respect to certain criteria. But when it comes to more intricate demands,
like counting or sampling a random element, very little is understood. We will put emphasis on counting, enumerating, and sampling methods for several of the mentioned graph classes. Related extremal
results (e.g. upper bounds on the number of triangulations) will also be considered for other classes, like string graphs of a fixed order k (intersection graphs of curves in the plane with at most k
intersections per pair) or visibility graphs in the presence of at most k connected obstacles. Aichholzer, Hurtado, and Welzl have been involved in recent progress on lower and upper bounds for the
number of several mentioned classes of geometric graphs and will cooperate with Pach (intersection graphs), Valtr, and Felsner (higher dimensions) on enumerating and counting.
Order types (rank 3 oriented matroids). Order types play a central role in the above mentioned problems, and constitute a useful tool to investigate the combinatorics of point sets. This is done,
e.g., by providing small instances of vertex sets for extremal geometric graphs in enumeration problems. Our goal is to generalize, and at the same time specialize, this concept. For example, we plan
to investigate the k-set problem as well as a generalization of the Erdős-Szekeres theorem for families of convex bodies in the plane. Typically, progress on the k-set problem has frequently been
achieved in the language of pseudoline arrangements, which are dual to order types. In particular we are interested in combinatorial results ranging from Sylvester-type results to counting certain
cells, and the number and structure of arrangements of n pseudo-lines. Felsner is an expert on pseudo-line arrangements and will collaborate here with Valtr, Pach, Welzl and Aichholzer on order
types. Moreover all PIs have been working on the kset problem individually and will make a joint afford.
This CRP tackles fundamental questions at the intersection of mathematics and theoretical computer science. It is well known that in this area some problems require only days to be solved, others may
take decades or even more. Thus, the working schedule with respect to obtaining the desired theoretical results must follow the standard approach: Continuation of work in progress, evaluation of the
results obtained by other authors and groups, and continuous identification of new directions for progress and exploration, hence always advancing the frontiers of knowledge. Since it is infeasible
to impose a proper temporal order on the objectives and milestones to be attained - the conceptual implications are manifold, and many of the stated objectives are strongly interrelated - it will be
the very progress of research and the obtained results that mark our progress in time. This is guaranteed by the competence of the team. The major 'visible' milestones will be the regular
presentations of joint papers in the main conferences of the field, the corresponding submissions to journals, and a series of progress reports that will help in keeping a clear and consistent
guidance and interaction with the other teams.
Several of the mentioned problems are long-standing open questions and known to be hard. Therefore we will consider several specific variants of them to determine how far state-of-the-art methods can
be used and where new approaches have to be found. This will definitely improve our understanding of the structure of these problems, with the goal of making major contributions towards their
solution or, in the ideal case, to finally settle them. Most of our approaches will be of theoretical nature. But we will also make intensive use of computers for enumeration and experiments, to get
initial insights into the structure of problems, or to support or refute conjectures.
It is well known that the mentioned problems have resisted several previous attacks and therefore require the cooperation of researchers with strong and complementary expertise. We consider
large-scale collaboration on these topics as one of the main ingredients for success. Thus we will not have individual projects running in parallel, but all participants will jointly work on the
topics, in a massive collaborative effort. To guarantee a strong interaction between the members of the group we will maintain regular exchanges of senior researchers and students, regular joint
research workshops (1-2 per year), and frequent visits.
Computational Geometry is dedicated to the algorithmic study of elementary geometric questions. Traditionally it deals with basic geometric objects like points, lines, and planes. For real world
applications, however, often reliable techniques for advanced geometric primitives like surfaces and location query structures are needed. The role of this project is twofold. On one hand it will
provide the theoretical background for advanced geometric algorithms and data structures for several other projects within this joint research project (JRP). These include geometric structures for
fast information retrieval, the generation and manipulation of triangular meshes, the computation of suitable distance functions to multidimensional objects, and the representation of advanced
geometric objects. Another aim of this project is to develop novel techniques for the manipulation and optimization of geometric structures. Here the emphasis is on geometric graphs
(triangulation-like and Voronoi diagram-like structures, spanning trees). Properties of these structures will be investigated, with the goal of designing more efficient geometric algorithms and data
structures. Existing geometric algorithms libraries (CGAL, LEDA) will be used to guarantee robustness of the developed algorithms.
Discrete mathematics studies the mathematical properties of structures that can be accurately represented by a computer. It is omnipresent in everyday life: encryption techniques, for example when
paying with a credit card or when surfing the Internet, are based on methods of discrete mathematics. Another example are optimization problems, for example when designing train timetables or when
planning industrial supply chains. More generally, discrete mathematics forms the theoretical backbone of computer science - an understanding of how an algorithm works is impossible without
mathematics. The consortium of our doc.fund brings together colleagues from TU Graz and the University of Graz and focuses on building bridges between sub-areas of discrete mathematics. Our
consortium emerges from the doctoral program “Discrete Mathematics”, which was financially supported by the FWF from 2010 to 2024 and has firmly anchored research in this area in Graz and made it
internationally visible. We concentrate on fundamental research without losing sight of application areas. We define the term discrete mathematics broadly, extending into the areas of number theory,
algebra and theoretical computer science, and thus cover a wide range of research fields. The specific topics in the doc.funds project range from the question of which polynomials can be represented
as a sum of squares, to computability in networks with limited information, to the problem of which surfaces can be made from textile material. Each doctoral position in this doc.funds project is
supervised equally by two members of the consortium. In most cases, the support takes place at different institutes and the proposed projects lie at the intersection of their expertise. This means we
work on innovative and highly relevant research topics with optimal team support. In addition, we are continuing our proven tools for excellent doctoral training, for example our lively weekly
seminar, the opportunity for long-term stays at foreign research institutions, and a successful mentoring program. This results in excellent training, both for an academic career and for many sectors
of the economy. In fact, graduates of our predecessor program hold responsible positions in a wide variety of areas, such as consulting, software development, insurance and data analysis.
Geometric objects such as points, lines and polygons are the key elements of a big variety of interesting research problems in computer science. With the rise of modern technologies, more and more of
these tasks are solved by computers, as opposed to the classic pen-and-paper approach. Over the last thirty years, researchers around the world have developed different techniques and algorithms that
take advantage of the structure provided by geometry to sIn this thesis we consider triangles in the colored Euclidean plane. We call a triangle monochromatic if all its vertices have the same color.
First, we study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a
monochromatic translated copy of the triangle. Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is
similar to the given triangle. Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring. We also show, that for every
triangle there exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic. Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points
with distance 1, which have the same color. olve these problems. This area of research, in between mathematics and computer science, is known as discrete and computational geometry. In this joint
seminar we plan to use tools from discrete aIn this thesis we consider triangles in the colored Euclidean plane. We call a triangle monochromatic if all its vertices have the same color. First, we
study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a monochromatic
translated copy of the triangle. Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is similar to the
given triangle. Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring. We also show, that for every triangle there
exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic. Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points with distance 1,
which have the same color. Computational geometry (such as order-type-like properties, see below) and apply them to problems that come motivated from the field of sensor networks.
Zentrales Thema dieser gemeinsamen Forschung ist die Untersuchung grundlegender Datenstrukturen aus dem Bereich der rechnerischen Geometrie (Computational Geometry), einem relativ jungen Teilgebiet
der (theoretischen) Informatik. Dabei sollen sowohl theoretische Aspekte untersucht werden, als auch deren konkrete Umsetzung im Rahmen einer allgemein verwendbaren Programmbibliothek. Auf Seite der
theoretischen Untersuchungen sollen sowohl klassische Datenstrukturen, wie Voronoi-Diagramme oder Triangulierungen, aber auch relativ neue Datenstrukturen, wie Pseudo-Triangulierungen oder
Straight-Skeletons, untersucht werden. Genauere Einzelheiten werden in den nachfolgenden Abschnitten beschrieben. Gleichzeitig soll aber auch für alle untersuchten Strukturen deren tatsächliche
Verwendbarkeit in der Praxis berücksichtigt werden. So ist es geplant, jeweils spezielle Aspekte dieser Strukturen in konkreten Implementationen umzusetzen. Um einen bestmöglichen Nutzen der
erzielten Ergebnisse zu gewährleisten, sollen die Implementationen in einer standardisierten Bibliothek der gesamten CG-Community zur Verfügung gestellt werden. Dazu ist die Umsetzung in CGAL
(Computational Geometry Algorithms Library, siehe { www.cgal.org}) geplant. Es handelt sich dabei um ein ursprünglich von der EU gefördertes Projekt, das insbesondere von unserem französichen
Projektpartner an zentraler Stelle mitbetrieben wird.
The general topic of this project is the investigation of geometric graphs, i.e., graphs where the vertex set is a point set in the plane and the edges are straight line segments spanned by these
points. Throughout we assume the points to be in general position, that is no three of them lie on a common line, and to be labeled. Geometric graphs are a versatile data structure, as they include
triangulations, Euclidean spanning trees, spanning paths, polygonalizations, plane perfect matchings and so on. The investigation of geometric graphs belongs to the field of (combinatorial)
mathematics, graph theory, as well as to discrete and computational geometry. The alliance of our two research groups will perfectly cover these fields. For example this will allow us to use an
interesting combination of enumerative investigations (lead by the Austrian team) and theoretical research (coordinated by the Spanish group). Let us point out that this combination of theoretical
knowledge and practical experience, which is perfectly provided by the combination of these two teams, will be essential for the success of this project. There are many classic as well as new
tantalizing problems on geometric graphs, and the investigation of seemingly unrelated questions often leads to new relations and deeper structural insight. So the focus of this project is to
investigate several classes of problems with the common goal of optimizing properties of geometric graphs.
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Cost calculator mortgage refinancing
Cost calculator mortgage refinancing
Based on the input of the incurred charges, the calculator calculates the cost of mortgage refinancing.
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ROCE - an indicator of the efficiency of investment
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The calculation wynagordzenia - wage calculator (service contract) in Poland
jak wyliczyć wynagrodzenie? jak obliczyć pensję? jak obliczyć wypłatę?
Wage Calculator (the service contract and work - power advances) calculates the net remuneration of civil contracts from the following gross salary.
Net return on sales (ROS)
Calculator based on the values entered and net profit calculated return on sales ROS. ROS announces net income (after tax), attributable to every penny of products, goods and services.
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Exploring Autoregressive (AR) Models: Techniques for Effective Time Series Analysis
Auto-regressive models aim to predict a time series by relying exclusively on its previous values, also known as lags. They are based on the assumption that the current value of a time series
variable depends linearly on its past values. In other words, an auto-regressive model predicts future values of a variable by using a linear combination of its previous observations.
The “auto-regressive” part of the model refers to the fact that it regresses the variable against itself. The order of the auto-regressive model denoted as AR(p), indicates the number of lagged
observations used to predict the current value. For example, an AR(1) model uses only the immediate past observation, while an AR(2) model uses the past two observations.
AR(p) model can be represented as:
• y(t) is variable at time t.
• c is a constant term.
• φ are the auto-regressive coefficients fitted on lags.
• ε_t is the error term, which captures the random fluctuations or unexplained variability.
Each observation in the series, except for the initial one, refers back to the preceding observation. This recursive relationship extends to the beginning of the series, giving rise to what are known
as long memory models.
Recursive dependency:
Order Selection
The order selection is a crucial step in building an AR model as it determines the number of lagged values to include in the model, thereby capturing the relevant dependencies in the time series
The primary goal of order selection is to strike a balance between model complexity and model accuracy. A model with a few lagged observations may fail to capture important patterns in the data,
leading to underfitting. On the other hand, a model with too many lagged observations may suffer from overfitting, resulting in poor generalization to unseen data.
There are a few commonly used methods to determine the appropriate order (p) for an AR model.
Auto Correlation (ACF — PACF)
The autocorrelation functions are statistical tools to measure and identify the correlation structure in a time series data set.
The Autocorrelation Function (ACF) measures the correlation between an observation and its lagged values. It helps in understanding how the observations at different lags are related.
The Partial Autocorrelation Function (PACF) removes the effects of intermediate lags. It measures the direct correlation between two observations.
ACF & PACF:
Suppose we observe a positive correlation at a lag of one day. This indicates that there is some correlation between the current day’s price and the previous day’s price. It suggests that the price
tends to exhibit a certain degree of persistence from one day to the next.
For example, the reputation of a company is increasing, as its reputation increases, more and more people buy shares of this company. Wednesday’s price is correlated to Tuesday’s which was correlated
with Monday's. As the lag increases, the correlation may decrease.
In the case of PACF, there is no influence on the days in between. For example, let’s say an ice cream shop runs a campaign on Mondays and gives free ice cream coupons valid on Wednesdays to anyone
who buys ice cream that day. In this case, there will be a direct correlation between the sales on Monday and Wednesday. However, the sales on Tuesday do not affect this situation.
We can use the statsmodels library to plot ACF and PACF in Python.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# Generate a random time series data
n_samples = 100
time = pd.date_range('2023-01-01', periods=n_samples, freq='D')
data = np.random.normal(0, 1, n_samples).cumsum()
# Create a DataFrame with the time series data
df = pd.DataFrame({'Time': time, 'Data': data})
df.set_index('Time', inplace=True)
# Plot the time series data
plt.figure(figsize=(10, 4))
plt.plot(df.index, df['Data'])
plt.title('Example Time Series Data')
Time series data plot:
from statsmodels.graphics.tsaplots import plot_acf
# ACF plot
plot_acf(df['Data'], lags=20, zero=False)
plt.title('Autocorrelation Function (ACF)')
ACF Plot:
The plot_acf method accepts several parameters:
• x is the array of time series data.
• lags specifies the lags to calculate the autocorrelation.
• alpha is the confidence level, default is 0.05 which corresponds to a 95% confidence level. It means that the confidence intervals in the ACF plot represent the range within which we expect the
true population autocorrelation coefficients to fall with 95% probability.
In the ACF plot, the confidence intervals are typically displayed as shaded regions, often in light blue. These intervals indicate the range of values that would be expected for autocorrelations
computed from a large number of random samples from a white noise process (where no autocorrelation is present). This means that these intervals represent the range of values that we would expect to
see if we were to repeatedly generate random data that does not correlate.
A correlation is considered statistically significant if it falls outside the confidence interval, indicating that it is unlikely to have occurred by chance alone. However, a correlation within the
confidence interval does not necessarily imply a lack of correlation, it simply means that the observed correlation is within the range of what we would expect from a white noise process.
• use_vlines indicates whether plot vertical lines at each lag to indicate the confidence intervals.
• vlines_kwargs is used to define additional parameters for appearance.
from statsmodels.graphics.tsaplots import plot_pacf
plot_pacf(df['Data'], lags=20, zero=False)
plt.title('Partial Autocorrelation Function (PACF)')
PACF Plot:
The plot_pacf accepts the same arguments as parameters, and in addition, we can select which method to use:
• “ywm” or “ywmle”: Yule-Walker method without adjustment. It is the default option.
• “yw” or “ywadjusted”: Yule-Walker method with sample-size adjustment in the denominator for autocovariance estimation.
• “ols”: Ordinary Least Squares regression of the time series on its own lagged values and a constant term.
• “ols-inefficient”: OLS regression of the time series on its lagged values using a single common sample to estimate all partial autocorrelation coefficients.
• “ols-adjusted”: OLS regression of the time series on its lagged values with a bias adjustment.
• “ld” or “ldadjusted”: Levinson-Durbin recursion method with bias correction.
• “ldb” or “ldbiased”: Levinson-Durbin recursion method without bias correction.
When evaluating the ACF and PACF plots, firstly look for correlation coefficients that significantly deviate from zero.
Then, notice the decay pattern in the ACF plot. As the lag increases, the correlation values should generally decrease and approach zero. A slow decay suggests a persistent correlation, while a rapid
decay indicates a lack of long-term dependence. In the PACF plot, observe if the correlation drops off after a certain lag. A sudden cutoff suggests that the direct relationship exists only up to
that lag.
Information Criteria
Information criteria, such as the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC), are statistical measures that balance model fit and complexity. These criteria provide a
quantitative measure to compare different models with varying orders. The model with the lowest AIC or BIC value is often considered the best-fitting model.
The Akaike Information Criterion (AIC) takes into account how well a model fits the data and how complex the model is. It penalizes complex models to avoid overfitting by including a term that
increases with the number of parameters. It seeks to find the model that fits the data well but is as simple as possible. The lower the AIC value, the better the model. It tends to prioritize models
with better goodness of fit, even if they are relatively more complex.
The Bayesian Information Criterion (BIC) imposes a stronger penalty on model complexity. It includes a term that grows with the number of parameters in the model, resulting in a more significant
penalty for complex models. It strongly penalizes more complexity, favoring simpler models. It is consistent even with a smaller sample size and preferable when the sample size is limited. Again, the
lower the BIC value, the better the model.
Now, this time let’s generate more complex data and calculate the AIC and BIC:
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.ar_model import AutoReg
# Generate complex time series data
n = 500 # Number of data points
epsilon = np.random.normal(size=n)
data = np.zeros(n)
for i in range(2, n):
data[i] = 0.7 * data[i-1] - 0.2 * data[i-2] + epsilon[i]
# Plot the time series data
plt.figure(figsize=(10, 4))
plt.title('Time Series Data')
Time series data:
# Fit AutoReg models with different orders and compare AIC and BIC
max_order = 10
aic_values = []
bic_values = []
for p in range(1, max_order + 1):
model = AutoReg(data, lags=p)
result = model.fit()
# Plot AIC and BIC values
order = range(1, max_order + 1)
plt.plot(order, aic_values, marker='o', label='AIC')
plt.plot(order, bic_values, marker='o', label='BIC')
plt.xlabel('Order (p)')
plt.ylabel('Information Criterion Value')
plt.title('Comparison of AIC and BIC')
Comparison of AIC and BIC:
The AutoReg class is used to fit autoregressive models. It takes several parameters:
• endog is the dependent variable.
• lags parameter specifies the lag order. It can be an integer representing a single lag value, or a list/array of lag values.
• trend determines the trend component included in the model. It can take the following values:
‘c’: adds a constant value to the model equation, allowing for a non-zero mean. It assumes that the trend is a horizontal line.
‘nc’: no constant term. It excludes the constant from the model equation, assuming that the series has a mean of zero. The trend is expected to fluctuate around zero without a systematic upward or
downward movement.
‘t’: a linear trend. The model assumes that the trend follows a straight line upward or downward.
‘ct’: combines constant and linear trends.
• seasonal indicates whether the model should include a seasonal component.
• exog is the exogenous or external variables that we want to include in the model.
• hold_back specifies the number of initial observations to exclude from the fitting process.
• period is used when seasonal=True and represents the length of the seasonal cycle. It is an integer indicating the number of observations per season.
model.fit() returns an AutoRegResults object and we can get the AIC and BIC from the related attributes.
Since we intentionally generated more complex data, the BIC is expected to favor simpler models more strongly than the AIC. Therefore, the BIC selected a lower-order (simpler) model compared to the
AIC, indicating a preference for simplicity while maintaining a reasonable fit to the data.
Ljung-Box Test
The Ljung-Box test is a statistical test used to determine whether there is any significant autocorrelation in a time series.
The test fits an autoregressive model to the data. The residuals of this model capture the unexplained or leftover variation in the data. Then, the Ljung-Box statistic for different lags is
calculated. These quantify the amount of residual autocorrelation.
To determine whether the autocorrelation is statistically significant, the Ljung-Box statistic is compared with critical values from a chi-square distribution. These critical values depend on the
significance level we choose. If the statistic is higher than the critical value, it suggests the presence of significant autocorrelation, indicating that the model did not capture all the
dependencies in the data. On the other hand, if the statistic is lower than the critical value, it implies that the autocorrelation in the residuals is not significant, indicating a good fit for the
autoregressive model.
By examining the Ljung-Box test results for different lags, we can determine the appropriate order (p). The lag order with no significant autocorrelation in the residuals is considered suitable for
the model.
import numpy as np
from statsmodels.tsa.ar_model import AutoReg
from statsmodels.stats.diagnostic import acorr_ljungbox
max_order = 10
p_values = []
for p in range(1, max_order + 1):
model = AutoReg(data, lags=p)
result = model.fit()
residuals = result.resid
result = acorr_ljungbox(residuals, lags=[p])
p_value = result.iloc[0,1]
# Find the lag order with non-significant autocorrelation
threshold = 0.05
selected_order = np.argmax(np.array(p_values) < threshold) + 1
print("P Values: ", p_values)
print("Selected Order (p):", selected_order)
P Values: [0.0059441704188493635, 0.8000450392943186, 0.9938379305765744, 0.9928852354721004, 0.8439698152504373, 0.9979352709556297, 0.998574234602306, 0.9999969308975543, 0.9999991895465976, 0.9999997412756536]
Selected Order (p): 1
Cross Validation
Cross-validation is another technique for determining the appropriate order (p). Firstly, we divide our time series data into multiple folds or segments. A common choice is to use a rolling window
approach, where we slide a fixed-size window across the data.
For each fold, we fit an AR model with a specific lag order (p) using the training portion of the data. Then, we use the fitted model to predict the values for the test portion of the data.
Next, we calculate an evaluation metric (e.g., mean squared error, mean absolute error) and compute the average performance for each fold.
Finally, we choose the lag order (p) that yields the best average performance across the folds.
import numpy as np
from statsmodels.tsa.ar_model import AutoReg
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import TimeSeriesSplit
data = np.random.normal(size=100)
max_order = 9
best_order = None
best_mse = np.inf
tscv = TimeSeriesSplit(n_splits=5)
for p in range(1, max_order + 1):
mse_scores = []
for train_index, test_index in tscv.split(data):
train_data, test_data = data[train_index], data[test_index]
model = AutoReg(train_data, lags=p)
result = model.fit()
predictions = result.predict(start=len(train_data), end=len(train_data) + len(test_data) - 1)
mse = mean_squared_error(test_data, predictions)
avg_mse = np.mean(mse_scores)
if avg_mse < best_mse:
best_mse = avg_mse
best_order = p
print("Best Lag Order (p):", best_order)
Best Lag Order (p): 3
In this example, we will consider a dataset of stock prices. Specifically, we will focus on the most recent 40-day period and analyze the stock prices of the company AMD.
We will need these packages.
import datetime
import numpy as np
import yfinance as yf
import matplotlib.pyplot as plt
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import TimeSeriesSplit
from statsmodels.tsa.ar_model import AutoReg
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from statsmodels.stats.diagnostic import acorr_ljungbox
We can use the yfinance library in Python to access historical stock data from Yahoo Finance.
ticker = "AMD"
end_date = datetime.datetime.now().date()
start_date = end_date - datetime.timedelta(days=60)
data = yf.download(ticker, start=start_date, end=end_date)
closing_prices = data["Close"]
[*********************100%***********************] 1 of 1 completed
plt.figure(figsize=(10, 6))
plt.title("AMD Stock Closing Prices (Last 42 Days)")
plt.ylabel("Closing Price")
Closing prices:
We will split the dataset, designating the most recent record as the test data, while the remaining records will be used for training. Our objective is to build models that can predict the current
day’s price. We will evaluate and compare the performance of these models based on their predictions.
n_test = 1
train_data = closing_prices[:len(closing_prices)-n_test]
test_data = closing_prices[len(closing_prices)-n_test:]
2023-06-14 127.330002
Name: Close, dtype: float64
Now, we will proceed to train autoregressive (AR) models with lag orders ranging from 1 to 10. We will use these models to forecast the last price in the dataset and calculate the error for each
error_list = []
for i in range(1,11):
model = AutoReg(train_data, lags=i)
model_fit = model.fit()
predicted_price = float(model_fit.predict(start=len(train_data), end=len(train_data)))
actual_price = test_data.iloc[0]
error_list.append(abs(actual_price - predicted_price))
plt.figure(figsize=(10, 6))
plt.title("Errors (Actual - Predicted)")
Errors against p-order values:
We explored the concept of autoregressive (AR) models in time series analysis and forecasting. AR models provide a (starting point) framework for forecasting future values based on past observations.
Order selection (p) is a crucial step in AR modeling, as it determines the number of lagged observations to include in the model. We examined most common techniques for order selection; visual
inspection using with ACF and PACF plots, information criteria, Ljung-Box test and cross validation.
AR models offer valuable insights into the dynamics and patterns of time series data, making them invaluable in various domains such as finance, economics, and weather forecasting. By harnessing the
power of AR models, we can gain deeper understanding, make accurate predictions, and make informed decisions based on historical patterns and trends.
Read More
Time Series Analysis: Mastering the Concepts of Stationarity
Cross-Validation Techniques for Time Series Data
Forecasting Intermittent Demand with Croston’s Method
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Cape Town Maths
Two weeks ago, I had a very nice trip down to Cape Town. It is a very beautiful city indeed. However, I did a series of sessions mixing HP Graphing Calculators, GeoGebra and Data Streaming to groups
of maths teachers, trainee maths teachers and undergraduate engineering students at the Cape Penninsula Technical University and the University of the Western Cape. South Africa, in the post
apartheid era has been trying to bring all of its education systems up to the level of the former elite schools. As you can imagine, this is a tough task, although the government’s commitment is
clear having one of the highest proportional spends on education anywhere in the world. The universities I visited are excellent examples of that move to change and I was delighted to work with
really enthusiastic students and teachers. Additionally, the campuses are equipped with state-of-the-art facilities, including innovative sensory play equipment, further enhancing the learning
environment. If you’re interested in learning more about innovative sensory play equipment, you can check out this site at https://timbertrails.co.uk/why-timber-trim-trail-equipment-is-beneficial/
. For high-quality playground equipment UK, check out this article for more valuable information. Furthermore, if you’re looking for maintenance of your playground or wet pour installation, you can
click here for more information and services. For more resources on primary school education at https://www.primaryschoolresources.org.uk/outcome/psed.
Graphing Calculators are pretty much unknown in South Africa, so it is interesting to see how the technology just makes sense. In the end, giving students the opportunity to investigate the effect on
changing the coefficeinets in the terms of a linear or quadratic equation in order to gain control of the functions, is a vital part of becoming effective in mathematical modelling. Having a device
on which you can rapidly see the relationship between the graph, the function and the table of values, makes developing that understanding fast and secure. It could be a computer suite, or a laptop
or tablet PC, but then again, a cheap, reliable graphing calculator works just as well. The students engaged with the maths beacuse they had a tool to support their thinking. I could hand out enough
of these tools with the 30 GCs I had squeezed into my suitcase, which did fine in a seminar room with 15 trainee teachers and perfectly well in a large hall with 120 engineering students. A few
triple A batteries in the bag meant that everything worked.
GeoGebra 4 has just been released and is a major step up from its predecessor. It sems to me that the technology you use in the classroom is all about engaging with a mathematical story. However,
often we get stuck in worrying asbout learning the technology. So, I set up the activity using GeoGebra and then set students owrking on the task with the GCs. They had never seen or used either. But
in a few seconds I had explained the tasks and they were doing and thinking mathematics, never worrying about the technology. The best example of this was setting up a general gradient functions by
plotting the value of the slope of the gradient to a function. Tracing the points generates the derivative. If you do this for exponentials, you can see (a) that the derivative is itself an
exponential and (b) that somewhere between values in the range of (say) 2.4 to 3.1, there is a value of a in f(x)=a^x where the function and the derivative are the same. Students then take the GCs
and investigate different values of a to see when this happens. So, we get a definition for e, rather than starting by stating it.
The most excitement was had using the distance sensor to match distance time graphs to actual motion. It is great to see how everyone, from beginning students to experienced teachers, finds this
difficult, because we have never had the opportunity to experience distance/time. Quickly the conversation changes to one of rates of chage (faster/slower, further away/closer, …STOP!). I finish with
a sine curve and this is really tricky … when should you be going fastest, how should you slow down/speed up? One PGCE studnet teacher at UWC really excelled at this, starting fast, then slowing
down decreasing the decceleration to sttop, then slow start in the opposite direction, increasing acceleration to the fastest point …. beautifully describing the rate of change (i.e. the speed) as he
went. It hadn’t occured to me before this point how we can ‘feel’ the derivative and describe it in this way.
A great trip, some great institutions and some wonderful students. But the key message for me was to see how students (and teachers) can engage mathematically to a new level of depth, with some
supporting technology and that even when they’ve never encoutered the technology before, it doesn’t need to get in the way. We just treat it as the tool we are using and keep the focus on the maths
and some great stuff happens.
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Matrices and Determinants Class 11 Mathematics Solutions | Exercise - 5.1 - WebNotee
Matrices and Determinants Class 11 Mathematics Solutions | Exercise – 5.1
If you were searching for the notes of Class 11, Matrices and Determinants chapter, then your search is over now. You'll find the notes in this article. However, you'll only find the notes of first
exercise. Nevertheless, you can click on the button below to proceed to the next exercise.
Chapter – 5
Matrices and Determinants
In class 11 matrices and determinants chapter, we learn the addition, subtraction and multiplication of 3×3 matrices. We already have learned the process of addition and subtraction of 2×2 matrices
in class 10. The process is different for 3×3 matrices as the process is lengthy.
Before we move towards the solution of matrices and determinant chapter it is important for us to learn about the topics and subtopics involved in matrices and determinants chapter. So let’s have a
brief introduction of the matrices and determinants chapter.
Matrices and determinants are two related mathematical concepts used in linear algebra. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The numbers,
symbols, or expressions in the matrix are known as its elements or entries.
A determinant is a number that is derived from the elements in a matrix and it is used to describe the properties of the matrix. Determinants are used to solve systems of linear equations, calculate
the inverse of a matrix, and calculate the volume of a parallelepiped.
Some Special Types of Matrices
In class 10, we have learned only about 4 types of matrix but in class 11 we have to study about 6 different types of matrices. Let’s have a quick look at the type of matrices.
• Square Matrix
• Unit Matrix or Identity Matrix
• Zero Matrix or Null Matrix
• Triangle Matrix
• Symmetric Matrix
• Skew-symmetric Matrix
Transpose of a Matrix
The new matrix obtained from a given matrix A by interchanging its rows and columns is called the transpose of A. It is denoted by A’ or Aᵀ. You can just remember the format of finding the transpose
of the given matrix and it will be easier for you to solve any problem of same kind.
Matrices and Determinants Class 11 Solutions PDF
In this PDF, you’ll find the solution of class 11 matrices and determinants chapter. It contains all the solutions of exercise- 5.1, if you want the solutions of other exercises then you’ll find them
above this PDF. Just click on the button and you’ll reach your destination.
You are not allowed to post this PDF in any website or social platform without permission.
Is Class 11 Mathematics Guide Helpful For Student ?
I have published this Notes for helping students who can’t solve difficult maths problems. Student should not fully depend on this note for completing all the exercises. If you totally depend on this
note and simply copy as it is then it may affect your study.
Student should also use their own will power and try to solve problems themselves. You can use this mathematics guide PDF as a reference. You should check all the answers before copying because all
the answers may not be correct. There may be some minor mistakes in the note, please consider those mistakes.
How to secure good marks in Mathematics ?
As, you may know I’m also a student. Being a student is not so easy. You have to study different subjects simultaneously. From my point of view most of the student are weak in mathematics. You can
take me as an example, I am also weak in mathematics. I also face problems while solving mathematics questions.
If you want to secure good marks in mathematics then you should practise them everyday. You should once revise all the exercise which are already taught in class. When you are solving maths problems,
start from easy questions that you know already. If you do so then you won’t get bored.
Maths is not only about practising, especially in grade 11 you to have the basic concept of the problem. When you get the main concept of the problem then you can easily any problems in which similar
concept are applied.
When your teacher tries to make the concept clear by giving examples then all students tries to remember the same example but you should never do that. You can create your own formula which you won’t
forget later.
If you give proper time for your practise with proper technique then you can definitely score a good marks in your examination.
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By Sanjeev Senior Editor
Hello, I'm Sanjeev Mangrati. Writing is my way of sharing thoughts, perspectives, and ideas that empower me. I thoroughly enjoy writing and have published many informative articles.I believe
knowledge and understanding can put you one step ahead in the clamorous race of life!
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Rate of Change Formula - What it is + Examples - Grade Potential White Plains, NY
Rate of Change Formula - What Is the Rate of Change Formula? Examples
Rate of Change Formula - What Is the Rate of Change Formula? Examples
The rate of change formula is one of the most used mathematical formulas throughout academics, specifically in chemistry, physics and accounting.
It’s most frequently used when talking about momentum, however it has many applications throughout various industries. Due to its usefulness, this formula is a specific concept that learners should
This article will go over the rate of change formula and how you can solve it.
Average Rate of Change Formula
In mathematics, the average rate of change formula shows the change of one figure in relation to another. In practice, it's utilized to evaluate the average speed of a variation over a certain period
of time.
Simply put, the rate of change formula is expressed as:
R = Δy / Δx
This computes the variation of y compared to the change of x.
The variation through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is additionally portrayed as the difference between the first point and the
second point of the value, or:
Δy = y2 - y1
Δx = x2 - x1
Because of this, the average rate of change equation can also be described as:
R = (y2 - y1) / (x2 - x1)
Average Rate of Change = Slope
Plotting out these figures in a Cartesian plane, is helpful when reviewing dissimilarities in value A when compared to value B.
The straight line that links these two points is known as secant line, and the slope of this line is the average rate of change.
Here’s the formula for the slope of a line:
y = 2x + 1
To summarize, in a linear function, the average rate of change among two figures is equal to the slope of the function.
This is why the average rate of change of a function is the slope of the secant line intersecting two arbitrary endpoints on the graph of the function. In the meantime, the instantaneous rate of
change is the slope of the tangent line at any point on the graph.
How to Find Average Rate of Change
Now that we know the slope formula and what the figures mean, finding the average rate of change of the function is achievable.
To make understanding this topic easier, here are the steps you should keep in mind to find the average rate of change.
Step 1: Understand Your Values
In these equations, math questions typically offer you two sets of values, from which you solve to find x and y values.
For example, let’s take the values (1, 2) and (3, 4).
In this scenario, then you have to search for the values via the x and y-axis. Coordinates are generally provided in an (x, y) format, as you see in the example below:
x1 = 1
x2 = 3
y1 = 2
y2 = 4
Step 2: Subtract The Values
Find the Δx and Δy values. As you may remember, the formula for the rate of change is:
R = Δy / Δx
Which then translates to:
R = y2 - y1 / x2 - x1
Now that we have obtained all the values of x and y, we can input the values as follows.
R = 4 - 2 / 3 - 1
Step 3: Simplify
With all of our values plugged in, all that is left is to simplify the equation by deducting all the values. Thus, our equation becomes something like this.
R = 4 - 2 / 3 - 1
R = 2 / 2
R = 1
As shown, just by plugging in all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were provided.
Average Rate of Change of a Function
As we’ve stated earlier, the rate of change is applicable to numerous diverse scenarios. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula
can also be used in functions.
The rate of change of function observes a similar principle but with a distinct formula due to the different values that functions have. This formula is:
R = (f(b) - f(a)) / b - a
In this situation, the values given will have one f(x) equation and one X Y axis value.
Negative Slope
Previously if you remember, the average rate of change of any two values can be graphed. The R-value, then is, equivalent to its slope.
Every so often, the equation results in a slope that is negative. This denotes that the line is descending from left to right in the X Y axis.
This translates to the rate of change is decreasing in value. For example, rate of change can be negative, which results in a decreasing position.
Positive Slope
On the other hand, a positive slope denotes that the object’s rate of change is positive. This tells us that the object is gaining value, and the secant line is trending upward from left to right.
With regards to our aforementioned example, if an object has positive velocity and its position is increasing.
Examples of Average Rate of Change
In this section, we will review the average rate of change formula via some examples.
Example 1
Calculate the rate of change of the values where Δy = 10 and Δx = 2.
In this example, all we must do is a plain substitution since the delta values are already provided.
R = Δy / Δx
R = 10 / 2
R = 5
Example 2
Extract the rate of change of the values in points (1,6) and (3,14) of the X Y axis.
For this example, we still have to find the Δy and Δx values by using the average rate of change formula.
R = y2 - y1 / x2 - x1
R = (14 - 6) / (3 - 1)
R = 8 / 2
R = 4
As given, the average rate of change is equal to the slope of the line joining two points.
Example 3
Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].
The third example will be calculating the rate of change of a function with the formula:
R = (f(b) - f(a)) / b - a
When calculating the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply replace the values on the equation with the values provided in the
The interval given is [3, 5], which means that a = 3 and b = 5.
The function parts will be solved by inputting the values to the equation given, such as.
f(a) = (3)2 +5(3) - 3
f(a) = 9 + 15 - 3
f(a) = 24 - 3
f(a) = 21
f(b) = (5)2 +5(5) - 3
f(b) = 25 + 10 - 3
f(b) = 35 - 3
f(b) = 32
Once we have all our values, all we need to do is substitute them into our rate of change equation, as follows.
R = (f(b) - f(a)) / b - a
R = 32 - 21 / 5 - 3
R = 11 / 2
R = 11/2 or 5.5
Grade Potential Can Help You Get a Grip on Math
Math can be a challenging topic to study, but it doesn’t have to be.
With Grade Potential, you can get paired with an expert teacher that will give you personalized support tailored to your capabilities. With the quality of our teaching services, comprehending
equations is as easy as one-two-three.
Connect with us now!
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Comparing methods for detecting multilocus adaptation with multivariate genotype-environment associations
Identifying adaptive loci can provide insight into the mechanisms underlying local adaptation. Genotype-environment association (GEA) methods, which identify these loci based on correlations between
genetic and environmental data, are particularly promising. Univariate methods have dominated GEA, despite the high dimensional nature of genotype and environment. Multivariate methods, which analyze
many loci simultaneously, may be better suited to these data since they consider how sets of markers covary in response to environment. These methods may also be more effective at detecting adaptive
processes that result in weak, multilocus signatures. Here, we evaluate four multivariate methods, and five univariate and differentiation-based approaches, using published simulations of multilocus
selection. We found that Random Forest performed poorly for GEA. Univariate GEAs performed better, but had low detection rates for loci under weak selection. Constrained ordinations showed a superior
combination of low false positive and high true positive rates across all levels of selection. These results were robust across the demographic histories, sampling designs, sample sizes, and levels
of population structure tested. The value of combining detections from different methods was variable, and depended on study goals and knowledge of the drivers of selection. Reanalysis of genomic
data from gray wolves highlighted the unique, covarying sets of adaptive loci that could be identified using redundancy analysis, a constrained ordination. Although additional testing is needed, this
study indicates that constrained ordinations are an effective means of detecting adaptation, including signatures of weak, multilocus selection, providing a powerful tool for investigating the
genetic basis of local adaptation.
Analyzing genomic data for loci underlying local adaptation has become common practice in evolutionary and ecological studies (Hoban et al., 2016). These analyses can help identify mechanisms of
local adaptation and inform management decisions for agricultural, natural resources, and conservation applications. Genotype-environment association (GEA) approaches are particularly promising for
detecting these loci (Rellstab et al. 2015). Unlike differentiation outlier methods, which identify loci with strong allele frequency differences among populations, GEA approaches identify adaptive
loci based on associations between genetic data and environmental variables hypothesized to drive selection. Benefits of GEA include the option of using individual-based (as opposed to
population-based) sampling and the ability to make explicit links to the ecology of organisms by including relevant predictors. The inclusion of predictors can also improve power and allows for the
detection of selective events that do not produce high genetic differentiation among populations (De Mita et al., 2013; de Villemereuil et al., 2014; Rellstab et al., 2015).
Univariate statistical methods have dominated GEA since their first appearance (Mitton et al., 1977). These methods test one locus and one predictor variable at a time, and include generalized linear
models (e.g. Joost et al. 2007; Stucki et al. 2016), variations on linear mixed effects models (e.g. Coop et al. 2010; Frichot et al. 2013; Yoder et al. 2014; Lasky et al. 2014), and non-parametric
approaches (e.g. partial Mantel, Hancock et al. 2011). While these methods perform well, they can produce elevated false positive rates in the absence of correction for multiple comparisons, an issue
of increased importance with large genomic data sets. Corrections such as Bonferroni can be overly conservative (potentially removing true positive detections), while alternative correction methods,
such as false discovery rate (FDR, Benjamini & Hochberg 1995), rely on an assumption of a null distribution of p-values, which may often be violated for empirical data sets. While these issues should
not discourage the use of univariate methods (though corrections should be chosen carefully, see François et al. (2016) for a recent overview), other analytical approaches may be better suited to the
high dimensionality of modern genomic data sets.
In particular, multivariate approaches, which analyze many loci simultaneously, are well suited to data sets comprising hundreds of individuals sampled at many thousands of genetic markers. Compared
to univariate methods, these approaches are thought to more effectively detect multilocus selection since they consider how groups of markers covary in response to environmental predictors (Rellstab
et al. 2015). This is important because many adaptive processes are expected to result in weak, multilocus molecular signatures due to selection on standing genetic variation, recent/contemporary
selection that has not yet led to allele fixation, and conditional neutrality (Yeaman & Whitlock, 2011; Le Corre & Kremer, 2012; Savolainen et al., 2013; Tiffin & Ross-Ibarra, 2014). Identifying the
relevant patterns (e.g., coordinated shifts in allele frequencies across many loci) that underlie these adaptive processes is essential to both improving our understanding of the genetic basis of
local adaptation, and advancing applications of these data for management, such as conserving the evolutionary potential of species (Savolainen et al., 2013; Harrisson et al., 2014; Lasky et al.,
2015). While multivariate methods may, in principle, be better suited to detecting these shared patterns of response, they have not yet been tested on common data sets simulating multilocus
adaptation, limiting confidence in their effectiveness on empirical data.
Here we evaluate a set of these methods, using published simulations of multilocus selection (Lotterhos & Whitlock, 2014, 2015). We compare power using empirical p-values, and evaluate false positive
rates based on cutoffs used in empirical studies. We follow up with a test of three of these methods on their ability to detect weak multilocus selection, as well as an assessment of the common
practice of combining detections across multiple tests. We investigate the effects of correction for population structure in one ordination method, and follow up with an application of this test to
an empirical data set from gray wolves. We find that the constrained ordinations we tested maintain the best balance of true and false positive rates across a range of demographies, sampling designs,
sample sizes, and selection levels, and can provide unique insight into the processes driving selection and the multilocus architecture of local adaptation.
Multivariate approaches to GEA
Multivariate statistical techniques, including ordinations such as principal components analysis (PCA), have been used to analyze genetic data for over fifty years (Cavalli-Sforza, 1966).Indirect
ordinations like PCA (which do not use predictors) use patterns of association within genetic data to find orthogonal axes that fully decompose the genetic variance. Constrained ordinations extend
this analysis by restricting these axes to combinations of supplied predictors (Jombart et al., 2009; Legendre & Legendre, 2012). When used as a GEA, a constrained ordination is essentially finding
orthogonal sets of loci that covary with orthogonal multivariate environmental patterns. By contrast, a univariate GEA is testing for single locus relationships with single environmental predictors.
The use of constrained ordinations in GEA goes back as far as Mulley et al. (1979), with more recent applications to genomic data sets in Lasky et al. (2012), Forester et al. (2016), and Brauer et
al. (2016). In this analysis, we test two promising constrained ordinations, redundancy analysis (RDA) and distance-based redundancy analysis (dbRDA). We also test an extension of RDA that uses a
preliminary step of summarizing the genetic data into sets of covarying markers (Bourret et al., 2014). We do not include canonical correspondence analysis, a constrained ordination that is best
suited to modeling unimodal responses, although this method has been used to analyze microsatellite data sets (e.g. Angers et al. 1999; Grivet et al. 2008).
Random Forest (RF) is a machine learning algorithm that is designed to identify structure in complex data and generate accurate predictive models. It is based on classification and regression trees
(CART), which recursively partition data into response groups based on splits in predictors variables. CART models can capture interactions, contingencies, and nonlinear relationships among
variables, differentiating them from linear models (De’ath & Fabricius, 2000). RF reduces some of the problems associated with CART models (e.g. overfitting and instability) by building a “forest” of
classification or regression trees with two layers of stochasticity: random bootstrap sampling of the data, and random subsetting of predictors at each node (Breiman, 2001). This provides a built-in
assessment of predictive accuracy (based on data left out of the bootstrap sample) and variable importance (based on the change in accuracy when covariates are permuted). For GEA, variable importance
is the focal statistic, where the predictor variables used at each split in the tree are molecular markers, and the goal is to sort individuals into groups based on an environmental category
(classification) or to predict home environmental conditions (regression). Markers with high variable importance are best able to sort individuals or predict environments. RF has been used in a
number of recent GEA and GWAS studies (e.g. Holliday et al. 2012; Brieuc et al. 2015; Pavey et al. 2015; Laporte et al. 2016), but has not yet been tested in a GEA simulation framework.
We compare these multivariate methods to the two differentiation-based and three univariate GEA methods tested by Lotterhos & Whitlock (2015): the X^TX statistic from Bayenv2 (Günther & Coop, 2013),
PCAdapt (Duforet-Frebourg et al., 2014), latent factor mixed models (LFMM, Frichot et al. 2013), and two GEA-based statistics (Bayes factors and Spearman’s ρ) from Bayenv2. We also include
generalized linear models (GLM), a regression-based GEA that does not use a correction for population structure.
GEA implementation
Constrained ordinations
We tested RDA and dbRDA as implemented by Forester et al. (2016). RDA is a two-step process in which genetic and environmental data are analyzed using multivariate linear regression, producing a
matrix of fitted values. Then PCA of the fitted values is used to produce canonical axes, which are linear combinations of the predictors. We centered and scaled genotypes for RDA (i.e., mean = 0, s
= 1; see Jombart et al. 2009 for a discussion of scaling genetic data for ordinations). Distance-based redundancy analysis is similar to RDA but allows for the use of non-Euclidian dissimilarity
indices. Whereas RDA can be loosely considered as a PCA constrained by predictors, dbRDA is analogous to a constrained principal coordinate analysis (PCoA, or a PCA on a non-Euclidean dissimilarity
matrix). For dbRDA, we calculated the distance matrix using Bray-Curtis dissimilarity (Bray & Curtis, 1957), which quantifies the dissimilarity among individuals based on their multilocus genotypes
(equivalent to one minus the proportion of shared alleles between individuals). For both methods, SNPs are modeled as a function of predictor variables, producing as many constrained axes as
predictors. We identified outlier loci on the constrained ordination axes based on the “locus score”, which represent the coordinates/loading of each locus in the ordination space. We use rda for RDA
and capscale for dbRDA in the vegan, v. 2.3-5 package (Oksanen et al., 2013) in R v. 3.2.5 (R Development Core Team, 2015) for this and all subsequent analyses.
Redundancy analysis of components
This method, described by Bourret et al. (2014), differs from the approaches described above in using a preliminary step that summarizes the genotypes into sets of covarying markers, which are then
used as the response in RDA. The idea is to identify from these sets of covarying loci only the groups that are most strongly correlated with environmental predictors. We began by ordinating SNPs
into principal components (PCs) using prcomp in R on the scaled data, producing as many axes as individuals. Following Bourret et al. (2014), we used parallel analysis (Horn, 1965) to determine how
many PCs to retain. Parallel analysis is a Monte Carlo approach in which the eigenvalues of the observed components are compared to eigenvalues from simulated data sets that have the same size as the
original data. We used 1,000 random data sets to generate the distribution under the null hypothesis and retained components with eigenvalues greater than the 99^th percentile of the eigenvalues of
the simulated data (i.e., a significance level of 0.01), using the hornpa package, v. 1.0 (Huang, 2015).
Next, we applied a varimax rotation to the PC axes, which maximizes the correlation between the axes and the original variables (in this case, the SNPs). Note that once a rotation is applied to the
PC axes, they are no longer “principal” components (i.e. axes associated with an eigenvalue/variance), but simply components. We then used the retained components as dependent variables in RDA, with
environmental variables used as predictors. Next, components that were significantly correlated with the constrained axis were retained. Significance was based on a cutoff (alpha = 0.05) corrected
for sample sizes using a Fisher transformation as in Bourret et al. (2014). Finally, SNPs were correlated with these retained components to determine outliers. We call this approach redundancy
analysis of components (cRDA).
Random Forest
The Random Forest approach implemented here builds off of work by Goldstein et al. (2010), Holliday et al. (2012), and Brieuc et al. (2015). This three-step approach is implemented separately for
each predictor variable. The environmental variable used in this study was continuous, so RF models were built as regression trees. For categorical predictors (e.g. soil type) classification trees
would be used, which require a different parameterization (important recommendations for this case are provided in Goldstein et al. 2010).
First, we tuned the two main RF parameters, the number of trees (ntrees) and the number of predictors sampled per node (mtry). We tested a range of values for ntrees in a subset of the simulations,
and found that 10,000 trees were sufficient to stabilize variable importance (note that variable importance requires a larger number of trees for convergence than error rates, Goldstein et al. 2010).
We used the default value of mtry for regression (number of predictors/3, equivalent to ∼3,330 SNPs in this case) after checking that increasing mtry did not substantially change variable importance
or the percent variance explained. In a GEA/GWAS context, larger values of mtry reduce error rates, improve variable importance estimates, and lead to greater model stability (Goldstein et al. 2010).
Because RF is a stochastic algorithm, it is best to use multiple runs, particularly when variable importance is the parameter of interest (Goldstein et al., 2010). We begin by building three full RF
models using all SNPs as predictors, saving variable importance as mean decrease in accuracy for each model. Next, we sampled variable importance from each run with a range of cutoffs, pulling the
most important 0.5%, 1.0%, 1.5%, and 2.0% of loci. These values correspond to approximately 50/100/150/200 loci that have the highest variable importance. For each cutoff, we then created three
additional RF models, using the average percent variance explained across runs to determine the best starting number of important loci for step 3. This step removes clearly unimportant loci from
further consideration (i.e. “sparsity pruning”, Goldstein et al. 2010).
Third, we doubled the best starting number of loci from step 2; this is meant to accommodate loci that may have low marginal effects (Goldstein et al. 2010). We then built three RF models with these
loci, and recorded the mean variance explained. We removed the least important locus in each model, and recalculated the RF models and mean variance explained. This procedure continues until two loci
remain. The set of loci that explain the most variance are the final candidates. Candidates are then combined across runs to identify outliers.
Differentiation-based and univariate GEA methods
For the two differentiation-based and the Bayenv2-based GEA methods, we compared power directly from the results provided in Lotterhos & Whitlock (2015). PCAdapt is a differentiation-based method
that concurrently identifies outlier loci and population structure using latent factors (Duforet-Frebourg et al., 2014). The X^TX statistic from Bayenv2 (Günther & Coop, 2013) is an F[ST] analog that
uses a covariance matrix to control for population structure. The two Bayenv2 GEA statistics (Bayes factors and Spearman’s ρ) also use the covariance matrix to control for population structure, while
identifying candidate loci based on log-transformed Bayes factors and nonparametric correlations, respectively. Details on these methods and their implementation are provided in Lotterhos & Whitlock
We reran latent factor mixed models, a GEA approach that controls for population structure using latent factors, using updated parameters as recommended by the authors (O. François, pers. comm.). We
tested values of K (the number of latent factors) ranging from one to 25 using a sparse nonnegative matrix factorization algorithm (Frichot et al., 2014), implemented as function snmf in the package
LEA, v. 1.2.0 (Frichot & François, 2015). We plotted the cross-entropy values and selected K based on the inflection point in these plots; when the inflection point was not clear, we used the value
where additional cross-entropy loss was minimal. We parameterized LFMM models with this best estimate of K, and ran each model ten times with 5,000 iterations and a burnin of 2,500. We used the
median of the squared z-scores to rank loci and calculate a genomic inflation factor (GIF) to assess model fit (Frichot & François, 2015; François et al., 2016). The GIF is used to correct for
inflation of z-scores at each locus, which can occur when population structure or other confounding factors are not sufficiently accounted for in the model (François et al. 2016). The GIF is
calculated by dividing the median of the squared z-scores by the median of the chi-squared distribution. We used the LEA and qvalue, v. 2.2.2 (Storey et al., 2015) packages in R. Full K and GIF
results are presented in Table S1. Finally, we ran generalized linear models (GLM) on individual allele counts using a binomial family and logistic link function for comparison with LFMM; GIF results
are presented in Table S1.
We used a subset of simulations published by Lotterhos & Whitlock (2014, 2015). Briefly, four demographic histories are represented in these data, each with three replicated environmental surfaces
(Fig. S1): an equilibrium island model (IM), equilibrium isolation by distance (IBD), and nonequilibrium isolation by distance with expansion from one (1R) or two (2R) refugia. In all cases,
demography was independent of selection strength, which is analogous to simulating soft selection (Lotterhos & Whitlock, 2014). Haploid, biallelic SNPs were simulated independently, with 9,900
neutral loci and 100 under selection. Note that haploid SNPs will yield half the information content of diploid SNPs (Lotterhos & Whitlock 2015). The mean of the environmental/habitat parameter had a
selection coefficient equal to zero and represented the background across which selective habitat was patchily distributed (Fig. S1). Selection coefficients represent a proportional increase in
fitness of alleles in response to habitat, where selection is increasingly positive as the environmental value increases from the mean, and increasingly negative as the value decreases from the mean
(Lotterhos & Whitlock 2014, Fig.S1). This landscape emulates a weak cline, with a north-south trend in the selection surface. Of the 100 adaptive loci, most were under weak selection. For the IBD
scenarios, selection coefficients were 0.001 for 40 loci, 0.005 for 30 loci, 0.01 for 20 loci, and 0.1 for 10 loci. For the 1R, 2R, and IM scenario, selection coefficients were 0.005 for 50 loci,
0.01 for 33 loci, and 0.1 for 17 loci. Note that realized selection varied across demographies, so results across demographic histories are not directly comparable (Lotterhos & Whitlock 2015).
We used the following sampling strategies and sample sizes from Lotterhos & Whitlock (2015): random, paired, and transect strategies, with 90 demes sampled, and 6 or 20 individuals sampled per deme.
Paired samples (45 pairs) were designed to maximize environmental differences between locations while minimizing geographic distance; transects (nine transects with ten locations) were designed to
maximize environmental differences at transect ends (Lotterhos & Whitlock 2015). Overall, we used 72 simulations for testing. We assessed trend in neutral loci using linear models of allele
frequencies within demes as a function of coordinates. We evaluated the strength of local adaptation using linear models of allele frequencies within demes as a function of environment. Note that the
Lotterhos & Whitlock (2014, 2015) simulations assigned SNP genotypes to individuals within a population sequentially (i.e., the first few individuals would all get the same allele until its target
frequency was reached, the remaining individuals would get the other allele). This creates artifacts (e.g., artificially low observed heterozygosity) and may affect statistical error rates when
subsampling individuals or performing analyses at the individual level. As recommended by K. Lotterhos (pers. comm.), we avoided these problems by randomizing allele counts for each SNP among
individuals within each population. The habitat surface, which imposed a continuous selective gradient on nonneutral loci, was used as the environmental predictor.
Evaluation statistics
In order to equitably compare power (true positive detections out of the number of loci under selection) across these methods, we calculated empirical p-values using the method of Lotterhos &
Whitlock (2015). In this approach, we first built a null distribution based on the test statistics of all neutral loci, and then generated a p-value for each selected locus based on its cumulative
frequency in the null distribution. We then converted empirical p-values to q-values to assess significance, using the same q-value cutoff (0.01) as Lotterhos & Whitlock (2015). We used code provided
by K. Lotterhos to calculate empirical p-values (code provided in Supplemental Information).
Because false positive rates (FPRs) are not very informative for empirical p-values (rates are universally low, see Lotterhos & Whitlock 2015 for a discussion), we applied cutoffs (e.g. thresholds
for statistical significance) to assess both true and false positive rates across methods. While power is important, determining FPRs is also an essential component of assessing method performance,
since high power achieved at the cost of high FPRs is problematic. Because cutoffs differ across methods, we tested a range of commonly used thresholds for each method and chose the approach that
performed the best (i.e., best balance of TPR and FPR). Note that cutoffs can be adjusted for empirical studies based on research goals and the tolerance for TP and FP detections. For each cutoff
tested, we calculated the TPR as the number of correct positive detections out of the number possible, and the FPR as the number of incorrect positive detections out of 9900 possible. For the main
text, we present results from the best cutoff for each method; full results for all cutoffs tested are presented in the Supplemental Information. For constrained ordinations (RDA and dbRDA) we
identified outliers as SNPs with a locus score +/− 2.5 and 3 SD from the mean score of each constrained axis. For cRDA, we used cutoffs for SNP-component correlations of alpha = 0.05, 0.01, and
0.001, corrected for sample sizes using a Fisher transformation as in Bourret et al. (2014). For GLM and LFMM, we compared two Bonferroni-corrected cutoffs (0.05 and 0.01) and a FDR cutoff of 0.1.
Weak selection
We compared the best-performing multivariate methods (RDA, dbRDA, and cRDA) for their ability to detect signals of weak selection (s = 0.005 and s = 0.001). All tests were performed as described
above, after removing loci under strong (s = 0.1) and moderate (s = 0.01) selection from the simulation data sets. The number of loci under selection in these cases ranged from 43 to 76.
Combining detections
We compared the effects of combining detections (i.e., looking for overlap) using cutoff results from two of the best-performing methods, RDA and LFMM. We also included a scenario in which a second,
uninformative predictor (the x-coordinate of each individual) is included in the RDA and LFMM tests. This predictor is analogous to including an environmental variable hypothesized to drive selection
that covaries with longitude.
Correction for population structure in RDA
To determine how explicit modeling of population structure affects the performance of the best-performing multivariate method, RDA, we accounted for population structure using three approaches: (1)
partialling out significant spatial eigenvectors not correlated with the habitat predictor, (2) partialling out all significant spatial eigenvectors, and (3) partialling out ancestry coefficients.
The spatial eigenvector procedure uses Moran eigenvector maps (MEM) as spatial predictors in a partial RDA. MEMs provide a decomposition of the spatial relationships among sampled locations based on
a spatial weighting matrix (Dray et al., 2006). We used spatial filtering to determine which MEMs to include in the partial analyses (Dray et al., 2012). Briefly, this procedure begins by applying a
principal coordinate analysis (PCoA) to the genetic distance matrix, which we calculated using Bray-Curtis dissimilarity. We used the broken-stick criterion (Legendre & Legendre, 2012) to determine
how many genetic PCoA axes to retain. Retained axes were used as the response in a full RDA, where the predictors included all MEMs. Forward selection (Blanchet et al., 2008) was used to reduce the
number of MEMs, using the full RDA adjusted R^2 statistic as the threshold. In the first approach, retained MEMs that were significantly correlated with environmental predictors were removed (alpha =
0.05/number of MEMs), and the remaining set of significant MEMs were used as conditioning variables in RDA. Note that this approach will be liberal in removing MEMs correlated with environment. In
the second approach, all significant MEMs were used as conditioning variables, the most conservative use of MEMs. We used the spdep, v. 0.6-9 (Bivand et al., 2013) and adespatial, v. 0.0-7 (Dray et
al., 2016) packages to calculate MEMs. For the third approach, we used individual ancestry coefficients as conditioning variables. We used function snmf in the LEA package to estimate individual
ancestry coefficients, running five replicates using the best estimate of K, and extracting individual ancestry coefficients from the replicate with the lowest cross-entropy.
Empirical data set
To provide an example of the use and interpretation of RDA as a GEA, we reanalyzed data from 94 North American gray wolves (Canis lupus) sampled across Canada and Alaska at 42,587 SNPs (Schweizer et
al., 2016). These data show similar global population structure to the simulations analyzed here: wolf data Fst = 0.09; average simulation Fst = 0.05. We reduced the number of environmental
covariates originally used by Schweizer et al. (2016) from 12 to eight to minimize collinearity among them (e.g., |r| < 0.7). One predictor, land cover, was removed because the distribution of cover
types was heavily skewed toward two of the ten types. Missing data levels were low (3.06%). Because RDA requires complete data frames, we imputed missing values by replacing them with the most common
genotype across individuals. We identified candidate adaptive loci as SNPs loading +/− 3 SD from the mean loading of significant RDA axes (significance determined by permutation, p < 0.05). We then
identified the covariate most strongly correlated with each candidate SNP (i.e., highest correlation coefficient), to group candidates by potential driving environmental variables. Annotated code for
this example is provided in the Supplementary Information.
Empirical p-value results
Power across the three ordination techniques was comparable, while power for RF was relatively low (Fig. 1). Ordinations performed best in IBD, 1R, and 2R demographies, with the larger sample size
improving power for the IM demography. Within ordination techniques, RDA and cRDA had slightly higher detection rates compared to dbRDA; subsequent comparisons are made using RDA results.
Except for a few cases in the IM demography, the power of RDA was generally higher than univariate GEAs (Fig. 2). Of the univariate methods, GLM had the highest overall power, while LFMM had reduced
power for the IBD demography. Power from the Bayes Factor (Bayenv2) was generally lower than RDA across all demographies. Finally, RDA had overall higher power than the two differentiation-based
methods (Fig. 3), with the exception of the IBD demography, where power was high for all methods.
Among the methods with the highest overall power, all performed well at detecting loci under strong selection (Fig. 4 and S2). Detection rates for loci under moderate and weak selection were highest
for ordination methods, with RDA and cRDA having the overall highest detection rates. Detection of moderate and weakly selected loci was lower and more variable for univariate methods, especially
LFMM, where detection was dependent on demography and sampling scheme.
Weak selection
We compared the three ordination methods for their power to detect only weak loci in the simulations (Fig. 5). Power from RDA was higher when all selected loci were included, especially for the IM
demography. Power using only weakly selected loci was comparable between RDA and dbRDA, with power slightly higher for RDA in most cases. cRDA was comparable to RDA for the IBD and 2R demographies,
but had very low to no power in the IM demography, and the 1R demography with the larger sample size.
Cutoff results
We compared cutoff results for the methods with the highest overall power: RDA, dbRDA, cRDA, GLM, and LFMM. The best performing cutoffs were: RDA/dbRDA, +/- 3 SD; cRDA, alpha = 0.001; GLM, Bonferroni
= 0.05, and LFMM, FDR = 0.1. We did not choose the FDR cutoff for GLMs since GIFs indicated that the test p-values were not appropriately calibrated (i.e., GIFs > 1, Table S1). For some scenarios,
LFMM GIFs were less than one (indicating a conservative correction for population structure, Table S1). We reran LFMM models with the best estimate of K minus one (i.e., K-1) to determine if a less
conservative correction would influence LFMM results. Because there was no consistent improvement in power or TPR/FPRs using K-1 (Tables S2-S3), all subsequent results refer to LFMM runs using the
best estimate of K.
Full cutoff results for each method are presented in the Supplementary Information (Fig. S3-S6). Cutoff FPRs were highest for cRDA and GLM (Fig. 6). By contrast, RDA and dbRDA had mostly zero FPRs,
with slightly higher FPRs for LFMM. Within these three low-FPR methods, RDA maintained the highest TPRs, except in the IM demography, where LFMM maintained higher power. LFMM was more sensitive to
sampling design than the other methods, with more variation in TPRs across designs.
Combining detections
We compared the univariate LFMM and multivariate RDA cutoff results for overlap and differences in their detections using both the habitat predictor only, and the habitat and (uninformative)
x-coordinate predictor (Figs. 7 and S7). When the driving environmental predictor is known, RDA detections alone are the best choice, since FPRs are very low and RDA detects a large number of
selected loci that are not identified by LFMM (except in the IM demography, Fig. 7a). However, when a noninformative environmental predictor is included, combining test results yields greater overall
benefits, since the tests show substantial commonality in TP detections, but show very low commonality in FP detections (Fig. 7b). By retaining only overlapping loci, FPRs are substantially reduced
at some loss of power due to discarded RDA (and LFMM in the IM demography) detections.
Correction for population structure in RDA
No MEM-based corrections for RDA were applied to IM scenarios, due to low spatial structure (i.e., no PCoA axes were retained based on the broken-stick criterion). The more liberal approach to
correction using MEMs (removing retained MEMs significantly correlated with environment), resulted in removal of MEMs with correlation coefficients ranging from 0.07 to 0.72. Ancestry-based
corrections were only applied to IM scenarios with 20 individuals since 6 individual samples had K=1. All approaches that correct for population structure in RDA resulted in substantial loss of power
across all scenarios, both in terms of empirical p-values and cutoff TPRs (Table 1 and Table S4). False positive rates (which were already very low for RDA) increased slightly when correcting for
population structure. There were only two scenarios where FPRs improved (one and two fewer FP detections); however, these scenarios saw a reduction in TPR of 81% and 92%, respectively (Table S4).
Empirical data set
There were four significant RDA axes in the ordination of the wolf data set (Fig. 9), which returned 556 unique candidate loci that loaded +/− 3 SD from the mean loading on each axis: 171 SNPs
detected on RDA axis 1, 222 on RDA axis 2, and 163 on RDA axis 3 (Fig. 10). Detections on axis 4 were all redundant with loci already identified on axes 1-3. The majority of detected SNPs were most
strongly correlated with precipitation covariates: 231 SNPs correlated with annual precipitation (AP) and 144 SNPs correlated with precipitation seasonality (cvP). The number of SNPs correlated with
the remaining predictors were: 72 with mean diurnal temperature range (MDR); 79 with annual mean temperature (AMT); 13 with NDVI; 12 with elevation; 4 with temperature seasonality (sdT); and 1 with
percent tree cover (Tree).
Multivariate genotype-environment association (GEA) methods have been noted for their ability to detect multilocus selection (Rellstab et al., 2015; Hoban et al., 2016), although there has been no
controlled assessment of the effectiveness of these methods in detecting multilocus selection to date. Since these approaches are increasingly being used in empirical analyses (e.g. Bourret et al.
2014; Brieuc et al. 2015; Pavey et al. 2015; Hecht et al. 2015; Laporte et al. 2016; Brauer et al. 2016), it is important that these claims are evaluated to ensure that the most effective GEA methods
are being used, and that their results are being appropriately interpreted.
Here we compare a suite of methods for detecting selection in a simulation framework to assess their ability to correctly detect multilocus selection under different demographic and sampling
scenarios. We found that constrained ordinations had the best overall performance across the demographies, sampling designs, sample sizes, and selection levels tested here. The univariate LFMM method
also performed well, though power was scenario-dependent and was reduced for loci under weak selection (in agreement with findings by de Villemereuil et al., 2014). Random Forest, by contrast, had
lower detection rates overall. In the following sections we discuss the performance of these methods and provide suggestions for their use on empirical data sets.
Random Forest
Random Forest performed relatively poorly as a GEA. This poor performance is caused by the sparsity of the genotype matrix (i.e., most SNPs are not under selection), which results in detection that
is dominated by strongly selected loci (i.e., loci with strong marginal effects). This issue has been documented in other simulation and empirical studies (Goldstein et al., 2010; Winham et al., 2012
; Wright et al., 2016) and indicates that RF is not suited to identifying weak multilocus selection or interaction effects in these large data sets. Empirical studies that have used RF as a GEA have
likely identified a subset of loci under strong selection, but are unlikely to have identified loci underlying more complex genetic architectures. Note that the amount of environmental variance
explained by the RF model can be high (i.e., overall percent variance explained by the detected SNPs, which ranged from 79-91% for these simulations, Table S5), while still failing to identify most
of the loci under selection. Removing strong associations from the genotypic matrix can potentially help with the detection of weaker effects (Goldstein et al., 2010), but this approach has not been
tested on large matrices. Combined with the computational burden of this method (taking ∼10 days on a single core for the larger data sets), as well as the availability of fast and accurate
alternatives such as RDA (which takes ∼3 minutes on the same data), it is clear that RF is not a viable option for GEA analysis of genomic data.
Random Forest does hold promise for the detection of interaction effects in much smaller data sets (e.g., tens of loci, Holliday et al. 2012). However, this is an area of active research, and the
capacity of RF models in their current form to both capture and identify SNP interactions has been disputed (Winham et al., 2012; Wright et al., 2016). New modifications of RF models are being
developed to more effectively identify interaction effects (e.g. Li et al. 2016), but these models are computationally demanding and are not designed for large data sets. Overall, extensions of RF
show potential for identifying more complex genetic architectures on small sets of loci, but caution is warranted in using them on empirical data prior to rigorous testing on realistic simulation
Constrained ordinations
The three constrained ordination methods all performed well. RDA in particular had the highest overall power across all methods tested here (Figs. 1-3). Ordinations were relatively insensitive to
sample size (6 vs 20 individuals sampled per deme), with the exception of the IM demography, where larger sample sizes consistently improved TPRs, as previously noted by De Mita et al. (2013) and
Lotterhos & Whitlock (2015) for univariate GEAs. Power was lowest in the IM demography, which is typified by a lack of spatial autocorrelation in allele frequencies and a reduced signal of local
adaptation (Table S6), making detection more difficult. This corresponds with univariate GEA results from Lotterhos & Whitlock (2015), who found very low detection rates for loci under weak selection
in the IM demography. Power was highest for IBD, followed by the 2R and 1R demographies. Data from natural systems likely lie somewhere among these demographic extremes, and successful
differentiation in the presence of IBD and non-equilibrium conditions indicate that ordinations should work well across a range of natural systems.
All three methods were relatively insensitive to sampling design, with transects performing slightly better in 1R and random sampling performing worst in IM (Figs. 4, 6, and S2). Otherwise results
were consistent across designs, in contrast to the univariate GEAs tested by Lotterhos and Whitlock (2015), most of which had higher power with the paired sampling strategy. Ordinations are likely
less sensitive to sampling design since they take advantage of covarying signals of selection across loci, making them more robust to sampling that does not maximize environmental differentiation
(e.g., random or transect designs). All methods performed similarly in terms of detection rates across selection strengths (Figs. 4 and S2). As expected, weak selection was more difficult to detect
than moderate or strong selection, except for IBD, where detection levels were high regardless of selection.
High TPRs were maintained when using cutoffs for all three ordination methods (Fig. 6). False positives were universally low for RDA and dbRDA. By contrast, cRDA showed high FPRs for all demographies
except IM, tempering its slightly higher TPRs. These higher FPRs are a consequence of using component axes as predictors. Across all scenarios and sample sizes, cRDA detected component 1, 2, or both
as significantly associated with the constrained RDA axes (Table S7). Most selected loci load on these components (keeping TPRs high), but neutral markers also load on these axes, especially in cases
where there are strong trends in neutral loci (i.e., maximum trends in neutral markers reflect FPRs for cRDA, Table S6, Fig. 6). Given these results, we hypothesized that it might be challenging for
cRDA to detect weak selection in the absence of a covarying signal from loci with stronger selection coefficients. If the selection signature is weak, it may load on a lower-level component axis
(i.e., an axis that explains less of the genetic variance), or it may load on higher-level axes, but fail to be significantly associated with the constrained axes. Note that although cRDA contains a
step to reduce the number of components, parallel analysis resulted in retention of all axes in every simulation tested here (Table S7). This meant that cRDA could search for the signal of selection
across all possible components.
When tested on simulations with loci under weak selection only, RDA, which uses the genotype matrix directly, maintained similar power as in the full data set (except in the IM scenario, where power
was higher when all selected loci were included), indicating that selection signals can be detected with this method in the absence of loci under strong selection (Fig. 5, top row). By contrast, cRDA
detection was more variable, ranging from comparable detection rates with the full data set, to no/poor detections under certain demographies and sample sizes. In these latter cases, poor performance
is reflected in the component axes detected as significant (Table S7); instead of identifying the signal in the first few axes, a variable set of lower-variance axes are detected (or none are
detected at all). This indicates that the method is not able to “find” the selected signal in the component axes in cases where that signal is not driven by strong selection. This result, in addition
to higher FPRs for cRDA, builds a case for using the genotype matrix directly with a constrained ordination such as RDA or dbRDA, as opposed to a preliminary step of data conversion with PCA.
Should results from different tests be combined?
A common approach in local adaptation studies is to run multiple tests (GEA only, or a combination of GEA and differentiation methods) and look for overlapping detections across methods. This ad hoc
approach is thought to increase confidence in TPRs, while minimizing FPRs. The problem with this approach is that it can bias detection toward strong selective sweeps to the exclusion of other
adaptive mechanisms which may be equally important in shaping phenotypic variation (Le Corre & Kremer, 2012; François et al., 2016). If the goal is to detect other forms of selection such as recent
selection or selection on standing genetic variation, this approach will not be effective since most methods are unlikely to detect these weak signals. Additionally, this approach limits detections
to those of the least powerful method used, forcing overall detection rates to be a function of the weakest method implemented.
The complexities of this issue are illustrated by comparing results across two sets of RDA and LFMM results: one where the driving environmental variable is known (Fig. 7a), and another where the
environmental predictors represent hypotheses about the most important factors driving selection (Fig. 7b). In both cases, agreement on TPs is high, and RDA has a large number of true positive
detections that are unique to that method, while unique detections by LFMM are largely limited to the IM demography. The differences in the cases lies in FP detections: when selection is well
understood, and uninformative predictors are not used, retaining RDA detections only is the approach that will maximize TPRs (and detection of weak loci under selection) while maintaining minimal to
zero FPRs (Fig. 7a). Where GEA analyses are more exploratory (i.e., when selective gradients are unknown), combining detections can help reduce FPRs (Fig. 7b). If some FP detections are acceptable,
keeping only RDA detections will improve TPRs at the cost of slightly increased FPRs. A third approach, keeping all detections across both methods, would yield little improvement in TPRs in both
cases, since LFMM has high levels of unique FPs, and minimal unique TP detections.
The decision of whether and how to combine results from different tests will be specific to the study questions, the tolerance for false negative and false positive detections, and the capacity for
follow-up analyses on detected markers. For example, if the goal is to detect loci with strong effects while keeping false positive rates as low as possible, or GEA is being used as an exploratory
analysis, running multiple GEA methods and considering only overlapping detections could be a suitable strategy. However, if the goal is to detect selection on standing genetic variation or a recent
selection event, and the most important selective agents (or close correlates of them) are known, combining detections from multiple tests would likely be too conservative. In this case, the best
approach would be to use a single GEA method, such as RDA, that can effectively detect covarying signals arising from multilocus selection, while being robust to selection strength, sampling design,
and sample size.
Correction for population structure
All three methods used to correct for populations structure in RDA resulted in substantial loss of power and, in most cases, increased FPRs (Table 1 and S4). The effect of correcting for population
structure can be seen in ordination biplots from an example simulation scenario (Fig. 8). In this 1R demographic scenario, the selection surface (“Hab”) and the refugial expansion gradient coincide,
so any correction for population structure will also remove the signal of selection from the selected loci. The correction is most conservative when using all significant MEM predictors to account
for spatial structure (Fig. 8d), and is less conservative when using only MEMs not significantly correlated with environment (Fig. 8c), or ancestry coefficients (Fig. 8b). In all cases, however, the
loss of the selection signal is significant (Table 1), and is visible in the increasing overlap of selected loci with neutral loci.
While the simulations used here have overall low global Fst (average Fst = 0.05), population structure is significant enough in many scenarios to result in elevated FPRs for GLMs (univariate linear
models which do not correct for population structure, Fig. 6). Despite this, RDA and dbRDA (the multivariate analogue of GLMs) do not show elevated FPRs, even when selection covaries with a range
expansion front, as in the 1R and 2R demographies. This is likely because only loci with extreme loadings are identified as potentially under selection, leaving most neutral loci, which share a
similar, but weaker, spatial signature, loading less than +/− 3 SD from the mean. The generality of these results needs to be tested in a comprehensive manner using an expanded simulation parameter
space that includes stronger population structure and metapopulation dynamics; this work is currently in progress. In the meantime, we recommend that RDA be used conservatively in empirical systems
with higher population structure than is tested here, for example, by finding overlap between detections identified by RDA and LFMM (or another GEA that accounts for population structure).
Empirical example
Triplots of three of the four significant RDA axes for the wolf data show SNPs (dark gray points), individuals (colored circles), and environmental variables (blue arrows, Fig. 9). The relative
arrangement of these items in the ordination space reflects their relationship with the ordination axes, which are linear combinations of the predictor variables. For example, individuals from wet
and temperate British Columbia are positively related to high annual precipitation (AP) and low temperature seasonality (sdT, Fig. 9a). By contrast, Artic and High Arctic individuals are
characterized by small mean diurnal temperature range (MDR), low annual mean temperature (AMT), lower levels of tree cover (Tree) and NDVI (a measure of vegetation greenness), and are found at lower
elevation (Fig. 9a). Atlantic Forest and Western Forest individuals load more strongly on RDA axis 3, showing weak and strong precipitation seasonality (cvP) respectively (Fig. 9b), consistent with
continental-scale climate in these regions.
If we zoom into the SNPs, we can visualize how candidate SNPs load on the RDA axes (Fig. 10). For example, SNPs most strongly correlated with AP have strong loadings in the lower left quadrant
between RDA axes 1 and 2 along the AP vector, accounting for the majority of these 231 AP-correlated detections (Fig. 10a). Most candidates highly correlated with AMT and MDR load strongly on axes 1
and 2, respectively. Note how candidate SNPs correlated with precipitation seasonality (cvP) and elevation are located in the center of the plot, and will not be detected as outliers on axes 1 or 2 (
Fig. 10a). However, these loci are detected as outliers on axis3 (Fig. 10b). Overall, candidate SNPs on axis 1 represent multilocus haplotypes associated with annual precipitation and mean diurnal
range; SNPs on axis 2 represent haplotypes associated with annual precipitation and annual mean temperature; and SNPs on axis 3 represent haplotypes associated with precipitation seasonality.
Of the 1661 candidate SNPs identified by Schweizer et al., (2016) using Bayenv (Bayes Factor > 3), only 52 were found in common with the 556 candidates from RDA. Of these 52 common detections, only
nine were identified based on the same environmental predictor. If we include Bayenv detections using highly correlated predictors (removed for RDA) we find nine more candidates identified in common.
Additionally, only 18% of the Bayenv identifications were most strongly related to precipitation variables, which are known drivers of morphology and population structure in gray wolves (Geffen et
al., 2004; O’Keefe et al., 2013; Schweizer et al., 2016). By contrast, 67% of RDA detections were most strongly associated with precipitation variables, providing new candidate regions for
understanding local adaptation of gray wolves across their North American range.
Conclusions and recommendations
We found that constrained ordinations, especially RDA, show a superior combination of low FPRs and high TPRs across weak, moderate, and strong multilocus selection. These results were robust across
the levels of population structure, demographic histories, sampling designs, and sample sizes tested here. Additionally, RDA outperformed an alternative ordination-based approach, cRDA, especially
(and importantly) when the multilocus selection signature was completely derived from loci under weak selection. It is important to note that population structure was relatively low in these
simulations. Results may differ for systems with strong population structure or metapopulation dynamics, where it can be important to correct for structure or combine detections with another GEA that
accounts for structure. Continued testing of these promising methods is needed in simulation frameworks that include more population structure, multiple selection surfaces, and genetic architectures
that are more complex than the multilocus selection response modeled here. However, this study indicates that constrained ordinations are an effective means of detecting adaptive processes that
result in weak, multilocus molecular signatures, providing a powerful tool for investigating the genetic basis of local adaptation and informing management actions to conserve the evolutionary
potential of species of agricultural, forestry, fisheries, and conservation concern.
Data accessibility
Simulation data from Lotterhos & Whitlock (2015): Dryad: doi:10.5061/dryad.mh67v Supporting simulation data (coordinate files) for Lotterhos & Whitlock (2015) data provided by Wagner et al. (2017):
Dryad: doi:10.5061/dryad.b12kk. Wolf data from Schweizer et al. (2016): Dryad: doi.org/10.5061/ dryad.c9b25.
Author contributions
BRF and DLU conceived the study. BRF performed the analyses and wrote the manuscript. HHW contributed code. JRL, HHW, and DLU helped interpret the results and write the manuscript.
Supporting information
Forester_Simulation_Rcode.zip: Contains R code for data preparation and all of the methods tested.
Forester_Wolf_Rcode.zip: Contains R code for data preparation, analysis, interpretation, and plotting of the wolf data set with RDA.
Forester_SI.pdf: Supplemental Figures S1-S7 and Tables S1-S7.
We thank Katie Lotterhos for sharing her simulation data (Lotterhos & Whitlock 2015) and for additional spatial coordinate data and code. Tom Milledge with Duke Resource Computing provided invaluable
assistance with the Duke Compute Cluster. We also thank Olivier François for helpful advice with LFMM, and three reviewers for constructive feedback that greatly improved the manuscript. BRF was
supported by a Katherine Goodman Stern Fellowship from the Duke University Graduate School and a PEO Scholar Award.
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Climate literacy, page 1
Apologies for the title.
The premise is simple. When it comes to climate change, most people are clueless, the media is doing a poor job, but climate science itself is not to blame.
The good news: the data is there.
This is an attempt to present all US max temperature trends/changes in one statistic and still preserve most of the unique structure of change. I’ll explain in detail what the charts show and what
they can’t show, when the topic comes up.
Since it is bad form to start any argument with a proclamation, the proposition is, we can understand the climate, how it changes and how it affects the places where most people live. The data allows
us to ground-truth any concept or perception we might have accepted as true.
Pick the region you live in or one you’re familiar with ( for starters preferably in the US), and develop a set of questions you think are relevant. As long as it is related to temperature change, it
can be anything.
Depending how well we can reliably determine what exactly has changed, there is theoretically no limit to where the data can lead.
As an example,
For people living in United States, the most direct and easy to use source is still NOAA’s databank.
NOAA US surface temps
NOAA US trend maps highest warming trend in the US highest cooling trend in the US
With this data it is possible to put a temperature ‘profile’ together, to a reasonable degree, for any region. We can find out exactly how each season has changed and what that could mean for
anything that is affected by temperature changes.
Most of the basic checks can be done inside the data tool, but some of it must be plotted in a spreadsheet to bring out more details. To keep this opening post as short as possible, i posted a quick
guide on how to plot charts relatively easy, in a separate thread. It’s not a big deal, but it’s necessary for some further analysis.
how to plot guide
For people outside the US we can do the same with surface station data.
GISS surface stations
The tool is mostly self explanatory and there is a direct link to the data. When each month is plotted, we can look at any changes in the same detail as in the US.
There are some sources for min/max temperatures, the easiest way to get the data for a station directly is the climate explorer. (see chart plot thread)
Climate Explorer
Two main issues related to surface stations will inevitably come up. Data availability and maturity. I think it makes the most sense to discuss that with real world examples.
The thread title is only meant to be half serious. If all this sounds too much like school to you, i hope you are at least curious enough to find out if your perception of climate change matches
To understand how exactly the climate changes, we need to follow the data. This is an invitation to do this here.
Some additional tools that could help
Earth Nullschool
is awesome. It’s based reanalysis and for-casting algorithms not real time data, but it’s a fantastic tool to see the climate system in action.
Google Earth has some good overlays (climate zones etc.).
Köppen-Geiger climate zones *NOAA is having some data bank issues, the page is loading slow or sometimes fails to load the charts at all. This is a link to a folder with all max temp charts sorted by
month. US max charts
edit on 21-3-2019 by WilliamR because: lnk
edit on 21-3-2019 by WilliamR because: typo
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Nonlinear Wave Research Articles - R Discovery
In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\
usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional, by employing the
analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the
researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we
rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability
assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined
using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge
of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.
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atan, atanf, atanl - arc tangent function
#include <math.h>
double atan(double x);
float atanf(float x);
long double atanl( long double x);
Link with -lm.
Feature Test Macro Requirements for glibc (see feature_test_macros(7)):
atanf(), atanl(): _BSD_SOURCE || _SVID_SOURCE || _XOPEN_SOURCE >= 600
|| _ISOC99_SOURCE; or cc -std=c99
The atan() function calculates the principal value of the arc tangent
of x; that is the value whose tangent is x.
On success, these functions return the principal value of the arc
tangent of x in radians; the return value is in the range
[-pi/2, pi/2].
If x is a NaN, a NaN is returned.
If x is +0 (-0), +0 (-0) is returned.
If x is positive infinity (negative infinity), +pi/2 (-pi/2) is
No errors occur.
C99, POSIX.1-2001. The variant returning double also conforms to SVr4,
4.3BSD, C89.
acos(3), asin(3), atan2(3), carg(3), catan(3), cos(3), sin(3), tan(3)
This page is part of release 3.24 of the Linux man-pages project. A
description of the project, and information about reporting bugs, can
be found at http://www.kernel.org/doc/man-pages/.
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rime number
2011 is a prime number year, which makes this a prime number year. The next prime number year will be 2017. That means we have just two prime number dates — when day, month, and year are all prime
numbers — left this year: 11/23/2011 and 11/29/2011. Then we will have to wait until February 2, 2017, for another prime number date.
I figured I had better tell you in case you wanted to do something special on Wednesday, or a week from Tuesday.
Prime number days and consecutive odds days
Today’s date is made up entirely of prime numbers: 7, 5, and 2011. I’m sure you already noticed that, because you’re already aware that 2011 is a prime number, and so you’re watching for the
fifty-two dates this year made up entirely of prime numbers. Which means that you have also noticed that there are three prime number Sundays this month, which is the greatest number of prime number
Sundays you can have in any month.
However, you may not have thought about the fact that Saturday’s date is made up of consecutive odd numbers (if, that is, you define the number of the present year to be 11, as it is often written,
rather than 2011). Ron Gordon of Redwood City has thought about it, and has received national press in his efforts to promote what he calls Odd Day. I’d have to say that a more precise name would be
Consecutive Odds Days, but I recognize that “Odd Day” is a catchier name.
Using Gordon’s definition, there are six Odd Days per century. For purists who believe that a number is a number, dammit, and you can’t just arbitrarily chop off the digits to the left of the tens
place, there were only six true Odd Days ever using our present system of numbering years, and those happened even before our present system was in place. While this notion might disturb you, it is
probably more satisfying to the pure mathematician, for the pure mathematician prefers things that don’t actually exist.
I got on a BART train today at about two in the afternoon. An ad next to the door of the train proclaimed:
Judgment Day
May 12, 2011
THE BIBLE GUARANTEES IT!
At six o’clock, the predicted time when Judgment Day was going to come (725,000 days after Jesus was executed, or something like that), I was sitting eating dinner with some friends. “We’re still
here,” someone said.
I just went to check the Web site of Family Radio — that’s the Web site controlled by Harold Camping, the guy who’s been predicting the end of the world. Their Web site is still up and running, and
it still says:
Judgment Day
May 21, 2011
THE BIBLE GUARANTEES IT!
00 days left
And their radio station is still broadcasting (they stream it live on the Web site if you want to check it out) — and the announcer just said that he’ll back back again tomorrow.
I guess that means the Rapture is off. So what happened? Was it supposed to be 7,250,000 days, not 725,000 days? Does God count in hexadecimal? Or maybe God prefers prime numbers (this is a prime
number year after all) so it’s going to be the next largest prime, 725,009?
I’m sure they’ll come up with some reason or another why the Rapture didn’t come today. And I would love to hear your speculations on where they did their math wrong.
Happy prime new year
This is going to be a prime year, and by that I don’t mean it’s going to be first-rate (though I don’t rule that out) — rather, 2011 is a prime number.
Since 2011 is a prime number, that means we can look forward to having several dates that consist solely of prime numbers. The first one will be 2/2/2011, and the last 11/29/2011. I leave it as an
exercise to the student to determine how many of these dates will occur all year (translation: I’m too lazy to figure it out myself, and I hope someone will post a comment with the answer). *
The last prime number year was 2003, and the next one will be 2017. While searching for lists of primes on the Web, I discovered that 2011 and 2017 are so-called “sexy primes”; that is, they differ
by six (“sexy” from the Latin “sex” for six); if they differed by four, they would be cousin primes, and if by two, twin primes. Thus 2011 is a sexy prime number year.
I suspect I am fascinated by prime number years because I was born in the middle of the largest gap in prime number years in the twentieth century (1951 to 1973). I had to wait more than a decade to
live in a prime number year; I had a deprived childhood.
* Here’s the list of primes 31 and under: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. Don’t say I didn’t help you out. Oh, all right, the answer is 52.
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Creating a Safe, Supportive Math Classroom
Creating an Inclusive Math Classroom
In today’s diverse classrooms, it’s essential for educators to recognize and address cultural and gender considerations in mathematics education. By fostering an inclusive environment, teachers can
ensure that all students feel valued, respected, and empowered to succeed in math. In this blog post, we’ll explore practical strategies that teachers can use to prepare for cultural and gender
considerations in their math classrooms.
Understand Cultural and Gender Differences
The first step in preparing for cultural and gender considerations in math education is to gain an understanding of the diverse backgrounds and perspectives of your students. Take the time to learn
about the cultural norms, values, and beliefs that may influence students’ attitudes towards math. Similarly, consider how gender stereotypes and biases might impact students’ confidence and
participation in math activities.
To create a culturally responsive math curriculum, incorporate examples, problems, and teaching materials that reflect the diversity of your students’ backgrounds. Use culturally relevant contexts
and real-world scenarios that resonate with students’ lived experiences. By making math relatable and meaningful, you can engage students from diverse cultural backgrounds and foster a sense of
belonging in the classroom.
Provide Opportunities for Student Voice and Representation
Empower students to share their cultural perspectives and experiences in math discussions and activities. Encourage open dialogue and respectful exchanges of ideas, allowing students to see
themselves reflected in the math curriculum. Incorporate diverse role models and mathematicians from different cultural and gender backgrounds to inspire all students to pursue math-related careers
and interests.
Challenge Gender Stereotypes and Biases
Be mindful of the language and examples used in math instruction to avoid reinforcing gender stereotypes or biases. Encourage all students to participate actively in math activities and challenge
traditional gender roles in math-related tasks. Provide equitable opportunities for girls and boys to excel in math and pursue their interests without limitations or expectations based on gender.
Create a supportive and inclusive classroom culture where students feel safe to express themselves and take risks in their math learning. Establish norms for respectful communication and
collaboration, emphasizing the value of diverse perspectives and contributions. Celebrate diversity and promote empathy, understanding, and acceptance among students to build a strong sense of
community in the math classroom.
Integrating Social-Emotional Learning into Mathematics Education
Social-emotional learning (SEL) plays a crucial role in a student’s overall development, including their success in mathematics education. As educators, it’s essential to recognize the
interconnectedness between students’ emotional well-being and their academic performance in math. In this blog post, we’ll explore practical strategies that teachers can use to prepare for and
integrate social-emotional learning into their mathematics instruction.
Cultivate a Positive Classroom Climate
Creating a supportive and nurturing classroom environment is foundational to integrating SEL into mathematics education. Start by building strong relationships with your students based on trust,
respect, and empathy. Foster a sense of belonging and acceptance by promoting inclusivity and celebrating diversity within the classroom community.
Help students develop self-awareness by encouraging reflection on their emotions, strengths, and areas for growth in math. Teach them strategies for managing stress, anxiety, and frustration when
faced with challenging math problems. Incorporate mindfulness exercises, deep breathing techniques, or short brain breaks to promote emotional regulation and focus during math lessons.
Foster Growth Mindset and Resilience
Promote a growth mindset in mathematics by emphasizing the importance of effort, perseverance, and resilience in learning. Encourage students to embrace challenges, learn from mistakes, and persist
in the face of setbacks. Use inspirational stories of mathematicians who overcame obstacles and failures to achieve success as role models for cultivating resilience and determination in math.
Integrate opportunities for collaborative problem-solving and peer-to-peer discussions in math activities. Emphasize the value of teamwork, communication, and active listening skills in solving math
problems effectively. Provide structured opportunities for students to share their mathematical thinking, ask questions, and explain their reasoning to their peers, promoting deeper understanding and
engagement in math.
Promote Empathy and Social Awareness
Incorporate math activities that highlight real-world issues and social justice themes to promote empathy and social awareness in mathematics education. Encourage students to explore how math can be
used to address social and environmental challenges, fostering a sense of purpose and civic responsibility. Facilitate discussions about equity, fairness, and ethical considerations in mathematical
decision-making processes.
Provide Opportunities for Reflection and Goal-Setting
Allocate time for students to reflect on their learning experiences in math and set personal goals for improvement. Encourage them to identify their strengths, areas for growth, and strategies for
achieving their mathematical goals. Scaffold goal-setting activities and provide constructive feedback to support students’ progress towards their academic and social-emotional learning objectives in
Recently I had a conversation with some colleagues who teach in a university. They were very worried about something they had noticed about their undergraduate students – a fear of making mistakes.
They had noticed that these students were very reluctant to hand in assignments in case they were ‘wrong’ and were often spending time very unproductively in checking and re-checking their answers.
While it is, of course, important to encourage students to be careful about checking their work, and to help them to develop a repertoire of checking strategies, this conversation does seem to
reflect a growing problem, that more and more students are becoming afraid to try new things in case they fail, and/or become depressed and question their own self-worth if they do make mistakes.
Mathematics, with its emphasis on ‘right’ or ‘wrong’ answers can potentially reinforce these fears. On the other hand, however, the mathematics classroom can also be the perfect environment for
sensitive teachers to help their pupils to face up to and overcome these fears – and, of course, the earlier in the child’s school life that this support begins, the better.
The purpose of this article is to illustrate some ways in which mathematics teachers can help to create a secure, supportive classroom environment in which the pupils learn to not fear failure and to
value making mistakes as an opportunity to learn and grow. Each section begins with a quotation from the Sathya Sai Education in Human Values program, a world-wide, secular program designed to
support the integration of values education across the curriculum. The sources of these quotations have not specifically been acknowledge because they appear in similar form in many different places,
but the quotations have been printed in italics. More details about these ideas are discussed in my book ‘Education in Human Values Through Mathematics, Mathematics Through Education in Human
Values’. For further information about how to order this book, please contact [email protected]
True education should make a person compassionate and humane.
It is likely that unwillingness to participate in the mathematics classroom arises from lack of understanding and compassion, which can often be unconscious, by teachers and other pupils.
Consequently, we need to ask the question: how can we encourage more effective participation by any students not participating fully?
• Do not be angry if a child cannot understand something or makes a mistake, because this can lead to fear of failure.
• Show him/her how to recover from the mistake and try again.
• Tell them about famous people who were not afraid to make mistakes (see stories below), or about some of the mistakes you have made – but also encourage accuracy and patiently ask them to correct
their careless errors. A useful source of ideas is a book called “Mistakes That Worked” by Charlotte Foltz Jones.
Students should not allow success or failure to ruffle their minds unduly. Courage and self-confidence must be instilled in the students.
• Use positive visual and body-language cues (nodding, smiling) and prompts (ah ha, hmm) to encourage them to arrive at appropriate answers.
• Be careful not to frown if a child makes a mistake, and don’t allow other children to frown if a classmate makes a mistake either.
There is over emphasis on quick and easy gains rather than patience, fortitude and hard work.
Peter was a very clever eleven-year-old. In the final year of his primary schooling, there was only one test on which he scored less than 100%, and then he only lost half a mark. His classwork was
always done quickly and correctly. When he knew that he could succeed, he was confident and willing to work hard. To challenge his thinking, Peter’s teacher would give him some difficult problems. If
Peter could not immediately see a way to solve a problem, he became a different child. He would sit, drawing on his notepad, or wander around the room. He would even ask his teacher if he could spend
the time tidying the storeroom. Peter, who was normally so successful and confident, was afraid to tackle a difficult task because he was afraid that he might fail. So his solution was to quit, to
make the fears go away. Fortunately, the story had a happy ending, because Peter and his teacher worked together to help him to develop more courage to tackle difficult problems rather than taking
the easiest path of stopping.
Many writers have written about students such as Peter, who expect solutions to come to them quickly and easily and will give up rather than face negative emotions associated with trying the task.
Another concern is that they often are not aware of when it is worthwhile to keep on exploring an idea and when it is appropriate to abandon it because it is leading in a wrong direction. They need
to know when it is appropriate to use a particular approach to the task, and how to recover from making a wrong choice.
Clare, aged ten, was given the following problem to solve:
By changing six figures into zeros you can make this sum equal 1111.
Clare selected the strategy of changing numbers in all three columns simultaneously. She worked at the task with patience and fortitude for two hours. As she worked, she said to herself, “I know that
this is going to work. All I need is time, to find the right combination.” After she repeated the strategy 21 times, her teacher interrupted and suggested that it might be time to look for another
way to solve the problem.
In Peter’s case, it was not enough for his teacher to tell him that frustration, for example, is a normal part of problem solving, and to encourage him to spend more time working on the task. Clare,
on the other hand, was “over persevering”, locked into persistently pursuing one approach when it may be more appropriate when stuck to use other strategies, even such as help-seeking. One of the
responsibilities of a mathematics teacher is to help pupils to learn how to persevere when the problem-solving process becomes difficult. They also need to know how to make decisions about avoiding
time being wasted on “over perseverance”.
1. Equip learners with a range of strategies/techniques for solving different types of problems.
2. Encourage them to experience the full range of positive and negative emotions associated with problem solving.
3. Promote the desire to persevere.
4. Help them to make “managerial” decisions about whether to persevere with a possible solution path (when to keep trying, and when to stop).
5. Encourage them to find more than one way to approach the problem.
One sequence of strategies which is used frequently by successful, persevering problem solvers is the following:
1. Try an approach.
2. Try it 2-3 times in case using different numbers or correcting errors might work.
3. Try something different. (You might decide to come back to your old way later.)
One student used this sequence to persevere successfully with a problem.
Stories About Famous Mathematicians
When you are teaching the appropriate topic, take a minute to tell your pupils an anecdote about one of the famous mathematicians who contributed to this particular field of mathematics. It is
important for pupils to be aware of the ‘human’ side of these famous people. “Using biographies of mathematicians can successfully bring the human story into the mathematics class. What struggles
have these people undergone to be able to study mathematics?…” (Voolich, 1993, p.16)
Stories About Famous Mathematicians
Education should impart to students the capacity or grit to face the challenges of daily life.
For students who have tried but are still having difficulties, McDonough (1984) advised that the teacher:
• ask the pupils to restate the problem in their own words and if this indicates that they have mis-read or mis-interpreted the card, ask them to read the instructions again,
• to help with the understanding of the written instructions question the pupils carefully to find out if they know the meanings of particular words and phrases (i.e. mathematical terminology),
• have the pupils show the teacher what they have done, compare this to what is asked in the instructions, and question the pupils to see if they could think of another method, for example, “Could
you have done this another way?” or, “Have you ever done a task like this before?”
• if necessary, give the children a small hint but only after questioning them carefully to find out what stage they have reached.
If the teacher follows procedures such as those described above, the pupils will be encouraged to be more thoughtful and self-reliant. If pupils are panicking or unable to think what to do, introduce
them to the valuable technique of silent sitting – that is, sitting for a few minutes in a state of complete outer and inner silence.
You can tell them about famous mathematicians who have solved problems by using this technique. For example, Sir Isaac Newton.
By example and precept, in the classroom and the playground, the excellence of intelligent co-operation, of sacrifice for the team, of sympathy for the less gifted, of help…has to be emphasized.
Some teachers’ comments:
I was concerned about two things. One was the way I could use praise to develop self esteem. The other thing was the way in which I was involved in my pupils’ activities. I chose these issues because
I had got into the habit of teaching from the front of the room and responding to the students’ answers with comments such as “Okay”, “Good”, “Sensible”. I was also concerned that the girls were
outnumbered by boys in the class and there was an underlying assumption that the boys were better than the girls, made particularly evident by a vocal group of boys. I consciously placed myself with
different pupils in the classroom and moved to groups when asking or answering questions. I deliberately targeted the quieter children to encourage them to participate in group/class discussions. I
developed a repertoire of responses to students’ answers, including, “Good thinking strategy,” or “Can you clarify that response?” I allowed more response time, focused on permitting girls to respond
following incorrect answers and followed their answers immediately by further questions. Although I only had two weeks in which to implement these initiatives, I felt sufficiently positive about the
change in quality of the students’ responses to warrant continuing this approach. (Primary School Teacher)
four components of language skills: speaking, listening, reading and writing. Interactions are indeed the heartbeat of the mathematics classroom. Mathematics is learned best when students are
actively participating in that learning. One method of active participation is to interact with the teacher and peers about mathematics. (Primary School Teacher)
I chose to work with a group of children about whom I felt I knew very little. I realized that these children could have ability which was not being shown, so I decided to make a more concentrated
effort to provide a variety of experiences and activities, to allow some ‘non-performing’ children to demonstrate their skills. I also recognized the need to discourage a group of ‘noisy’ boys from
putting down the girls and their contributions. A colleague undertook a similar exercise with an older class. She was surprised that she knew the boys better as being more confident and responsive.
She intends to investigate this further by asking a colleague to observe her teach to find out whether her suspicions are true that she is responding more to the boys than to the girls. (Secondary
School Teacher)
Education must award self-confidence, the courage to depend on one’s own strength.
• Some of us may believe that it is acceptable to be untruthful if it is to avoid hurting somebody else’s feelings. On the other hand, some people can also be cruelly truthful and blunt if they do
not like something about another person. We need to realize that neither of these behaviors is really appropriate.
• If we are patient and consistent in our approach and give criticism with compassion, we will have a more significant influence on the child’s subconscious levels of thinking than we realize.
• This does not mean that you have to be blunt or to hurt somebody else’s feelings by telling them something unkind. For example, when correcting students you could say, “I don’t like the way you
answered that question. I like it better when you give me a sensible answer and I know that you have put thought into it.” Or you could say, “I don’t really like the way you have done this piece
of work. I prefer it when you do it more slowly and make fewer mistakes”. This means that you are making it very clear to the other person why you are not happy and how you would prefer her to
The teachers who wrote the comments above were asked to recommend ideas which they could try in their classrooms to encourage more understanding of those students who may not feel safe to participate
as fully as they should or could be. Recommendations included:
• give continuous encouragement, mainly verbally. Value everybody’s responses and have firm rules about interruptions and ‘put downs’,
• encourage a balance between co-operative and competitive teaching and learning styles,
• demonstrate an ‘expectation’ for students to participate,
• encourage group work and peer tutoring, particularly on activity-based and problem-solving tasks,
• allow students sufficient time to complete their work,
• encourage different strategies for approaching and solving problems,
• talk to the non-participators about their reasons for lack of participation – perhaps our perceptions are invalid.
Bell, E.T.(1937). Men of Mathematics (Touchstone Edition, 1986). New York: Simon & Schuster. ISBN 0-671-62818-6 PBK
Lovitt, C. & Clarke, D. (1992). The Mathematics Curriculum and Teaching Program, Professional Development Package, Activity Bank Volume 2. Victoria, Australia: Curriculum Corporation. ISBN 0 642
53279 6.
McDonough, A. (1984). ‘Fun maths in boxes’, in A.Maurer (Ed.). Conflict in Mathematics Education, (pp.60-70). Melbourne, Australia: Mathematical Association of Victoria.
Perl, T. (1993) Women and Numbers. San Carlos, California: Wide World Publishing/Tetra. ISBN: 0-933174-87-X
Voolich, E. (1993). ‘Using biographies to ‘humanize’ the mathematics class’. Arithmetic Teacher, 41(1),16-19.
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Package com.jme3.scene.shape
package com.jme3.scene.shape
generate meshes for various geometric shapes
• Class
A box with solid (filled) faces.
A static, indexed, Triangles-mode mesh for an axis-aligned rectangle in the X-Y plane.
is a visual, line-based representation of a
A simple cylinder, defined by its height and radius.
A simple line implementation with a start and an end.
A parameterized torus, also known as a pq torus.
Quad represents a rectangular plane in space defined by 4 vertices.
A static, indexed, Triangle-mode mesh that renders a rectangle or parallelogram, with customizable normals and texture coordinates.
Sphere represents a 3D object with all points equidistant from a center point.
A box with solid (filled) faces.
This class represents a surface described by knots, weights and control points.
An ordinary (single holed) torus.
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1. Density of states
Besides the band structure, the density of states (DOS) also provides a direct view on the electronic structure of a material. It is easy to construct it from a DFT calculation.
1.1. Density of states for Si
We perform our first DOS calculations for Si. This starts by obtaining a self-consistent density with an 8x8x8 k point mesh
<kPointMesh nx="8" ny="8" nz="8" gamma="F"/>
as it is typically generated with the default parameters (also use the other default parameters). We construct three different density of states on top of this calculation. Consider creating
subfolders for each of them and copy the results (including the cdn.hdf file) of your self-consistent calculation to each subfolder.
To construct a DOS the inp.xml file has to be modified:
1. output/@dos has to be set to "T".
2. output/densityOfStates/@ndir has to be set to "-1".
The other XML attributes in output/densityOfStates have the following use: With minEnergy and maxEnergy the energy window for the DOS is specified, with sigma one defines a broadening for each state
to obtain a smooth result. By default sigma is 0.015. The three different DOS calculations will use different values for sigma: 0.015, 0.005, 0.0015
Perform the three different DOS calculations on top of the already converged result. The calculations will generate a DOS.1 file, where the "1" relates to the spin. In spin-polarized calculations
also a DOS.2 file will be generated (and for bandstructure calculations a bands.2 file). The DOS.1 file is a readable text file with several columns. The first colum defines an energy mesh. The
second column is the total DOS and afterwards there are multiple columns for the projection of the DOS onto certain regions in the unit cell and onto certain orbital characters around each atom. A
detailed description is available at the bottom of the respective user-guide page on DOS calculations. We are interested in the energy mesh, the total DOS, and the $s$ (column 7) and $p$ (column 8)
projections at the Si atoms. It may also be interesting to plot the DOS in the interstitial region (column 3).
Generate for each of the three calculations plots showing the total DOS, $s$-, and $p$-projected DOS, each. The plots should feature on the y axis the energy and on the x axis the DOS. The three
plots should look similar to what is displayed in the following figure.
In combination with a bandstructure plot you can partially relate certain bands to the respective orbital character. Comparing the plots with each other you can observe results strongly varying with
respect to the local Gaussian averaging according to the specified sigma parameter. Obviously it is crucial to find a sigma that provides a good balance between a smoothing of the curves and a
resolution of the features of the electronic structure.
Of course, besides sigma the smoothness of the curves also depends on the $\vec{k}$-point set. After all the density of states is obtained on the basis of the eigenstates at each $\vec{k}$ point. The
Gaussian averaging is needed because the $\vec{k}$-point mesh is not infinitely dense. For the smallest sigma (0.0015) we will therefore also test other $\vec{k}$-point sets. It should be enough if
this is performed on top of the already obtained self-consistent density for the coarser $\vec{k}$-point mesh. Test
<kPointMesh nx="23" ny="23" nz="23" gamma="F"/>
<kPointMesh nx="47" ny="47" nz="47" gamma="F"/>
for the DOS calculation. How does it affect the result? How does it affect the computational demands? Your results should be similar to what is sketched in the following figure.
For the finest $\vec{k}$-point mesh: Do you think that we still see artifacts of the sigma parameter and the finiteness of the $\vec{k}$-point mesh in the plot? If so: Where in the plot is this most
visible and why is it most visible in that region of the plot?
2. Exercises
2.1. Band structure and DOS of a monatomic Cu wire (van Hove singularities)
Experimentalists are capable of producing monatomic wires of certain chemical elements, either on some substrate or free standing wires obtained with break junctions or by pulling scanning tunneling
microscope (STM) tips out of a sample. For each energy the conductivity along such a wire is limited by the conductance quantum $G_0 = \frac{2e^2}{h}$ times the number of bands at the respective
energy. Calculating the band structure of such a system therefore provides direct information on its ballistic electron transport properties.
We perform band structure and density of states calculations for a monatomic Cu wire. For this we set up a tetragonal unit cell with lattice parameters that provide a wide vacuum in two dimensions
and the nearest neighbor distance between adjacent Cu atoms in the third dimension. Use $a=12.5~a_0$ and $c=4.82247~a_0$. For the self-consistent density calculation use a $\vec{k}$-point set of
<kPointMesh nx="1" ny="1" nz="201" gamma="F"/>
Note: Band structure calculations and DOS calculations are two different calculations. Even though both related flags can be set to 'T' in a single calculation without complains of the code, most of
the time it is not reasonable to do this. These calculations require $\vec{k}$-point sets with different properties. While a band structure calculation needs a $\vec{k}$-point path, a DOS calculation
needs a $\vec{k}$-point set that nicely samples the whole volume of the (irreducible wedge of the) Brillouin zone.
For the band structure calculation we explicitly provide the $\vec{k}$-point path by specifying "special k points" along the path:
<kPointCount count="200" gamma="F">
<specialPoint name="g">0.0 0.0 0.0</specialPoint>
<specialPoint name="X">0.0 0.0 0.5</specialPoint>
Note that for band structure calculations the $\vec{k}$-point path within the "altKPointSet" XML element is used if the purpose attribute of this alternative $\vec{k}$-point set is set to "bands".
For the DOS calculation we use a very fine $\vec{k}$-point mesh such that the expected van Hove singularities at the band edges can easily be identified:
<kPointMesh nx="1" ny="1" nz="7001" gamma="F"/>
The number of energy mesh points for the DOS is fixed. As a consequence it is a good idea to refine the upper and lower limits of this mesh such that the mesh is not too coarse for our needs. Find
out the Fermi energy obtained for the self-consistent density (grep -i fermi out) and adjust the energy window such that it only covers the interesting part of the band structure, e.g., from 0.1 Htr
below the Fermi energy to 0.2 Htr above the Fermi energy. The energy window is specified by the parameters in 'output/densityOfStates/@minEnergy' and 'output/densityOfStates/@maxEnergy'. The sigma
parameter should also be very small. Choose 0.0003.
Plot the "total DOS" (column 2), the $s$- (column 7), and the $d$-projected (column 9) DOS. The van Hove singularities schould nicely be visible on every lower and upper band edge. Why is the $s$
-projected DOS so much smaller than the $d$-projected DOS?
Results to be delivered: 1. The plot of the band structure, 2. The plot with the DOS.
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TH‐D‐M100F‐06: Surface Smoothing of a Tubular Structure Using a Non‐Shrinking Algorithm
Purpose: The modeling of a 3D structure by interpolating a stack of 2D contours may result in an unrealistic faceted shape, even though each contour is smooth. A difficulty with geometric smoothing
is that the surface can shrink after a number of iterations. This work investigates the use of a non‐shrinking smoothing algorithm in structure delineation for radiotherapy treatment planning.
Materials and methods: The surface of a tubular structure is parameterized by an interpolating function using original contour data points in cylindrical coordinates. The center of the polar
coordinates for each axial contour is placed on a smooth fitting function while the contour is still a single‐valued function. By interpolation the surface is re‐sampled into a set of evenly spaced
vertices. In an iterative process each vertex is shifted by an average displacement vector from its neighbor vertices and scaled by a factor. Each step of iteration involves two shifts for every
vertex with the scaling factor in opposite signs, in order to avoid shrinkage. The iterative process stops after a desired smoothness is achieved with all average displacement vectors smaller than a
specified tolerance. This method is tested on five prostate IMRT cases. The axial contours of smoothed structure are displayed with the original contours for validation by three physicians. Results:
The resulting surface appeared smooth in all projections. The physician approved the new contours for all five patients. The volume change for each structure was less than 2%. Treatment planning
using smoothed CTVs and PTVs reduced the numbers of MU and MLC segments by 8 – 11%. Conclusions: A technique was developed for smoothing a structure surface constructed using 2D contours. The
calculation was fast for 3D contouring. Our planning results suggested that unrealistically irregular target shapes can have adverse effects on dose conformity and delivery efficiency.
Dive into the research topics of 'TH‐D‐M100F‐06: Surface Smoothing of a Tubular Structure Using a Non‐Shrinking Algorithm'. Together they form a unique fingerprint.
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Trend Trading Strategy Based on Price Extremum
1. Trend Trading Strategy Based on Price Extremum
Trend Trading Strategy Based on Price Extremum
, Date: 2023-12-12 14:36:14
This strategy calculates the maximum and minimum price points over a certain period to form upper and lower bands. When the current price breaks through the upper or lower band, long or short
positions are taken. The strategy mainly judges the trend of prices and trades when the trend strengthens.
Strategy Logic
The core indicator of this strategy is to calculate the maximum and minimum price points over a period. The specific calculation methods are:
Upper Band: Scan the K-line in the period from left to right to find a maximum high point, and then determine whether the 1st K-line on its left to the utmost left and the 1st K-line on its right to
the utmost right are both lower than this maximum high point. If so, this point is confirmed as the top of the range.
Lower Band: Scan the K-line in the period from left to right to find a minimum low point, and then determine whether the 1st K-line on its left to the utmost left and the 1st K-line on its right to
the utmost right are both higher than this minimum low point. If so, this point is confirmed as the bottom of the range.
By repeating this calculation, the upper and lower bands of prices over a period can be obtained. Take long positions when prices break through the upper band and take short positions when prices
break through the lower band. This forms a trend trading strategy based on determining trend by price extremum points.
Advantage Analysis
The way this strategy judges the trend is quite straightforward by determining the strengthening part of the trend through price extremum points, which can effectively filter out consolidation
scenarios and avoid trading in consolidations. The signal generation position of the strategy has advantages and can easily form trend tracking. In addition, the strategy takes signals in a
relatively strict way, which can reduce erroneous signals.
Risk Analysis
The strategy takes signals quite strictly, which may miss more trading opportunities. In addition, extremum points need some time to accumulate and form, which will be relatively lagging. The
parameters need proper optimization. When the parameters are improper, erroneous signals are also very likely to occur.
The strictness of judging the extremum points can be moderately reduced to allow some fluctuations to reduce the risk of misjudgment. In addition, confirmation can be made with other indicators to
avoid wrong signals.
Optimization Directions
The cycle for determining the upper and lower bands can be properly optimized to better capture the trend. In addition, the scanning range for judging extremum points can also be adjusted.
To reduce the possibility of missing trading opportunities, the conditions for determining extremum points can be moderately loosened to allow some fluctuation.
Attempts can be made to confirm with other indicators such as volume indicators, moving averages, etc. to avoid the risk of wrong signals resulting from single indicator judgment.
The way this strategy judges trend characteristics by price extremum points is quite straightforward and effective. It can effectively filter out consolidation and determine the strengthening time of
trends for trend trading. The advantage of the strategy lies in good signal generation position to chase trends. The shortcoming is that the signals may have some lag and it is difficult to capture
turns. Through the optimization of parameters and conditions, this strategy can become a relatively reliable trend judging tool.
start: 2022-12-05 00:00:00
end: 2023-12-11 00:00:00
period: 1d
basePeriod: 1h
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// Copyright by HPotter v1.0 19/02/2018
// Stock market moves in a highly chaotic way, but at a larger scale, the movements
// follow a certain pattern that can be applied to shorter or longer periods of time
// and we can use Fractal Chaos Bands Indicator to identify those patterns. Basically,
// the Fractal Chaos Bands Indicator helps us to identify whether the stock market is
// trending or not. When a market is trending, the bands will have a slope and if market
// is not trending the bands will flatten out. As the slope of the bands decreases, it
// signifies that the market is choppy, insecure and variable. As the graph becomes more
// and more abrupt, be it going up or down, the significance is that the market becomes
// trendy, or stable. Fractal Chaos Bands Indicator is used similarly to other bands-indicator
// (Bollinger bands for instance), offering trading opportunities when price moves above or
// under the fractal lines.
// The FCB indicator looks back in time depending on the number of time periods trader selected
// to plot the indicator. The upper fractal line is made by plotting stock price highs and the
// lower fractal line is made by plotting stock price lows. Essentially, the Fractal Chaos Bands
// show an overall panorama of the price movement, as they filter out the insignificant fluctuations
// of the stock price.
// You can change long to short in the Input Settings
// WARNING:
// - For purpose educate only
// - This script to change bars colors.
fractalUp(pattern) =>
p = high[pattern+1]
okl = 1
okr = 1
for i = pattern to 1
okl := iff(high[i] < high[i+1] and okl == 1 , 1, 0)
for i = pattern+2 to pattern*2+1
okr := iff(high[i] < high[i-1] and okr == 1, 1, 0)
res = iff(okl == 1 and okr == 1, p, res[1])
fractalDn(pattern) =>
p = low[pattern+1]
okl = 1
okr = 1
for i = pattern to 1
okl := iff(low[i] > low[i+1] and okl == 1 , 1, 0)
for i = pattern+2 to pattern*2+1
okr := iff(low[i] > low[i-1] and okr == 1, 1, 0)
res = iff(okl == 1 and okr == 1, p, res[1])
strategy(title="Fractal Chaos Bands", overlay = true)
Pattern = input(1, minval=1)
reverse = input(false, title="Trade reverse")
xUpper = fractalUp(Pattern)
xLower = fractalDn(Pattern)
pos = iff(close > xUpper, 1,
iff(close < xLower, -1, nz(pos[1], 0)))
possig = iff(reverse and pos == 1, -1,
iff(reverse and pos == -1, 1, pos))
if (possig == 1)
strategy.entry("Long", strategy.long)
if (possig == -1)
strategy.entry("Short", strategy.short)
barcolor(possig == -1 ? red: possig == 1 ? green : blue )
plot(xUpper, color=red, title="FCBUp")
plot(xLower, color=green, title="FCBDn")
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Python Math Pi
In mathematics, “Pi” is a constant representing the ratio of a circle’s circumference to its diameter. The “math” module supports various methods and constants, including the “Pi” value in Python. It
can be obtained from the “math” module using the “math.pi” constant in a program.
The outcomes from this tutorial are:
What is “math.pi” in Python?
The “math.pi” constant of the “math” module is used to retrieve the value of Pi. To use this, first, we need to import the “math” module at the start of the program and then call the method. To get
more understanding, let’s take the following examples:
Example 1: Printing the Value of “math.pi” in Python
In the below-provided example, first, we imported the “math” module and then used the “math.pi” to print the Pi value:
import math
The Pi value has been retrieved to the console successfully:
Example 2: Determining the Circle’s Circumference Using “math.pi”
The “math.pi” return value can be used in various operations, such as determining the circle circumferences. In the below code, first, we have imported the “math” module, and then an integer radius
value is assigned to the variable. The “C=2PiR” circle circumference has been calculated in the following code:
import math
radius = 6
After executing the above-stated code, the following output will retrieve:
Example 3: Using Numpy and Scipy for Printing the Value of Pi
The “Numpy” and “Scipy” modules can also be used to retrieve the value of Pi. For example, in the following code, the “scipy.pi” and “numpy.pi” are used to retrieve the Pi value:
import scipy, numpy
The above code generates the following output:
Example 4: Getting the Value of Pi Using the Radians
We can also get/retrieve the value of Pi using the radians by utilizing the “math.radians()” method. This method takes the radians value as an argument and retrieves the value of Pi. Let’s utilize
the below code as an example:
import math
This code retrieves the following output:
In Python, we utilized the “math.pi” constant of the “math” module to print the value of “pi” and used it for various mathematical operations. The “math.pi” constant value can be used to find the
Circle’s Circumference. This write-up demonstrates the “math.pi” constant of the “math” module using multiple examples.
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Principles of Constant Mass Flow
How does the Dräger Dolphin Work?
Nevil Adkins
Note: the following information is mainly theoretical in nature, has not been checked by others, is not issued by Dräger and may not be 100% correct. As an engineer I have applied my knowledge of
thermodynamics and hopefully this will give an insight as to how the Dolphin works.
The Dräger Dolphin and its stable mate the Ray are recreational Semi-closed Circuit Rebreathers (SCRs) using a constant mass-flow design to add gas to the breathing loop. They are very similar in
nature so I will talk of the Dolphin and highlight where the Ray is different where I can (I use the Dolphin, not the Ray so may not know all the differences). The basic nature of the beast is taught
on the relevant user course and will not be dwelt on here. For further reference read the user manuals supplied with the units and “Rebreather Diving”[1] by Bob Cole.
Many questions have been raised on various e-mail groups about how the Dolphin will react when changing the interstage pressure, diving deep, using different gas mixes, etc. The explanations that
have ensued have been vague, confusing or even wrong in some circumstances so I will attempt to explain the engineering theory that drives the Dolphin. The theory is quite simple in principle but to
really get to terms with it you need some heavy-duty maths. For those who want the inside story refer to “Engineering Thermodynamics”[2] by Rogers and Mayhew. The relevant chapter is Chapter 18 – One
Dimensional Steady Flow and Jet Propulsion.
Background theory
Let us take a basic look at the theory of gas flow in nozzles. Consider the situation in figure 1
Gas in a large container has an initial pressure p[1], temperature T[1] and velocity v[1] which is negligible. The ambient pressure at the outlet of the nozzle is p[2]. Since p[2] is lower than p[1],
the gas feels a pressure difference resulting in an accelerating force. The gas begins to move towards and through the nozzle. The gas’s potential energy in the form of pressure is gradually
converted to kinetic energy in the form of velocity. As the inlet pressure is increased, the gas feels a greater acceleration force and the velocity achieved in the throat of the nozzle increases. At
a certain point however the difference in pressure between p[1] and p[2] is sufficient to accelerate the gas to the speed of sound at local conditions in the throat of the nozzle (p[t], T[t]), a
barrier that cannot be passed. A further small increase in the initial pressure p[1] (potential energy) cannot increase the velocity (kinetic energy) in the throat so the gas still has some potential
energy left – the throat pressure p[t] rises above the ambient pressure p[2]. As the gas exits the nozzle it becomes unconstrained and expands irreversibly to the ambient pressure p[2], decelerating
instantly to low velocity. If at this stage the ambient pressure were to be lowered then the gas conditions in the nozzle do not change and no extra flow occurs. Similarly if the ambient pressure is
raised, as long as it does not rise to match the throat pressure again, no influence on the throat conditions are felt and the flow rate remains constant. This situation is known as choked flow and
achieves constant mass flow for given inlet conditions, regardless of how the outlet conditions vary.
What happens if the inlet pressure is increased further? Since only a fraction of the initial potential (pressure) energy is needed to accelerate the gas to the speed of sound (kinetic energy), more
potential energy is left over and the pressure in the throat rises. This increases both the local speed of sound and the gas density, which results in a greater mass flow. Thus, once choked flow is
achieved, variations in outlet (ambient) pressure will have no effect on the mass flow rate, but variations in inlet pressure will cause corresponding variations in the mass flow rate.
The rigorous mathematical analysis in Engineering Thermodynamics leads to the equation
Equation 1
From this equation it can be seen that the critical mass flow for a particular nozzle (i.e. fixed A[t]) is a function of the inlet pressure and temperature and the nature of the gas passing through
Critical mass flow is achieved provided that
equation 2
where p[2] is the nozzle outlet pressure.
From this equation it can be seen that the critical pressure ratio (p[2]/p[1]) is a function solely of the nature of the gas passing through the nozzle.
What happens if the critical pressure ratio is exceeded?
Provided that the pressure ratio p[2]/p[1 ]is less than the critical value then the nozzle will be choked and a constant mass flow regime will be achieved. What happens if the critical ratio is
exceeded? The answer is that the velocity in the throat of the nozzle will not reach local sonic and the nozzle will not be choked. The flow rate will gradually reduce until no flow occurs when the
pressure ratio is unity, i.e. there is no pressure difference. The characteristic is illustrated in the graph, figure 2.
What is γ?
The physical property of the gas known as γ, is the ratio of the specific heat capacities at constant pressure and volume and can be measured for each gas. The γ for a mixture of gases such as nitrox
can be calculated if the composition of the gas is known and the γ for each component is known.
For example:
Table 1
│Oxygen │ │ γ = 1.416│
│Nitrogen│ │ γ = 1.403│
│Air │(0.21 x 1.146)+(0.79 x 1.403) │ γ = 1.406│
│EAN32 │(0.32 x 1.146)+(0.68 x 1.403) │ γ = 1.407│
│EAN40 │(0.4 x 1.146)+(0.6 x 1.403) │ γ = 1.408│
│EAN50 │(0.5 x 1.146)+(0.5 x 1.403) │ γ = 1.410│
│EAN60 │(0.6 x 1.146)+(0.4 x 1.403) │ γ = 1.411│
How does this theory relate to the Dolphin?
The gas contained in the cylinder of the Dolphin is reduced to a fixed interstage pressure by the first stage regulator. The output of this first stage is held constant at 250psi/17.2bar gauge,
irrespective of depth. This contrasts to the usual open circuit and Ray regulator design, which maintains the interstage pressure at a fixed margin (typically 10bar/140psi) above the ambient
pressure. The gas is then taken to the dosage jets which each contain two laser drilled rubies that act as nozzles according to the above theory. The holes in the rubies are sized to give a certain
flow rate with a certain gas mix, within a certain tolerance band.
From the basic theory outlined above let us take Equation 1
and deduce the consequences.
1. T[1] is the absolute inlet temperature to the jets, measured in Kelvin. Due to expansion through the regulator it is likely to be a few degrees cooler than ambient, but probably not significantly.
For the most part we dive in water temperatures between say 273 K (0°C) and 303 K (30°C), a variation of ±5% from the average 288 K (15°C). This variation will only change the mass flow by ±2.5% so
can probably be ignored totally, and certainly over the duration of a single dive.
2. The mass flow is related in a complex way to the properties of the mix flowing through the nozzle. Whilst considering nitrox, the variation in γ between EAN32 and EAN60 is about 0.2%, which leads
to a variation in the mass flow of about 0.08%. Even dramatically changing nitrox mixes will not affect the mass flow rate flowing through any particular jet.
3. The mass flow is directly proportional to the inlet (interstage) pressure, p[1]. Any variation in the interstage pressure, whether by intent or mishap, will directly affect the mass flow rate. For
example, dropping the interstage pressure from 18.2 bar absolute to 13.6 bar absolute will reduce the mass flow rate of each of the jets by 25%.
Now let us take Equation 2
and deduce the consequences.
1. If the interstage pressure p[1] is fixed at 18.2 bar absolute, then the mass flow will remain constant provided the ambient pressure p[2] is less than a certain value, dependant on γ. For nitrox
mixes p[2]/p[1] must be less than 0.526. This implies that the maximum ambient pressure for critical mass flow is 9.6bar absolute, or approximately 86m. This is way in excess of the 40m limit for
recreational use or 52m limit for military use recommended by Dräger.
2. If the interstage pressure p[1] is allowed to vary, either by accident or design, the maximum depth achievable whilst retaining critical flow will also vary.
How does this theory relate to the Ray?
Unlike the Dolphin, the first stage of the Ray reduces the pressure in the cylinder to an interstage pressure at a fixed margin above ambient pressure – this is the same as most open circuit
regulators. The gas is then taken to the dosage jet, there is only one, which like the Dolphin contains two laser drilled rubies that act as nozzles according to the above theory. The Ray is
specified for diving to a maximum of 22m with EAN50.
From the basic theory outlined above let us take Equation 1
and deduce the consequences.
1. As for the Dolphin, the variation in flow rate with temperature is miniscule and can effectively be ignored.
2. As with the Dolphin, the mass flow is related in a complex way to the properties of the mix flowing through the nozzle. Since the Ray is specified for use only with a single mix there should be no
variation in γ but as we have seen for the Dolphin the variations in γ, and the subsequent variations in the flow rates are also negligible with varying nitrox mixes. The Ray could therefore be used
with a number of different mixes, albeit with the limitation of choice of flow rates.
3. The mass flow is still directly proportional to the inlet (interstage) pressure, p[1], but unlike the Dolphin this is not fixed. The jet inlet pressure at is related to the depth in the following
equation 3
which for a typical interstage pressure of 10bar becomes
equation 4
Referenced to the flow rate at the surface, the flow rate at the specified maximum depth of 22m will be 20% higher. The gas consumption of the Ray is therefore not independent of depth as the Dolphin
is but the reduction in unit duration from 1½ hours at the surface to 1¼ hours at 20m is insignificant for the majority of recreational users at whom this unit is aimed.
Now let us take Equation 2
which for nitrox reduces to
equation 5
Solving Equations 3 & 5 gives
which for our typical interstage pressure of 10 bar means P[ambient] ≤ 11.1bar or 101m
The Ray will not stray out of the critical ratio limitation until extreme depth – but don’t forget the mass flow is not constant across the depth range! The Ray is still not suitable for diving much
beyond the recommended maximum of 22m.
Does the Dolphin/Ray use “supersonic” flow?
No. The gas passing through the nozzle accelerates from negligible velocity until it achieves sonic velocity at local conditions of temperature and pressure. Theoretically the gas could be further
accelerated as it leaves the nozzle, to achieve supersonic velocities, but this would not increase the mass flow. Furthermore the nozzle would have to be very carefully (and expensively) engineered,
and very minor variations in conditions would quickly bring the nozzle off its design point. The gasses leaving the nozzle expand instantaneously and irreversibly, and decelerate back to negligible
velocity. Supersonic velocities are not needed.
How will using trimix affect the Dolphin?
Modern thinking is that diving beyond 40m should be done using trimix. The section on the Dolphin above shows that the maximum theoretical depth for the Dolphin should be in the range of 80m. How
will using trimix in the Dolphin in the 40-80m range affect performance?
Returning to equation 2 reminds us that the critical pressure ratio is dependant on γ, which is itself a function of gas composition. Extending table 1 to include helium and trimix gives
│Helium │ │γ = 1.630│
│TMX27/30│(0.27 x 1.146)+(0.43 x 1.403)+(0.3 x 1.630) │γ = 1.475│
│TMX16/45│(0.16 x 1.146)+(0.39 x 1.403)+(0.45 x 1.630) │γ = 1.508│
│TMX10/80│(0.10 x 1.146)+(0.1 x 1.403)+(0.8 x 1.630) │γ = 1.586│
table 2
It can be seen that even very high helium mixes such as TMX10/80 only have a γ about 12% higher than EAN32. The increase in γ reduces the critical pressure ratio slightly, making the maximum ambient
pressure for critical mass flow 9.1bar absolute, or approximately 81m – not greatly different from the 86m theoretical limit calculated for nitrox. The difference is that 81m is a feasible dive on
trimix but not for nitrox!
The increase in γ will also be reflected by an increase in mass flow rate of about 4% for any particular jet. This is well within the tolerance band of individual dosage jets and below the
measurement accuracy of most people and will not realistically affect dive planning. This theory is confirmed by several people’s experience in using the unit on trimix.
The conclusions from these calculations match up interestingly with the reports that Dräger have developed a modification pack to allow the Dolphin to be used to 80m on trimix.
How would feeding the dosage jets from a standard regulator affect the Dolphin?
It has been suggested to feed gas to the dosage jets from a second source via a standard regulator. In this way it would be possible to feed a rich nitrox or pure oxygen to the loop during
decompression. As noted in the section regarding the Ray, the use of depth compensating regulators will result in variable flow rates. Since decompression using rich mixes is usually only carried out
over a small range of depths, the variation would not be that great and could be coped with. If however the gas fed to the jets was a bottom gas, say from a side-slung cylinder acting as either a
bail-out or as a main cylinder, then the system would be very dynamic in nature. If set up to give correct flow rates at the surface, large quantities of gas would be wasted at depth. If set up to
give correct flow rates at depth then the oxygen mass flow rate at shallow depths might not be sufficient to meet demands with dire consequences. As per the Ray, use of depth-compensating regulators
to feed gasses to the dosage jets should not be considered deeper than about 22m.
I hope this look at the physics driving the Dolphin and Ray has been readable, and more importantly correct. If anybody wishes to offer corrections, suggestions for further areas of coverage, etc.
then please contact me at adkinsoman@yahoo.com
[1] Rebreather Diving, Bob Cole, published by Sub-Aqua Association 1998, ISBN 0 9519337 9 5.
[2] Engineering Thermodynamics – Work & Heat Transfer 4^th Edition, Rogers & Mayhew, published by Longman Scientific & Technical, ISBN 0-582-04566-6.
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